The dynamical fluctuations of kaon--to--pion ratios have been studied over a wide range of center--of--mass energies $\sqrt{s}$. Based on changing phase space volume which apparently is the consequence of phase transition from hadrons to quark--gluon plasma at large $\sqrt{s}$, single--particle distribution function $f$ is assumed to be rather modified. Varying $f$ and phase space volume are implemented in the grand--canonical partition function, especially at large $\sqrt{s}$, so that hadron resonance gas model, when taking into account the experimental acceptance and quark phase space occupation factor, turns to be able to reproduce the dynamical fluctuations over the entire range of $\sqrt{s}$.
We review our recent description of high energy forward hadronic cross sections and in particular of pion scattering data by means of Regge theory. We obtain a prediction for the total proton-proton cross section at LHC. In addition, and contrary to some suggestions in the literature we show that standard Regge behavior can accommodate both crossing symmetry and factorization. Some consequences for low energy pion-pion scattering are briefly discussed.
We carry out a theoretical and Monte Carlo study on the $J/ψ$ decays into $ωππ$ and $K\bar{K}π$ through intermediate subthreshold $ρ$ meson by using SU(3)-symmetric Lagrangian approach. It is found that the subthreshold $ρ$ contribution is not negligible and may have significant influence on partial wave analysis of resonances in these channels, especially near the $ωπ$ and $K \bar{K}$ thresholds.
First simultaneous global QCD analysis of kaon and pion parton distributions with lattice QCD constraints
Argonne National Laboratory（阿贡国家实验室）；North Carolina State University（北卡罗来纳州立大学）；Jefferson Lab（杰弗逊实验室）；Helmholtz-Institut für Strahlen- und Kernphysik, University of Bonn（波恩大学亥姆霍兹辐射与核物理研究所）；Bethe Center for Theoretical Physics, University of Bonn（波恩大学贝特理论物理中心）
We present the first lattice-QCD calculation of the kaon valence-quark distribution functions using the large-momentum effective theory (LaMET) approach. The calculation is performed with multiple pion masses with the lightest one around 220 MeV, 2 lattice spacings $a=0.06$ and 0.12 fm, $(M_π)_\text{min} L \approx 5.5$, and high statistics ranging from 11,600 to 61,312 measurements. We also calculate the valence-quark distribution of pion and find it to be consistent with the FNAL E615 experimental results, and our ratio of the $u$ quark PDF in the kaon to that in the pion agrees with the CERN…
The interplay of strong and weak decay amplitudes for B -> pion-pion-kaon and B -> kaon-antikaon-kaon, with the pion-pion and kaon-antikaon pairs interacting in isospin-0 S-wave, is analyzed for pion-pion effective mass from threshold to 1.2 GeV. To improve agreement with experiment of a factorization approach with some QCD corrections, addition of long-distance contributions, called charming penguins is necessary.
We consider strong and electromagnetic isospin violation in near-threshold pion-kaon scattering. At tree level, such effects are small for all physical channels. We work out the complete one-loop corrections to the process pi^- K^+ -> pi^0 K^0. They come out rather small. We also show that the corresponding radiative cross section is highly suppressed at threshold.
A rainbow-ladder truncation of QCD's Dyson-Schwinger equations, constrained by existing applications to hadron physics, is employed to compute the valence-quark parton distribution functions of the pion and kaon. Comparison is made to pi-N Drell-Yan data for the pion's u-quark distribution and to Drell-Yan data for the ratio u_K(x)/u_π(x): the environmental influence of this quantity is a parameter-free prediction, which agrees well with existing data. Our analysis unifies the computation of distribution functions with that of numerous other properties of pseudoscalar mesons.
