Three-Body Hadron Dynamics from Lattice QCD

Abstract

Presently, lattice quantum chromodynamics is the only available tool that allows for the calculation of hadron properties in terms of their constituents, quarks and gluons, incorporating the non-perturbative nature of the strong interaction in the low-energy regime. However, information about few-hadron dynamics is not directly accessible. Instead, the finite-volume energy spectra, determined in lattice calculations, have to be related to the infinite-volume scattering- and decay-amplitudes. <br /> In this thesis, by application of non-relativistic effective field theory techniques, methods for the analysis of data from lattice quantum chromodynamics are developed that allow for the extraction of three-body scattering- and decay properties. The fundamental concepts of lattice quantum chromodynamics, focusing on the methods of hadron spectroscopy, are outlined. The framework of non-relativistic effective field theories is introduced and the role of relativistic invariance is discussed. <br /> A relation between finite volume decay matrix elements and infinite volume decay amplitudes is derived at the leading order in the non-relativistic effective field theory power counting for the weak decay of a scalar particle into three identical likewise scalar particles. This equation establishes a generalization of the Lellouch-Lüscher formalism to the three-body sector. <br /> Furthermore, a novel formulation of the non-relativistic effective field theory formalism is suggested, which is devoid of some shortcomings of the existing approaches related to the explicit non-covariance of the three-particle propagator. The three-particle quantization condition, relating the finite-volume energy spectra to the infinite-volume scattering matrix elements, is written down in a manifestly relativistic-invariant form within this modified formalism, such that data from different moving frames can be combined in a global analysis. <br /> Finally, the three-body analog of the Lellouch-Lüscher equation is generalized to higher orders and the systematic inclusion of higher partial waves is discussed. In contrast to the leading order expression derived in an earlier chapter, it is expressed in a manifestly relativistic-invariant form. This setup is of particular importance, since in practice the extraction of weak decay amplitudes from lattice calculations requires the inclusion of data from different moving frames.