title: Exploring quantization conditions: relating finite-volume data to infinite-volume observables
Creators: name: Peterken, Toby
Publisher: name: The University of Edinburgh
Year: 2024
Subjects: subject: scattering amplitude
subject: Lattice Field Theory
subject: standing waves
subject: compositeness

description: We explore the formalisms, originally developed by Martin Luscher and others, that relate finite-volume spectra to infinite-volume scattering and transition amplitudes. This is particularly relevant for numerical Lattice Field Theory calculations. Such calculations necessitate a Wick rotation to imaginary time and it is numerically infeasible to analytically continue the resulting Euclidean correlators back to real-time, and thus we cannot directly calculate the desired observables. This is particularly relevant for multi-hadron scattering and decay amplitudes. By contrast, energy levels and time-independent matrix elements of the finite-volume system carry no information about the space-time signature and can be used to constrain these observables. This thesis starts by exploring the relevant mathematical results and techniques necessary to compare correlators defined in a finite- and infinite-volume theory. The rest of the work is then divided into two parts. The first of these parts presents the derivation and formalism relating finite-volume spectra to infinite-volume 2-particle scattering amplitudes. We assume a generic relativistic scalar field theory and use an all-orders diagrammatic approach, as developed by Luscher and Kim, Sachrajda, and Sharpe, to relate the finite-volume energy levels to the infinite-volume scattering amplitude. This result is then extended to include the effects of intrinsic spin, following work by Brice\~no, before the formalism is explicitly shown in some example situations, including situations where resonances occur. The formalism is extended further to include non-zero lattice-spacing effects. The new quantization condition takes discretized energies as an input and returns a version of the discretized scattering amplitude whose definition is transparent in the context of a Symanzik effective field theory. This extension is new to this work and represents one of the core new derivations reached in the course of the PhD. The second part studies generalizations to the conversion factor derived by Lellouch and Luscher that can be used to calculate infinite-volume K-&gt; pi pi decay amplitudes from finite-volume matrix elements. We give two separate results that are applicable when either the incoming state has no hadrons or a single hadron. Following work by Brice\~no, Hansen and Walker-Loud, both derivations proceed using a generic relativistic scalar field theory and then via an all-orders diagrammatic expansion. In this paradigm, it is possible to calculate a conversion factor relating the desired infinite-volume amplitude, evaluated at one of the finite-volume energy levels, to a finite-volume matrix element. Both derivations then extend the result to include the effects of particles with intrinsic spin and explicitly show how the result could be used in practice. Due to the breaking of rotation symmetry, all of the previous results have contributions caused by partial-wave mixing. Although this infinite tower of partial-wave components affects the conversion factor, the S-wave (l=0) component is expected to dominate and often, in practice, all other partial-wave amplitudes are formally set to zero. However, as the precision of lattice calculations increases, assessing the systematic uncertainty arising from this approximation may become important. Therefore, we compare, for the specific case of 1 -&gt; 2 transitions, the S-wave-only results with those truncated at the next contaminating partial wave. We quantify the effect of higher partial waves for a range of systems, presenting both generic numerical examples and results relevant to realistic K -&gt; pi pi amplitudes. We also consider the accidental degeneracy occurring in the 8th excited state of the zero-momentum system. This exhibits qualitatively new features at l=4, not seen in the l=0 truncation. This detailed treatment of partial-wave contamination is an additional new result of this work. The final chapter of the thesis is a novel attempt to define a notion of compositeness compatible with Quantum Field Theory. By considering the analytic properties of the S-matrix, we conclude that there is no satisfactory definition of compositeness compatible with Quantum Field Theory. Without this notion, one must claim that all bound states are equally fundamental, that is, one cannot rigorously claim that everyday objects are made of atoms or that atoms are made of protons and neutrons.