We present a calculation of the connected-diagram contributions to the first three non-trivial Mellin moments for the pion and kaon, extracted using local operators with up to 3 covariant derivatives. We use one ensemble of gauge configurations with two degenerate light, a strange and a charm quark ($N_f$=2+1+1) of maximally twisted mass fermions with clover improvement. The ensemble has a pion mass $\sim$260 MeV, and a kaon mass $\sim$530 MeV. We reconstruct the $x$-dependence of the PDFs via fits to our results, and find that our lattice data favor a $(1-x)^2$-behavior in the large-$x$ regio…
Borrowing the structures of the hydrogen atom, molecular ion, and diatomic molecule, we predict the nature of a new type of hydrogenlike charm-pion or charm-kaon matter that could be obtained by replacing the proton and electron in hydrogen matter with a charmed meson and a pion or a kaon, respectively. We find that the spectra of the atom, molecular ion, and diatomic molecule can be obtained simultaneously with the Coulomb potential for the hydrogen, the charm-pion, and the charm-kaon systems. The predicted charm-pion matter also allows us to explore the mass shift mediated by the strong inte…
We study the contributions from the connected and disconnected contraction diagrams to the pion-kaon scattering amplitude within the framework of SU$(4|1)$ partially-quenched chiral perturbation theory. Combining this with a finite-volume analysis, we demonstrate that a lattice calculation of the easier computable connected correlation functions is able to provide valuable information of the noisier disconnected correlation functions, and may serve as a theory guidance for the future refinement of the corresponding lattice techniques.
We study the contribution of the strong interaction between the two pions in S- and P-waves to the weak B --> pion pion kaon decay amplitudes. The interference between these two waves is analyzed in the pion-pion effective mass range of the rho(770) and f0(980) resonances. We use a unitary pion-pion and antikaon-kaon coupled-channel model to describe the S-wave interactions and a Breit-Wigner function for the P-wave amplitude. The weak B-decay amplitudes, obtained from QCD factorization, are supplemented with charming penguin contributions in both waves. The four complex parameters of these lo…
We describe expressions for pion and kaon dressed-quark distribution functions that incorporate contributions from gluons which bind quarks into these mesons and hence overcome a flaw of the commonly used handbag approximation. The distributions therewith obtained are purely valence in character, ensuring that dressed-quarks carry all a meson's momentum at a characteristic hadronic scale and vanishing as $(1-x)^2$ when Bjorken-$x\to 1$. Comparing such distributions within the pion and kaon, it is apparent that the size of SU(3)-flavour symmetry breaking in meson parton distribution functions i…
In high energy Heavy Ion Collisions, the onset of the quark-gluon plasma is the colour glass condensate, dominated by gluons. The final state is hadronic, and dominated by pions and kaons. Here we investigate an effective approach of QCD with these bosonic fields and which can help to describe the transition of gluons into light mesons. Formally, our approach consists in integrating out the quark fields from the QCD path integral, utilizing auxiliary fields. In this way the fermionic fields are replaced by light mesons, such as the pions and sigma field. We apply our effective action to comput…
We present a coupled channel partial wave analysis of nondiffractive S-wave pi+ pi- and K+ K- photoproduction focusing on the KK threshold. Final state interactions are included. We calculate total cross sections, angular and effective mass distributions in both pion-pion and kaon-antikaon channels. Our results indicate that these processes are experimentally measurable and valuable information on the f0(980) resonance structure can be obtained.
We discuss the recent progress in the study of semileptonic kaon and pion decays, including new experimental results, improved electroweak radiative corrections, form factor calculations and isospin-breaking effects. As a result, we obtain $|V_{us}|=0.22309(40)(39)(3)$ from kaon semileptonic decays and $|V_{us}/V_{ud}|=0.22908(66)(41)(40)(2)(1)$ from the ratio between the kaon and pion semileptonic decay rates. We report an apparent violation of the top-row Cabibbo-Kobayashi-Maskawa matrix unitarity at a $3.2\sim 5.6σ$ level, and a discrepancy at a $2.2σ$ level between the value of $|V_{us}/V_…
Within the three-flavor PNJL and EPNJL chiral quark models we have obtained pseudoscalar meson properties in quark matter at finite temperature $T$ and baryochemical potential $μ_B$. We compare the meson pole (Breit-Wigner) approximation with the Beth-Uhlenbeck (BU) approach that takes into account the continuum of quark-antiquark scattering states when determining the partial densities of pions and kaons. We evaluate the kaon-to-pion ratios along the (pseudo-)critical line in the $T-μ_B$ plane as a proxy for the chemical freezeout line, whereby the variable $x=T/μ_B$ is introduced that corres…
Pion and kaon structural properties provide insights into the emergence of mass within the Standard Model and attendant modulations by the Higgs boson. Novel expressions of these effects, in impact parameter space and in mass and pressure profiles, are exposed via $π$ and $K$ generalised parton distributions, built using the overlap representation from light-front wave functions constrained by one-dimensional valence distribution functions that describe available data. Notably, e.g. $K$ pressure profiles are spatially more compact than $π$ profiles and both achieve near-core pressures of simil…
Based on our analysis of the contributions from the connected and disconnected contraction diagrams to the pion-kaon scattering amplitude, we provide the first determination of the only free low-energy constant at $\mathcal{O}(p^4)$, known as $L_0^r$, in SU$(4|1)$ Partially-Quenched Chiral Perturbation Theory using the data from the Extended Twisted Mass Collaboration, $L_0^r(μ=M_ρ) = 0.77(20)(25)(7)(7)(2)\cdot 10^{-3}$. The theory uncertainties originate from the unphysical scattering length, the physical low-energy constants, the higher-oder chiral corrections, the (lattice) meson masses and…
We have calculated the Bjorken-x dependence of the kaon and pion valence quark distributions in a statistical model. Each meson is described by a Fock state expansion in terms of quarks, antiquarks and gluons. Although Drell-Yan experiments have measured the pion valence quark distributions directly, the kaon valence quark distributions have only been deduced from the measurement of the ratio $\bar{u}_K(x)/\bar{u}_π(x)$. We show that, using no free parameters, our model predicts the decrease of this ratio with increasing x.
