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In the standard model the rare kaon decay 𝙆_𝐿 → 𝜇⁺𝜇⁻ is a highly suppressed, ``strangeness changing neutral current process'' that requires the exchange of two weak bosons with an accurately measured branching fraction 𝐵(𝙆_𝐿 → 𝜇⁺𝜇⁻) = (6.84 ∓ 0.11 ) ✕ 10⁻⁹ [1]. For this measurement to become an important short-distance test of the standard model, the competing 𝑂(𝛼²_𝙴𝙼𝐺_𝙵) two-photon contribution must be computed and removed from the total decay amplitude. While the imaginary part of this contribution can be obtained from the 𝙆_𝐿 → 𝜇⁺𝜇⁻ decay rate and the optical theorem, the real part must be computed in QCD [2]. Depending on a relative sign, a 10% calculation of the real part of the 𝑂(𝛼²_𝙴𝙼𝐺_𝙵) two-photon contribution would lead to a 6% or 17% test of the standard model. As a first step in developing a strategy for computing the two-photon contribution to the 𝙆_𝐿 → 𝜇⁺𝜇⁻ decay, we examine a simpler process 𝜋⁰ → 𝓮⁺𝓮⁻. Here no weak interaction vertex is involved and, more importantly, there is no intermediate hadronic state with a mass smaller than that of the initial pion. The sole complication arises from the presence of the two-photon intermediate state, only one of the difficulties offered by the 𝙆_𝐿 → 𝜇⁺𝜇⁻ decay. We show that the 𝜋⁰ → 𝓮⁺𝓮⁻ amplitude can be calculated with an analytic continuation method where the entire decay amplitude including the imaginary part is preserved. The real part involves non-perturbative QCD contribution and is of substantial interest, while the imaginary part of calculated amplitude can be compared with the prediction of optical theorem to demonstrate the effectiveness of this method. We obtain Re𝓐 = 18.60(1.19)(1.04) eV, Im𝓐 = 32.59(1.50)(1.65) e𝐕 and a more precise value for their ratio Re𝓐/Im𝓐 = 0.571(10)(4) from continuum extrapolation of two lattice ensembles, where 𝓐 is the decay amplitude, the error in the first parenthesis is statistical and the error in the second parenthesis is systematic. Next, we develop a computational strategy to determine the 𝙆_𝐿 → 𝛾 𝛾 decay amplitude. It involves the same hadronic matrix element as the 𝙆_𝐿 → 𝜇⁺𝜇⁻ decay as well as all the intermediate states whose energies are lower than or close to the initial kaon sate except for the |𝜋𝜋𝜇〉that is difficult to deal with. While the lattice QCD calculation is carried out in finite volume, the emitted photons are treated in infinite volume and the resulting finite-volume errors decrease exponentially in the linear size of the lattice volume. Only the 𝑪𝑷-conserving contribution to the decay is computed and we must subtract unphysical contamination resulting from single pion and eta intermediate states which grow exponentially (or fall slowly) as the time separation between the initial and final lattice operators is increased. Results from a calculation without disconnected diagrams on a 24³ ✕ 64 lattice volume with 1/𝛼 =1 Ge𝐕 and physical quark masses are presented.

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.