Kaon to two-pion decay and pion-pion scattering from lattice QCD

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.