We investigate the unpolarized pion and kaon fragmentation functions using the nonlocal chiral-quark model. In this model the interactions between the quarks and pseudoscalar mesons is manifested nonlocally. In addition, the explicit flavor SU(3) symmetry breaking effect is taken into account in terms of the current quark masses. The results of our model are evaluated to higher $Q^2$ value $Q^2=4\,\mathrm{GeV}^2$ by the DGLAP evolution. Then we compare them with the empirical parametrizations. We find that our results are in relatively good agreement with the empirical parametrizations and the…
We present the state-of-the-art lattice QCD calculation of the pion and kaon light-cone distribution amplitudes (DAs) using large-momentum effective theory. The calculation is done at three lattice spacings $a\approx\{0.06,0.09,0.12\}$ fm and physical pion and kaon masses, with the meson momenta $P_z = \{1.29,1.72,2.15\}$ GeV. The result is non-perturbatively renormalized in a recently proposed hybrid scheme with self renormalization, and extrapolated to the continuum as well as the infinite momentum limit. We find a significant deviation of the pion and kaon DAs from the asymptotic form, and …
Using the tadpole improved Wilson quark action on small, coarse and anisotropic lattices, $Kπ$ scattering length in the $I=3/2$ channel is calculated within quenched approximation. The results are extrapolated towards the chiral and physical kaon mass region. Finite volume and finite lattice spacing errors are also analyzed and a result in the infinite volume and continuum limit is obtained. Our result is compared with the results obtained using Roy equations, Chiral Perturbation Theory, dispersion relations and the experimental data.
We analyze the phase diagram of a three-flavor Nambu-Jona-Lasinio model at finite temperature $T$ and chemical potentials $μ_u, μ_d, μ_s$. We study the competition of pion and kaon condensation and we propose a physical situation in which kaon condensation could be led only by light quark finite densities.
We present an overview of searches fo violation of lepton flavor universality with focus on low energy precision probes using pions, kaons, tau leptons, and nuclear beta decays. The current experimental results are reviewed, the theoretical status within the context of the Standard Model is summarized, and future prospects (both experimental and theoretical) are discussed. We review the implications of these measurements for physics beyond the Standard Model by performing a global model-independent fit to modified $W$ couplings to leptons and four-fermion operators. We also discuss new physics…
Strangeness was discovered roughly seventy years ago, lodged in a particle now known as the kaon, $K$. Kindred to the pion, $π$; both states are massless in the absence of Higgs-boson couplings. Kaons and pions are Nature's most fundamental Nambu-Goldstone modes. Their properties are largely determined by the mechanisms responsible for emergent mass in the standard model, but modulations applied by the Higgs are crucial to Universe evolution. Despite their importance, little is known empirically about $K$ and $π$ structure. This study delivers the first parameter-free predictions for all $K$ d…
We calculate the pion and kaon electromagnetic form factors in finite volume for a generic momentum transfer in space-like region, using SU(3) chiral perturbation theory. To this end, we first find the hadronic matrix element in a new form which is suitable for our calculation in finite volume. We present our numerical findings for the chiral behavior and finite volume corrections and compare these for pion and kaon form factors. As a result, for the pion electromagnetic form factor we find the finite volume correction $ΔF^π_{em} = 0.0041$ at momentum transfer $q^2 = -0.1358$ and for the kaon …
