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Register a new user$Kto(pipi)_{I=2}$ decays and twisted boundary conditions
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We propose a new method to evaluate the Lellouch-Luscher factor which relates the $Delta I=3/2$ $Ktopipi$ matrix elements computed on a finite lattice to the physical (infinite-volume) decay amplitudes. The method relies on the use of partially twisted boundary conditions, which allow the s-wave $pipi$ phase shift to be computed as an almost continuous function of the centre-of-mass relative momentum and hence for its derivative to be evaluated. We successfully demonstrate the feasibility of the technique in an exploratory computation.
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The pi+pi+ s-wave scattering phase-shift is determined below the inelastic threshold using Lattice QCD. Calculations were performed at a pion mass of m_pi~390 MeV with an anisotropic n_f=2+1 clover fermion discretization in four lattice volumes, with spatial extent L~2.0, 2.5, 3.0 and 3.9 fm, and with a lattice spacing of b_s~0.123 fm in the spatial direction and b_t b_s/3.5 in the time direction. The phase-shift is determined from the energy-eigenvalues of pi+pi+ systems with both zero and non-zero total momentum in the lattice volume using Luschers method. Our calculations are precise enough to allow for a determination of the threshold scattering parameters, the scattering length a, the effective range r, and the shape-parameter P, in this channel and to examine the prediction of two-flavor chiral perturbation theory: m_pi^2 a r = 3+O(m_pi^2/Lambda_chi^2). Chiral perturbation theory is used, with the Lattice QCD results as input, to predict the scattering phase-shift (and threshold parameters) at the physical pion mass. Our results are consistent with determinations from the Roy equations and with the existing experimental phase shift data.
We describe the computation of the amplitude A_2 for a kaon to decay into two pions with isospin I=2. The results presented in the letter Phys.Rev.Lett. 108 (2012) 141601 from an analysis of 63 gluon configurations are updated to 146 configurations giving Re$A_2=1.381(46)_{textrm{stat}}(258)_{textrm{syst}} 10^{-8}$ GeV and Im$A_2=-6.54(46)_{textrm{stat}}(120)_{textrm{syst}}10^{-13}$ GeV. Re$A_2$ is in good agreement with the experimental result, whereas the value of Im$A_2$ was hitherto unknown. We are also working towards a direct computation of the $Kto(pipi)_{I=0}$ amplitude $A_0$ but, within the standard model, our result for Im$A_2$ can be combined with the experimental results for Re$A_0$, Re$A_2$ and $epsilon^prime/epsilon$ to give Im$A_0/$Re$A_0= -1.61(28)times 10^{-4}$ . Our result for Im,$A_2$ implies that the electroweak penguin (EWP) contribution to $epsilon^prime/epsilon$ is Re$(epsilon^prime/epsilon)_{mathrm{EWP}} = -(6.25 pm 0.44_{textrm{stat}} pm 1.19_{textrm{syst}}) times 10^{-4}$.
The possibility of imposing partially twisted boundary conditions in the lattice study of the resonance states is investigated by using the effective field theory (EFT) methods. In particular, it is demonstrated that - in certain cases - it is possible to use partial twisting even in the presence of the quark annihilation diagrams.
We present new results for the amplitude $A_2$ for a kaon to decay into two pions with isospin $I=2$: Re$A_2 = 1.50(4)_mathrm{stat}(14)_mathrm{syst}times 10^{-8}$ GeV; Im$A_2 = -6.99(20)_mathrm{stat}(84)_mathrm{syst}times 10^{-13}$ GeV. These results were obtained from two ensembles generated at physical quark masses (in the isospin limit) with inverse lattice spacings $a^{-1}=1.728(4)$ GeV and $2.358(7)$ GeV. We are therefore able to perform a continuum extrapolation and hence largely to remove the dominant systematic uncertainty from our earlier results, that due to lattice artefacts. The only previous lattice computation of $Ktopipi$ decays at physical kinematics was performed using an ensemble at a single, rather coarse, value of the lattice spacing ($a^{-1}simeq 1.37(1)$ GeV). We confirm the observation that there is a significant cancellation between the two dominant contributions to Re$A_2$ which we suggest is an important ingredient in understanding the $Delta I=1/2$ rule, Re$A_0$/Re$A_2simeq 22.5$, where the subscript denotes the total isospin of the two-pion final state. Our result for $A_2$ implies that the electroweak penguin contribution to $epsilon^prime/epsilon$ is Re($epsilon^prime/epsilon)_textrm{EWP}=-(6.6pm 1.0)times 10^{-4}$.
Phase shifts for $s$-wave $pipi$ scattering in both the $I=0$ and $I=2$ channels are determined from a lattice QCD calculation performed on 741 gauge configurations obeying G-parity boundary conditions with a physical pion mass and lattice size of $32^3times 64$. These results support our recent study of direct CP violation in $Ktopipi$ decay cite{Abbott:2020hxn}, improving our earlier 2015 calculation cite{Bai:2015nea}. The phase shifts are determined for both stationary and moving $pipi$ systems, at three ($I=0$) and four ($I=2$) different total momenta. We implement several $pipi$ interpolating operators including a scalar bilinear $sigma$ operator and paired single-pion bilinear operators with the constituent pions carrying various relative momenta. Several techniques, including correlated fitting and a bootstrap determination of p-values have been used to refine the results and a comparison with the generalized eigenvalue problem (GEVP) method is given. A detailed systematic error analysis is performed which allows phase shift results to be presented at a fixed energy.
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