In this dissertation, a low energy theory approach is applied to the studies of Dark Matter direct detection experiments and two-dimensional Quantum Chromodynamics (QCD) spectra. We build a general framework of non-relativistic effective field theory of Dark Matter direct detection using non-relativistic operators. Any Dark Matter particle theory can be translated into the coefficients of an effective operator and any effective operator can be related to a most general description of the nuclear response. Response functions are evaluated for common Dark Matter targets. Based on the effective field theory we perform an analysis of the experimental constraints on the full parameter space of elastically scattering Dark Matter. We also formulate an analytic approach to solving two-dimensional gauge theories. We find that in theories with confinement, in a conformal operator basis, the decoupling of high scaling-dimension operators from the low-energy spectrum occurs exponentially fast in their scaling-dimension. Consequently the low-energy spectrum of a strongly coupled system like QCD can be calculated using a truncated conformal basis, to an accuracy parametrized exponentially by the cutoff dimension. We apply the conformal basis approach in two models, a two-dimensional QCD with an adjoint fermion at large N, and a two-dimensional QCD with a fundamental fermion at finite N. It is shown that the low energy spectrum converges efficiently in both cases.

In this thesis, we extend the numerical S-matrix bootstrap program to 1+1d theories with a boundary, where we bootstrap the 1-to-1 reflection matrix (R-matrix). We review the constraints that a physical R-matrix must obey, namely unitarity, analyticiy and crossing symmetry. We then carve out the allowed space of 2d R-matrices with the O(N) nonlinear sigma model and the periodic Yang Baxter solution in the bulk. We find a variety of integrable R-matrices along the boundary of the allowed space both with and without free parameters. The integrable models without a free parameter appear at vertices of the allowed space, while those with a free parameter occupy the whole boundary. We also introduce the extended analyticity constraint where we increase the domain of analyticity beyond the physical region. In some cases, the allowed space of R-matrices shrinks drastically and we observe new vertices which correspond to integrable theories. We also find a new integrable R-matrix through our numerics, which we later obtained by solving the boundary Yang--Baxter equation. Finally, we derive the dual to the extended analyticity problem and find that the formalism allows for R-matrices which do not saturate unitarity to lie on the boundary of the allowed region.