We study the low $Q^2$ wave-functions of pions and kaons as an expansion in terms of hadron-like Fock state fluctuations. In this formalism, pion and kaon wave-functions are related one another. Consequently, the knowledge of the pion structure allows the determination of parton distributions in kaons. In addition, we show that the intrinsic (low $Q^2$) sea of pions and kaons are different due to their different valence quark structure. Finally, we analize the feasibility of a method to extract kaon's parton distribution functions within this approach and compare with available experimental da…
We determine the leading Fock state light front wave functions (LFWFs) of the pion and kaon via light front projections of the covariant Bethe-Salpeter wave function. Using these LFWFs we study the multi-dimensional images of the valence quarks in the pion and kaon that are provided by their generalized parton distribution functions (GPDs) and transverse momentum dependent parton distribution functions (TMDs). Moments of the GPDs are taken to obtain the electromagnetic and gravitational form factors of the pion and kaon, and comparisons to available experimental and lattice data are made. High…
The Lattice results for the pion and kaon decay constants are analysed within the Resonance Chiral Theory framework in the large NC limit. The approximately linear behaviour of the observable at large light-quark mass is explained through the interaction with the lightest multiplet of scalar resonances. The analysis of the Lattice results allows to obtain the resonance mass MS=1049 +- 25 MeV and the Chiral Perturbation Theory parameters at leading order in 1/NC.
In the present work the Mott effect for pions and kaons is described within a Beth- Uhlenbeck approach on the basis of the PNJL model. The contribution of these degrees of freedom to the thermodynamics is encoded in the temperature dependence of their phase shifts. A comparison with results from $N_f = 2 + 1$ lattice QCD thermodynamics is performed.
The pion-nucleon and kaon-nucleon scattering lengths are studied in the QCD sum rule. We show that the leading and next-to-leading order terms of the OPE give rise to the Tomozawa-Weinberg and sigma terms, respectively. We also show that in the kaon-nucleon system the $Λ(1405)$ contribution has to be subtracted from the OPE side in order to obtain the scattering length. The odd components of the $T$-matrices are in agreement with the experimental values not only in the pion-nucleon channel but also in the kaon-nucleon channel after the $Λ(1405)$ contribution subtracted. The even components dis…
We investigate the parton-distribution functions (PDFs) for the positively charged pion and kaon at a low-renormalization scale ~ 1 GeV. To this end, we employ the gauge-invariant effective chiral action from the nonlocal chiral-quark model, resulting in that the vector currents are conserved. All the model parameters are determined phenomenologically with the normalization condition for PDF and the empirical values for the pseudoscalar-meson weak-decay constants. We consider the momentum dependence of the effective quark mass properly within the model calculations. It turns out that the leadi…
In this presentation, we briefly review recent investigations on pion and kaon structures from the instanton vacuum. Starting from the low-energy QCD partition function, we have computed the Gasser-Leutwyler low-energy constants, electromagnetic form factors of the pion and kaon, semileptonic decay form factors of the kaon, and light-cone distribution amplitues of the pion and kaon. The results are in good agreement with the experimental and empirical data.
