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In this work, we present a lattice QCD calculation of two closely related quantities: 1). The 𝜋𝜋 scattering phase shift for both 𝑰=0 and 𝑰=2 channels at seven energies in total, and 2). The 𝜟𝑰=1/2, 𝛫 → 𝜋𝜋 decay amplitude 𝐴₀ and 𝜖′, the measure of direct CP violation. These two results improve our earlier calculation presented in 2015 [1]. The calculation is performed on an ensemble of 32³ × 64 lattice with 𝛼⁻¹=1.3784(68)GeV. This is a physical calculation, where the chiral symmetry breaking is controlled by the 2+1 flavor Möbius Domain Wall Fermion, and we take the physical value for both kaon and pion. The G-parity boundary condition is used and carefully tuned so that the ground state energy of the 𝜋𝜋₁₌₀ state matches the kaon mass. Three sets of 𝜋𝜋 interpolating operators are used, including a scalar bilinear ``σ" operator and paired single-pion bilinear operators with the constituent pions carrying various relative momenta. Several techniques, including correlated fits and a bootstrap determination of the 𝑝-value have been used, and a detailed analysis of all major systematic error is performed. The 𝜋𝜋 scattering phase shift results are presented in Fig. 5.10 and Tab. 5.12. For the Kaon decay amplitude, we finally get Re(𝐴₀) = 2.99(0.32)(0.59) × 10⁻⁷GeV, which is consistent with the experimental value of Re(𝐴₀) = 3.3201(18) × 10⁻⁷GeV, and Im(𝐴₀) = -6.98(0.62)(1.44) × 10⁻¹¹GeV. Combined with our earlier lattice calculation of 𝐴₂ [2], we obtained Re(𝜖′/𝜖) = 21.7(2.6)(6.2)(5.0) × 10⁻⁴, which agrees well with the experimental value of Re(𝜖′/𝜖) = 16.6(2.3) × 10⁻⁴, and Re(𝐴₀)/Re(𝐴₂) = 19.9(2.3)(4.4), consistent with the experimental value of Re(𝐴₀)/Re(𝐴₂) = 22.45(6), known as the 𝜟𝑰=1/2 rule.

In this work, we study two problems in quantum field theory from a boundary point of view. Our perspective is motivated by the bootstrap philosophy, which aims to understand how principles such as kinematics, unitarity, and symmetry constrain physical observables. Regarding kinematics, we actually first relax the unitarity constraint and investigate thenon-unitary representations of the boundary superconformal algebra for AdS4 with N = 2 supercharges. In particular, we identify multiplets containing partially massless (PM) fields, as well as other exotic shortening conditions and structures exclusive to the nonunitary regime. Then, turning on interactions, we study a problem centered in dynamics: we investigate the structure of the flat space wavefunctional in scalar field theories with nonlinearly realized symmetries. In particular, we highlight the so-called exceptional scalar field theories, which are the nonlinear sigma model, Dirac-Born-Infeld, and (special) galileon theories. We find that nonlinearly realized symmetries imply soft theorems which must be obeyed by the wavefunction. Moreover, we develop bootstrap techniques utilizing this information along with the singularity structure of the wavefunction to fix its form. In addition, we systematize this construction into a novel set of recursion relations.