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The kaon physics programme, long heralded as a cutting-edge frontier by the European Strategy for Particle Physics, continues to stand at the intersection of discovery and innovation in high-energy physics (HEP). With its unparalleled capacity to explore new physics at the multi-TeV scale, kaon research is poised to unveil phenomena that could reshape our understanding of the Universe. This document highlights the compelling physics case, with emphasis on exciting new opportunities for advancing kaon physics not only in Europe but also on a global stage. As an important player in the future of HEP, the kaon programme promises to drivemore » transformative breakthroughs, inviting exploration at the forefront of scientific discovery.« less

Abstract Despite the observation of significant suppressions of $$$$b\rightarrow s\mu ^+\mu ^-$$$$ $b\to s{\mu }^{+}{\mu }^{-}$ branching ratios no clear sign of New Physics (NP) has been identified in $$$$\Delta F=2$$$$ $\Delta F=2$ observables $$$$\Delta M_{d,s}$$$$ $\Delta M_{d,s}$ , $$$$\varepsilon _K$$$$ ${\epsilon }_K$ and the mixing induced CP asymmetries $$$$S_{\psi K_S}$$$$ $S_{\psi K_S}$ and $$$$S_{\psi \phi }$$$$ $more »$ S ψ ϕ . Assuming negligible NP contributions to these observables allows to determine CKM parameters without being involved in the tensions between inclusive and exclusive determinations of $$$$|V_{cb}|$$$$ $|V_{\mathrm{cb}}|$ and $$$$|V_{ub}|$$$$ $|V_{\mathrm{ub}}|$ . Furthermore this method avoids the impact of NP on the determination of these parameters present likely in global fits. Simultaneously it provides SM predictions for numerous rare K and B branching ratios that are most accurate to date. Analyzing this scenario within $$$$Z^\prime $$$$ $Z^{\prime }$ models we point out, following the 2009 observations of Monika Blanke and ours of 2020, that despite the absence of NP contributions to $$$$\varepsilon _K$$$$ ${\epsilon }_K$ , significant NP contributions to $$$$K^+\rightarrow \pi ^+\nu {\bar{\nu }}$$$$ $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ , $$$$K_{L}\rightarrow \pi ^0\nu {\bar{\nu }}$$$$ $K_L\to {\pi }^0\nu \overline{\nu }$ , $$$$K_S\rightarrow \mu ^+\mu ^-$$$$ $K_S\to {\mu }^{+}{\mu }^{-}$ , $$$$K_L\rightarrow \pi ^0\ell ^+\ell ^-$$$$ $K_L\to {\pi }^0{\ell }^{+}{\ell }^{-}$ , $$$$\varepsilon '/\varepsilon $$$$ ${\epsilon }^{\prime }/\epsilon$ and $$$$\Delta M_K$$$$ $\Delta M_K$ can be present. In the simplest scenario, this is guaranteed, as far as flavour changes are concerned, by a single non-vanishing imaginary left-handed $$$$Z^\prime $$$$ $Z^{\prime }$ coupling $$$$g^L_{sd}$$$$ $g_{\mathrm{sd}}^L$ . This scenario implies very stringent correlations between the Kaon observables considered by us. In particular, the identification of NP in any of these observables implies automatically NP contributions to the remaining ones under the assumption of non-vanishing flavour conserving $$$$Z^\prime $$$$ $Z^{\prime }$ couplings to $$$$q{\bar{q}}$$$$ $q\overline{q}$ , $$$$\nu {\bar{\nu }}$$$$ $\nu \overline{\nu }$ , and $$$$\mu ^+\mu ^-$$$$ ${\mu }^{+}{\mu }^{-}$ . A characteristic feature of this scenario is a strict correlation between $$$$K^+\rightarrow \pi ^+\nu {\bar{\nu }}$$$$ $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ and $$$$K_{L}\rightarrow \pi ^0\nu {\bar{\nu }}$$$$ $K_L\to {\pi }^0\nu \overline{\nu }$ branching ratios on a branch parallel to the Grossman-Nir bound. Moreover, $$$$\Delta M_K$$$$ $\Delta M_K$ is automatically suppressed as seems to be required by the results of the RBC-UKQCD lattice QCD collaboration. Furthermore, there is no NP contribution to $$$$K_L\rightarrow \mu ^+\mu ^-$$$$ $K_L\to {\mu }^{+}{\mu }^{-}$ which otherwise would bound NP effects in $$$$K^+\rightarrow \pi ^+\nu {\bar{\nu }}$$$$ $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ . Of particular interest are the correlations of $$$$K^+\rightarrow \pi ^+\nu {\bar{\nu }}$$$$ $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ and $$$$K_{L}\rightarrow \pi ^0\nu {\bar{\nu }}$$$$ $K_L\to {\pi }^0\nu \overline{\nu }$ branching ratios and of $$$$\Delta M_K$$$$ $\Delta M_K$ with the ratio $$$$\varepsilon '/\varepsilon $$$$ ${\epsilon }^{\prime }/\epsilon$ . We investigate the impact of renormalization group effects in the context of the SMEFT on this simple scenario.« less

Quantum Chromodynamics, the theory of quarks and gluons, whose interactions can be described by a local SU(3) gauge symmetry with charges called “color quantum numbers”, is reviewed; the goal of this review is to provide advanced Ph.D. students a comprehensive handbook, helpful for their research. When QCD was “discovered” 50 years ago, the idea that quarks could exist, but not be observed, left most physicists unconvinced. Then, with the discovery of charmonium in 1974 and the explanation of its excited states using the Cornell potential, consisting of the sum of a Coulomb-like attraction and a long range linear confining potential,more » the theory was suddenly widely accepted. This paradigm shift is now referred to as the November revolution. It had been anticipated by the observation of scaling in deep inelastic scattering, and was followed by the discovery of gluons in three-jet events.« less

We review and update our results for K → π π decays and K⁰- K¯⁰ mixing obtained by us in the 1980s within an approach based on the dual representation of QCD as a theory of weakly interacting mesons for large N colours. In our analytic approach the dynamics behind the enhancement of ReA₀ and suppression of ReA₂, the so-called ΔI = 1/2 rule for K → π π decays, has a simple structure: the usual octet enhancement through quark-gluon renormalization group evolution down to the scales O(1 GeV) is continued as a meson evolution down to zero momentum scalesmore » at which the factorization of hadronic matrix elements is at work. The inclusion of lowest-lying vector meson contributions in addition to the pseudoscalar ones and of Wilson coefficients in a momentum scheme improves significantly the matching between quark-gluon and meson evolutions. In particular, the anomalous dimension matrix governing the meson evolution exhibits the structure of the known anomalous dimension matrix in the quark-gluon evolution. The recent results on ReA₂ and ReA₀ from the RBC-UKQC collaboration give support for our approach. In particular, the signs of the two main contractions found numerically by these authors follow uniquely from our analytic approach. At NLO in 1/N we obtain R = ReA₀/ReA₂= 16.0±1.5 which amounts to an order of magnitude enhancement over the strict large N limit value √2. QCD penguins contribute at 15% level to this result. We also find B^_K = 0.73± 0.02, with the smallness of 1/N corrections to the large N value B^_K = 3/4 resulting within our approach from an approximate cancellation between pseudoscalar and vector meson one-loop contributions. We summarize the status of ΔM_K in this approach.« less