In this work, we investigate the gluon distribution functions for the pion and kaon, in addition to the improved result of the valence-quark ones, in the gauge-invariant nonlocal chiral-quark model (NL$χ$QM), in which the momentum dependence of the quark interactions is properly taken into account. We then analyze the gluon distribution functions, generated dynamically through the splitting functions in the DGLAP QCD evolution. By comparing with the recent lattice QCD results and JAM global analyses, it is found that the present numerical results for the gluon parton distribution functions for…
We develop a unitarity approach to consider the final state interaction corrections to the tree level graphs calculated from Chiral Perturbation Theory ($χPT$) allowing the inclusion of explicit resonance fields. The method is discussed considering the coupled channel pion and kaon vector form factors. These form factors are then matched with the one loop $χPT$ results. A very good description of experimental data is accomplished for the vector form factors and for the $ππ$ P-wave phase shifts up to $\sqrt{s}\lesssim 1.2$ GeV, beyond which multiparticle states play a non negligible role. In pa…
We present a calculation of the scalar, vector and tensor pion and kaon form factors using one ensemble of two degenerate light, a strange and a charm quark ($N_f=2+1+1$) of maximally twisted mass fermions with clover improvement. The quark masses are chosen so that they produce a pion mass of about 265 MeV, and a kaon mass of 530 MeV. The lattice spacing of the ensemble is 0.093 fm and the lattice has a spatial extent of 3 fm. We use a rest frame, as well as a boosted frame to obtain the form factors for a wider and denser set of four-vector momentum transfer squared, $Q^2$. To assess and eli…
The different ground state energies of N-pion and M-kaon systems for N+M <=12 are studied in lattice QCD. These energies are then used to extract the various two- and three- body interactions that occur in these systems. Particular attention is paid to additional thermal states present in the spectrum because of the finite temporal extent. These calculations are performed using one ensemble of 2+1 flavor anisotropic lattices with a spatial lattice spacing a_s ~ 0.125 fm, an anisotropy factor ξ=a_s/a_t=3.5, and a spatial volume L^3 ~ (2.5 fm)^3. The quark masses used correspond to pion and kaon…
We present new sets of pion, kaon, proton and inclusive charged hadron fragmentation functions obtained in NLO combined analyses of single-inclusive hadron production in electron-positron annihilation, proton-proton collisions, and deep-inelastic lepton-proton scattering with either particles identified in the final state. At variance with all previous fits, the present analyses take into account data where hadrons of different electrical charge are identified, which allow to discriminate quark from anti-quark fragmentation functions.
In this talk we present our recent studies for the unpolarized fragmentation functions for the pion and kaon, employing the nonlocal chiral quark model, which manifests the nonlocal interaction between the quarks and pseudoscalar mesons, in the light-cone frame. It turns out that the nonlocal interaction produces considerable differences in comparison to typical local-interaction models.
We investigate the parton distribution functions (PDFs) of the pion and the kaon by combining quantum chromodynamics (QCD) evolution with the basis light front quantization. The initial PDFs result from the light front wave functions obtained by diagonalizing the effective Hamiltonian consisting of the holographic QCD confinement potential, a complementary longitudinal confinement potential, and the color-singlet Nambu--Jona-Lasinio interactions. The valence-quark PDF of the pion, after QCD evolution, is consistent with the result from the E-0615 experiment at Fermilab. Meanwhile, the pion str…
One of the biggest challenges in contemporary physics is understanding the origin and dynamics of the internal structure of hadrons which, at a fundamental level, is described by quantum chromodynamics (QCD). Taking great prominence amongst hadrons are pions and kaons which, despite being the lightest hadrons in nature, their very existence is intimately connected to those mechanisms responsible for almost all of the mass of the visible matter. In this manuscript we discuss many aspects of the pion and kaon structure via light front wave functions and generalized parton distributions, and a co…
We discuss a general diagrammatic description of n-point functions in the QCD instanton vacuum that resums planar diagrams, enforces gauge invariance and spontaneously broken chiral symmetry. We use these diagrammatic rules to derive the pion and kaon quasi-parton amplitude and distribution functions at leading order in the instanton packing fraction for large but finite momentum. The instanton and anti-instanton zero modes and non-zero modes are found to contribute to the quasi-distributions, but the latter are shown to drop out in the large momentum limit. The pertinent pion and kaon parton …
We present a new determination of unpolarised charged pion and kaon fragmentation functions from a set of single-inclusive electron-positron annihilation and lepton-nucleon semi-inclusive deep-inelastic scattering data. The determination includes next-to-next-to-leading order QCD corrections to both processes, and is carried out in a framework that combines a neural-network parametrisation of fragmentation functions with a Monte Carlo representation of their uncertainties. We discuss the quality of the determination, in particular its dependence on higher order corrections.
We present a detailed assessment of uncertainties in parton-to-pion and parton-to-kaon fragmentation functions obtained in recent global QCD analyses of single-inclusive hadron production data at next-to-leading order accuracy. We use the robust Lagrange multiplier approach for determining uncertainties to validate the applicability of the simpler but approximate Hessian method. Extensive comparisons of the results obtained within both methods are presented for the individual parton-to-pion and kaon fragmentation functions. We provide Hessian eigenvector sets of pion and kaon fragmentation fun…
We discuss the rescattering effects in decays $B\to π+K$. The picture we take is very simple: first, B decay into $K*ρ$, then $K*$ and $ρ$ go to kaon and pion by exchanging a pion. We find a up to ten percent CP violation asymmetry rate. We also discuss its correction to the constraint of angle $γ$ proposed recently.
