In this dissertation, we present work towards characterizing various conformal and nearly conformal quantum field theories nonperturbatively using a combination of numerical and analytical techniques. A key area of interest is the conformal window of four dimensional gauge theories with Dirac fermions and its potential applicability to beyond the standard model physics.
In the first chapter, we review some of the history of models of composite Higgs scenarios in order to motivate the study of gauge theories near the conformal window. In the second chapter we review lattice studies of a specific theory, SU(3) gauge theory with eight flavors of Dirac fermions in the fundamental representation of the gauge group. We place a particular emphasis on the light flavor-singlet scalar state appearing in the spectrum of this model and its possible role as a composite Higgs boson. We advocate an approach to characterizing nearly conformal gauge theories in which lattice calculations are used to identify the best low energy effective field theory (EFT) description of such nearly conformal gauge theories, and the lattice and EFT are then used as complementary tools to classify the generic features of the low energy physics in these theories. We present new results for maximal isospin ππ → ππ scattering on the lattice computed using Lüscher's finite volume method. This scattering study is intended to provide further data for constraining the possible EFT descriptions of nearly conformal gauge theory. In Chapter 3, we review the historical development of chiral effective theory from current algebra methods up through the chiral Lagrangian and modern effective field theory techniques. We present a new EFT framework based on the linear sigma model for describing the low lying states of nearly conformal gauge theories. We place a particular emphasis on the chiral breaking potential and the power counting of the spurion field.
In Chapter 4, we report on a new formulation of lattice quantum field theory suited for studying conformal field theories (CFTs) nonperturbatively in radial quantization. We demonstrate that this method is not only applicable to CFTs, but more generally to formulating a lattice regularization for quantum field theory on an arbitrary smooth Riemann manifold. The general procedure, which we refer to as quantum finite elements (QFE), is reviewed for scalar fields. Chapter 5 details explicit examples of numerical studies of lattice quantum field theories on curved Riemann manifolds using the QFE method. We discuss the spectral properties of the finite element Laplacian on the 2-sphere. Then we present a study of interacting scalar field theory on the 2-sphere and show that at criticality it is in close agreement with the exact c = 1/2 minimal Ising CFT to high precision. We also investigate interacting scalar field theory on [special characters omitted] x [special characters omitted]2, and we report significant progress towards studying the 3D Ising conformal fixed point in radial quantization with the QFE method. In the near future, we hope for the QFE method to be used to characterize the four dimensional conformal fixed points considered in the first half of this dissertation.
|Advisor:||Appelquist, Thomas, Fleming, George Tamminga|
|School Location:||United States -- Connecticut|
|Source:||DAI-B 80/07(E), Dissertation Abstracts International|
|Keywords:||Conformal, Effective Field Theory, Lattice, Nonperturbative, Numerical, Strong Coupling|
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