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Register a new userLong-distance contributions to flavour-changing processes
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C. Sachrajda
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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.
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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.
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.
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.
We explore the possibility that the masses for the first two generations of fermions and the quark flavor violation are generated radiatively in the Minimal Supersymmetric Standard Model. We assume that the source of all flavor violation resides in the the supersymmetry breaking sector and is transmitted radiatively to the Standard Model fermion sector through finite corrections at low energy. The approximate radiative alignment between the Yukawa and soft supersymmetry breaking matrices helps to suppress some of the supersymmetric contributions to flavor changing processes, overcoming current experimental constraints. This mechanism may also explain the non-observation of proton decay, since flavor conservation in the superpotential would imply the suppression of dimension five operators in supersymmetric grand unified theories.
In Ref. [1], a method was proposed to calculate QED corrections to hadronic self energies from lattice QCD without power-law finite-volume errors. In this paper, we extend the method to processes which occur at second-order in the weak interaction and in which there is a massless (or almost massless) leptonic propagator. We demonstrate that, in spite of the presence of the propagator of an almost massless electron, such an infinite-volume reconstruction procedure can be used to obtain the amplitude for the rare kaon decay $K^+topi^+ ubar u$ from a lattice quantum chromodynamics computation with only exponentially small finite-volume corrections.
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