Algebraic Ansaetze for the Poincaré-covariant Bethe-Salpeter wave functions of the pion and kaon are used to calculate their light-front wave functions (LFWFs), parton distribution amplitudes (PDAs), quasi-PDAs (qPDAs), valence parton distribution functions (PDFs), and quasi-PDFs (qPDFs). The LFWFs are broad, concave functions; and the scale of flavour-symmetry violation in the kaon is roughly 15%, being set by the ratio of emergent masses in the $s$-and $u$-quark sectors. qPDAs computed with longitudinal momentum $P_z =1.75\,$GeV provide a semiquantitatively accurate representation of the obj…
We study S- and P-wave pion-pion and kaon-pion distributions for the three-body production processes $J/ψ\toωπ^{+}π^{-}$, $D^{+}\to K^{-}π^{+}π^{+}$, $D^{+}\to K^{+}π^{+}π^{-}$, and $D^{+}_{s}\to K^{+}π^{+}π^{-}$, by applying the Resonance-Spectrum Expansion to the two-body subamplitudes. The results are compared to the corresponding signals reported by the BES, E791 and FOCUS collaborations. We find that the studied S- and P-wave channels can largely account for the observed structures in the data up to 1 GeV.
Light constituent quark masses and the corresponding dynamical quark masses are determined by data, the quark-level linear sigma model, and infrared QCD. This allows to define effective nonstrange and strange current quark masses, which reproduce the experimental pion and kaon masses very accurately, by simple additivity. In contrast, the usual nonstrange and strange current quarks employed by the Particle Data Group and Chiral Perturbation Theory do not allow a straightforward quantitative explanation of the pion and kaon masses.
The CMS experiment at the Large Hadron Collider has recently presented data on the average transverse momentum of protons, kaons and pions as a function of particle multiplicity. We relate the particle multiplicity to saturation momentum using recently published computations of the interaction radius determined from the theory of the Color Glass Condensate. We show that the pp and the pA experimental data scale in terms of these saturation momenta. Computing transverse momentum spectra for identified particles using Boltzmann-type distributions and relating different associated multiplicities …
In order for a Sullivan-like process to provide reliable access to a meson target as $t$ becomes spacelike, the pole associated with that meson should remain the dominant feature of the quark-antiquark scattering matrix and the wave function describing the related correlation must evolve slowly and smoothly. Using continuum methods for the strong-interaction bound-state problem, we explore and delineate the circumstances under which these conditions are satisfied: for the pion, this requires $-t \lesssim 0.6\,$GeV$^2$, whereas $-t\lesssim 0.9\,$GeV$^2$ will suffice for the kaon. These results …
We calculate the Collins fragmentation function in the framework of a spectator model with pseudoscalar pion-quark coupling and a Gaussian form factor at the vertex. We determine the model parameters by fitting the unpolarized fragmentation function for pions and kaons. We show that the Collins function for the pions in this model is in reasonable agreement with recent parametrizations obtained by fits of the available data. In addition, we compute for the first time the Collins function for the kaons.
In this paper we study the next-to-leading order corrections to deeply virtual pion and kaon production in neutrino experiments. We estimate these corrections in the kinematics of the Minerva experiment at FERMILAB, and find that they are sizable and increase the leading order cross-section by up to a factor of two. We provide a code, which can be used for the evaluation of the cross-sections, taking into account these corrections and employing various GPD models.
Non-universality of color transparency onset in pion and kaon electroproduction
Center for Exotic Nuclear Studies, Institute for Basic Science（基础科学研究所奇异核研究中心）；Research Institute of Basic Science, Korea Aerospace University（韩国航空大学基础科学研究所）；Department of Physics and Engineering Physics, Yonsei University（延世大学物理与工程物理系）
We discuss the pion and kaon distribution amplitudes up to twist-3 in the context of the random instanton vacuum (RIV). We construct explicitly the pertinent quasi-pion and quasi-kaon distributions in the RIV, and analyze them in leading order in the diluteness factor, at a resolution fixed by the inverse instanton size. The distribution amplitudes (DA) follow from the large momentum limit. The results at higher resolution are discussed using QCD evolution, and compared to their asymptotic limits and some lattice results.
