The rare kaon decays $Ktopiell^+ell^-$ and $Ktopi ubar{ u}$ are flavor changing neutral current (FCNC) processes and hence promising channels with which to probe the limits of the standard model and to look for signs of new physics. In this paper we
demonstrate the feasibility of lattice calculations of $Ktopiell^+ell^-$ decay amplitudes for which long-distance contributions are very significant. We show that the dominant finite-volume corrections (those decreasing as powers of the volume) are negligibly small and that, in the four-flavor theory, no new ultraviolet divergences appear as the electromagnetic current $J$ and the effective weak Hamiltonian $H_W$ approach each other. In addition, we demonstrate that one can remove the unphysical terms which grow exponentially with the range of the integration over the time separation between $J$ and $H_W$. We will now proceed to exploratory numerical studies with the aim of motivating further experimental measurements of these decays. Our work extends the earlier study by Isidori, Turchetti and Martinelli which focussed largely on the renormalization of ultraviolet divergences. In a companion paper we discuss the evaluation of the long-distance contributions to $Ktopi ubar{ u}$ decays; these contributions are expected to be at the level of a few percent for $K^+$ decays.
We report on the first complete calculation of the $K_L-K_S$ mass difference, $Delta M_K$, using lattice QCD. The calculation is performed on a 2+1 flavor, domain wall fermion ensemble with a 330MeV pion mass and a 575 MeV kaon mass. We use a quenche
d charm quark with a 949 MeV mass to implement Glashow-Iliopoulos-Maiani cancellation. For these heavier-than-physical particle masses, we obtain $Delta M_K =3.19(41)(96)times 10^{-12}$ MeV, quite similar to the experimental value. Here the first error is statistical and the second is an estimate of the systematic discretization error. An interesting aspect of this calculation is the importance of the disconnected diagrams, a dramatic failure of the OZI rule.
The RBC and UKQCD collaborations have recently proposed a procedure for computing the K_L-K_S mass difference. A necessary ingredient of this procedure is the calculation of the (non-exponential) finite-volume corrections relating the results obtaine
d on a finite lattice to the physical values. This requires a significant extension of the techniques which were used to obtain the Lellouch-Luscher factor, which contains the finite-volume corrections in the evaluation of non-leptonic kaon decay amplitudes. We review the status of our study of this issue and, although a complete proof is still being developed, suggest the form of these corrections for general volumes and a strategy for taking the infinite-volume limit. The general result reduces to the known corrections in the special case when the volume is tuned so that there is a two-pion state degenerate with the kaon.
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 g
iving 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 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 extend the Rome-Southampton regularization independent momentum-subtraction renormalization scheme(RI/MOM) for bilinear operators to one with a nonexceptional, symmetric subtraction point. Two-point Greens functions with the insertion of quark bil
inear operators are computed with scalar, pseudoscalar, vector, axial-vector and tensor operators at one-loop order in perturbative QCD. We call this new scheme RI/SMOM, where the S stands for symmetric. Conversion factors are derived, which connect the RI/SMOM scheme and the MSbar scheme and can be used to convert results obtained in lattice calculations into the MSbar scheme. Such a symmetric subtraction point involves nonexceptional momenta implying a lattice calculation with substantially suppressed contamination from infrared effects. Further, we find that the size of the one-loop corrections for these infrared improved kinematics is substantially decreased in the case of the pseudoscalar and scalar operator, suggesting a much better behaved perturbative series. Therefore it should allow us to reduce the error in the determination of the quark mass appreciably.
We present a calculation of the renormalization coefficients of the quark bilinear operators and the K-Kbar mixing parameter B_K. The coefficients relating the bare lattice operators to those in the RI/MOM scheme are computed non-perturbatively and t
hen matched perturbatively to the MSbar scheme. The coefficients are calculated on the RBC/UKQCD 2+1 flavor dynamical lattice configurations. Specifically we use a 16^3 x 32 lattice volume, the Iwasaki gauge action at beta=2.13 and domain wall fermions with L_s=16.
