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Prospects for a lattice computation of rare kaon decay amplitudes: $Ktopiell^+ell^-$ decays

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 نشر من قبل Antonin Portelli
 تاريخ النشر 2015
  مجال البحث
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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.



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The rare kaon decays $Ktopi ubar{ u}$ are strongly suppressed in the standard model and widely regarded as processes in which new phenomena, not predicted by the standard model, may be observed. Recognizing such new phenomena requires precise standar d model prediction for the braching ratio of $Ktopi ubar{ u}$ with controlled uncertainty for both short-distance and long-distance contributions. In this work we demonstrate the feasibility of lattice QCD calculation of the long-distance contribution to rare kaon decays with the emphasis on $K^+topi^+ ubar{ u}$. Our methodology covers the calculation of both $W$-$W$ and $Z$-exchange diagrams. We discuss the estimation of the power-law, finite-volume corrections and two methods to consistently combine the long distance contribution determined by the lattice methods outlined here with the short distance parts that can be reliably determined using perturbation theory. It is a subsequent work of our first methodology paper on $Ktopiell^+ell^-$, where the focus was made on the $gamma$-exchange diagrams.
The rare decays of a kaon into a pion and a charged lepton/antilepton pair proceed via a flavour changing neutral current and therefore may only be induced beyond tree level in the Standard Model. This natural suppression makes these decays sensitive to the effects of potential New Physics. To discern such New Physics one must be able to control the errors on the Standard Model prediction of the decay amplitude. These particular decay channels however are dominated by a single photon exchange; this involves a sizeable long-distance hadronic contribution which represents the current major source of theoretical uncertainty. Here we outline our methodology for the computation of the long distance contributions to these rare decay amplitudes using lattice QCD, and present the numerical results of some exploratory studies using the Domain Wall Fermion ensembles of the RBC and UKQCD collaborations.
The rare decays of a kaon into a pion and a charged lepton/antilepton pair proceed via a flavour changing neutral current and therefore may only be induced beyond tree level in the Standard Model. This natural suppression makes these decays sensitive to the effects of potential New Physics. The CP conserving $Ktopi ell^+ell^-$ decay channels however are dominated by a single photon exchange; this involves a sizeable long-distance hadronic contribution which represents the current major source of theoretical uncertainty. Here we outline our methodology for the computation of the long-distance contributions to these rare decay amplitudes using lattice QCD and present the numerical results of the first exploratory studies of these decays in which all but the disconnected diagrams are evaluated. The domain wall fermion ensembles of the RBC and UKQCD collaborations are used, with a pion mass of $M_{pi}sim 430,mathrm{MeV}$ and a kaon mass of $M_{K}sim 625,mathrm{MeV}$. In particular we determine the form factor, $V(z)$, of the $K^+topi^+ell^+ell^-$ decay from the lattice at small values of $z=q^2/M_{K}^{2}$, obtaining $V(z)=1.37(36),, 0.68(39),, 0.96(64)$ for the three values of $z=-0.5594(12),, -1.0530(34),, -1.4653(82)$ respectively.
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