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Register a new user$K rightarrow pipi$ $Delta I=3/2$ decay amplitude in the continuum limit
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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($epsilon^prime/epsilon)_textrm{EWP}=-(6.6pm 1.0)times 10^{-4}$.
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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$(epsilon^prime/epsilon)_{mathrm{EWP}} = -(6.25 pm 0.44_{textrm{stat}} pm 1.19_{textrm{syst}}) times 10^{-4}$.
We review the status of recent calculations by the RBC-UKQCD collaboration of the complex amplitude $A_2$, corresponding to the decay of a kaon to a two pion state with total isospin 2. In particular, we present preliminary results from two new ensembles: $48^3 times 96$ with $a^{-1}=1.73$ GeV and $64^3 times 128$ with $a^{-1}=2.3$ GeV, both at physical kinematics. Both ensembles were generated Iwasaki gauge action and domain wall fermion action with 2+1 flavours. These results, in comparison to our earlier ones on a $32^3$ DSDR lattice with $a^{-1}=1.36$ GeV, enable us to significantly reduce the discretization errors. The partial cancellation between the two dominant contractions contributing to Re($A_2$) has been confirmed and we believe that this cancellation is a major contribution to the $Delta I=1/2$ rule.
We report a direct lattice calculation of the $K$ to $pipi$ decay matrix elements for both the $Delta I=1/2$ and 3/2 amplitudes $A_0$ and $A_2$ on 2+1 flavor, domain wall fermion, $16^3times32times16$ lattices. This is a complete calculation in which all contractions for the required ten, four-quark operators are evaluated, including the disconnected graphs in which no quark line connects the initial kaon and final two-pion states. These lattice operators are non-perturbatively renormalized using the Rome-Southampton method and the quadratic divergences are studied and removed. This is an important but notoriously difficult calculation, requiring high statistics on a large volume. In this paper we take a major step towards the computation of the physical $Ktopipi$ amplitudes by performing a complete calculation at unphysical kinematics with pions of mass 422,MeV at rest in the kaon rest frame. With this simplification we are able to resolve Re$(A_0)$ from zero for the first time, with a 25% statistical error and can develop and evaluate methods for computing the complete, complex amplitude $A_0$, a calculation central to understanding the $Delta =1/2$ rule and testing the standard model of CP violation in the kaon system.
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.
We improve a previous quenched result for heavy-light pseudoscalar meson decay constants with the light quark taken to be the strange quark. A finer lattice resolution (a ~ 0.05 fm) in the continuum limit extrapolation of the data computed in the static approximation is included. We also give further details concerning the techniques used in order to keep the statistical and systematic errors at large lattice sizes L/a under control. Our final result, obtained by combining these data with determinations of the decay constant for pseudoscalar mesons around the D_s, follows nicely the qualitative expectation of the 1/m-expansion with a (relative) 1/m-term of about -0.5 GeV/m_PS. At the physical b-quark mass we obtain F_{B_s} = 193(7) MeV, where all errors apart from the quenched approximation are included.
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