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Register a new userProspects for a lattice computation of rare kaon decay amplitudes II $Ktopi ubar{ u}$ 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 standard 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.
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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.
We report a first, complete lattice QCD calculation of the long-distance contribution to the $K^+topi^+ ubar{ u}$ decay within the standard model. This is a second-order weak process involving two four-Fermi operators that is highly sensitive to new physics and being studied by the NA62 experiment at CERN. While much of this decay comes from perturbative, short-distance physics there is a long-distance part, perhaps as large as the planned experimental error, which involves nonperturbative phenomena. The calculation presented here, with unphysical quark masses, demonstrates that this contribution can be computed using lattice methods by overcoming three technical difficulties: (i) a short-distance divergence that results when the two weak operators approach each other, (ii) exponentially growing, unphysical terms that appear in Euclidean, second-order perturbation theory, and (iii) potentially large finite-volume effects. A follow-on calculation with physical quark masses and controlled systematic errors will be possible with the next generation of computers.
In Ref [1] we have presented the results of an exploratory lattice QCD computation of the long-distance contribution to the $K^+topi^+ ubar{ u}$ decay amplitude. In the present paper we describe the details of this calculation, which includes the implementation of a number of novel techniques. The $K^+topi^+ ubar{ u}$ decay amplitude is dominated by short-distance contributions which can be computed in perturbation theory with the only required non-perturbative input being the relatively well-known form factors of semileptonic kaon decays. The long-distance contributions, which are the target of this work, are expected to be of O(5%) in the branching ratio. Our study demonstrates the feasibility of lattice QCD computations of the $K^+topi^+ ubar{ u}$ decay amplitude, and in particular of the long-distance component. Though this calculation is performed on a small lattice ($16^3times32$) and at unphysical pion, kaon and charm quark masses, $m_pi=420$ MeV, $m_K=563$ MeV and $m_c^{overline{mathrm{MS}}}(mbox{2 GeV})=863$ MeV, the techniques presented in this work can readily be applied to a future realistic calculation.
The rare kaon decay $K^+topi^+ ubar{ u}$ is an ideal process in which to search for signs of new physics and is the primary goal of the NA62 experiment at CERN. In this paper we report on a lattice QCD calculation of the long-distance contribution to the $K^+topi^+ ubar{ u}$ decay amplitude at the near-physical pion mass $m_pi=170$ MeV. The calculations are however, performed on a coarse lattice and hence with a lighter charm quark mass ($m_c^{bar{mathrm{MS}}}(mbox{3 GeV})=750$ MeV) than the physical one. The main aims of this study are two-fold. Firstly we study the momentum dependence of the amplitude and conclude that it is very mild so that a computation at physical masses even at a single kinematic point would provide a good estimate of the long-distance contribution to the decay rate. Secondly we compute the contribution to the branching ratio from the two-pion intermediate state whose energy is below the kaon mass and find that it is less than 1% after its exponentially growing unphysical contribution has been removed and that the corresponding non-exponential finite-volume effects are negligibly small.
The rare pion decays ${pi}^+{rightarrow}{mu}^+{ u}_{mu}{ u}bar{ u}$ and ${pi}^+{rightarrow}e^+{ u}_{e}{ u}bar{ u}$ are allowed in the Standard Model but highly suppressed. These decays were searched for using data from the PIENU experiment. A first result for ${Gamma}({pi}^+{rightarrow}{mu}^+{ u}_{mu}{ u}bar{ u})/{Gamma}({pi}^+{rightarrow}{mu}^+{ u}_{mu})<8.6{times}10^{-6}$, and an improved measurement ${Gamma}({pi}^+{rightarrow}{e}^+{ u}_{e}{ u}bar{ u})/{Gamma}({pi}^+{rightarrow}{mu}^+{ u}_{mu})<1.6{times}10^{-7}$ were obtained.
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