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Register a new userLong distance contributions to the rare kaon decay $Ktopiell^{+}ell^{-}$
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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.
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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.
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.
Neutrinoless double beta decay, if detected, would prove that neutrinos are Majorana fermions and provide the direct evidence for lepton number violation. If such decay would exist in nature, then $pi^-pi^-to ee$ and $pi^-topi^+ ee$ (or equivalently $pi^-e^+topi^+ e^-$) are the two simplest processes accessible via first-principle lattice QCD calculations. In this work, we calculate the long-distance contributions to the $pi^-topi^+ee$ transition amplitude using four ensembles at the physical pion mass with various volumes and lattice spacings. We adopt the infinite-volume reconstruction method to control the finite-volume effects arising from the (almost) massless neutrino. Providing the lattice QCD inputs for chiral perturbation theory, we obtain the low energy constant $g_ u^{pipi}(m_rho)=-10.89(28)_text{stat}(74)_text{sys}$, which is close to $g_ u^{pipi}(m_rho)=-11.96(31)_text{stat}$ determined from the crossed-channel $pi^-pi^-to ee$ decay.
Standard lattice calculations in flavour physics or in studies of hadronic structure are based on the evaluation of matrix elements of local composite operators between hadronic states or the vacuum. In this talk I discuss developments aimed at the computation of long-distance, and hence non-local, contributions to such processes. In particular, I consider the calculation of the $K_L$-$K_S$ mass difference $Delta m_K=m_{K_L}-m_{K_S}$ and the amplitude for the rare-kaon decay processes $Ktopiell^+ell^-$, where the lepton $ell=e$ or $mu$. Lattice calculations of the long-distance contributions to the indirect $CP$-violating parameter $epsilon_K$ and to the rare decays $Ktopi ubar u$ are also beginning. Finally I discuss the possibility of including $O(alpha)$ electromagnetic effects in computations of leptonic and semileptonic decay widths, where the novel feature is the presence of infrared divergences. This implies that contributions to the width from processes with a real photon in the final state must be combined with those with a virtual photon in the amplitude so that the infrared divergences cancel by the Bloch-Nordsieck mechanism. I present a proposed procedure for lattice computations of the $O(alpha)$ contributions with control of the cancellation of the infrared divergences.
The largest contribution to the CP violating K_L-K_S mixing parameter epsilon_K comes from second order weak interactions at short distances and can be accurately determined by a combination of electroweak perturbation theory and the calculation of the parameter B_K from lattice QCD. However, there is an additional long distance contribution to epsilon_K which is estimated to be of order 5%. Here recently introduced lattice techniques for computing the long-distance component of the K_L-K_S mass difference are generalized to this long-distance contribution to epsilon_K.
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