2011 Theses Doctoral

Delta I Equals Three Halfs Kaon To Two Pion Decays Using Lattice Quantum Chromodynamics with Domain Wall Fermions

Lightman, Matthew

We calculate matrix elements for kaon to two pion decays in the 𝛥𝛪 = 3/2 channel using lattice gauge theory simulations. From these we can extract the decay amplitude 𝐴₂, for which the real part is related to the decay rate and can be compared to the experimental result Re(𝐴₂) = 1.484 x 10⁻⁸ GeV, and for which the imaginary part is related to direct charge-parity violation in the neutral kaon system. We report the results of one simulation with nearly physical particle masses and kinematics, specifically m_𝜅 = 509.0(9.1) MeV, m_π = 142.8(2.5) MeV, and E_ππ = 485.7(8.0) MeV.

This simulation was performed on RBC/UKQCD 32³ x 64, 𝐿_s = 32 lattices, using 2+1 dynamical flavors of domain wall fermions and a Dislocation Suppressing Determinant Ratio plus Iwasaki gauge action, and with an inverse lattice spacing a⁻¹ = 1.373(24) GeV so that the spatial extent of the lattice is 4.60 fm and m_π𝐿 = 3.3. We find that Re(𝐴₂) = 1.461(87)_stat(200)_sys x 10⁻⁸ GeV, in good agreement with the experimental value. We also find Im(𝐴₂) = -8.67(45)_stat(1.95)_sys x 10⁻¹³ GeV, and Im(𝐴₂)/Re(𝐴2) = -5.93(27)_stat(1.42)_sys x 10⁻⁵, however the value of Im(𝐴₂) depends on a rough hypothesis for some of the renormalization constants which have not yet been calculated, and thus we quote a large systematic error.

We also report the results of a simulation involving a variety of kaon and pion masses and momenta, which was conducted in order to study the dependence of the decay amplitude on particle masses and kinematics, and to study the effect of not having exactly physical masses and kinematics in the first simulation. The use of the quenched approximation and smaller spatial volume in this second simulation allowed for multiple masses to be simulated in a reasonable amount of time, but introduced an uncontrolled approximation and forced us to use pion masses a bit larger than the physical mass. The study was conducted on 24³ x 64, 𝐿_s = 16 lattices, with the quenched Doubly Blocked Wilson 2 gauge action, and an inverse lattice spacing of 𝛼⁻¹ = 1.31(2) GeV.

We find that an extrapolation to physical masses and kinematics yields values Re(𝐴₂) = 2.25(18)_stat x 10⁻⁸ GeV and Im(𝐴₂) = -13.44(84)stat x 10⁻¹³ GeV. These results are significantly larger than those of the full dynamical simulation and of experiment. We attribute this mainly to the an inaccurate determination of the lattice spacing a using the rho mass, since it comes in as a⁻³ in the calculation of 𝐴₂. Finally, a third simulation is performed with 2+1 dynamical flavors of domain wall fermions on a finer 32³ x 64, 𝐿_s = 16 lattice, but only with pions that have nearly zero momentum. It, and the quenched simulation, are used mainly to estimate the systematic error in the first simulation, which is taken as the final result.