abstract

We reemphasize the strong dependence of the branching ratios \(\mathcal {B}(K^+\to \pi ^+\nu \bar \nu )\) and \(\mathcal {B}(K_{\mathrm {L}}\to \pi ^0\nu \bar \nu )\) on \(|V_{cb}|\) that is stronger than in rare \(B\) decays, in particular for \(K_{\mathrm {L}}\to \pi ^0\nu \bar \nu \). Thereby the persistent tension between inclusive and exclusive determinations of \(|V_{cb}|\) weakens the power of these theoretically clean decays in the search for new physics (NP). We demonstrate how this uncertainty can be practically removed by considering within the SM suitable ratios of the two branching ratios between each other and with other observables like the branching ratios for \(K_{\mathrm {S}}\to \mu ^+\mu ^-\), \(B_{s,d}\to \mu ^+\mu ^-\) and \(B\to K(K^*)\nu \bar \nu \). We use as basic CKM parameters \(V_{us}\), \(|V_{cb}|\), and the angles \(\beta \) and \(\gamma \) in the unitarity triangle (UT) with the latter two determined through the measurements of tree-level \(B\) decays. This avoids the use of the problematic \(|V_{ub}|\). A ratio involving \(\mathcal {B}(K^+\to \pi ^+\nu \bar \nu )\) and \(\bar {\mathcal {B}}(B_{s}\to \mu ^+\mu ^-)\) while being \(|V_{cb}|\)-independent exhibits sizable dependence on the angle \(\gamma \). It should be of interest for several experimental groups in the coming years. We point out that the \(|V_{cb}|\)-independent ratio of \(\mathcal {B}(B^+\to K^+\nu \bar \nu )\) and \( \bar {\mathcal {B}}(B_{s}\to \mu ^+\mu ^-)\) from Belle II and LHCb signals a \(1.8\sigma \) tension with its SM value. As a complementary test of the Standard Model, we propose to extract \(|V_{cb}|\) from different observables as a function of \(\beta \) and \(\gamma \). We illustrate this with \(\varepsilon _K\), \(\Delta M_d\), and \(\Delta M_s\) finding tensions between these three determinations of \(|V_{cb}|\) within the SM. We point out that from \(\Delta M_s\) and \(S_{\psi K_{\mathrm {S}}}\) alone, one finds \(|V_{cb}|=41.8(6)\times 10^{-3}\) and \(|V_{ub}|=3.65(12)\times 10^{-3}\). We stress the importance of a precise measurement of \(\gamma \). Assuming no NP in \(|\varepsilon _K|\) and \(S_{\psi K_{\mathrm {S}}}\), we determine independently of \(|V_{cb}|\) and \(\gamma \): \(\mathcal {B}(K^+\rightarrow \pi ^+\nu \bar \nu )_{\mathrm {SM}}= (8.60\pm 0.42)\times 10^{-11}\) and \(\mathcal {B}(K_{\mathrm {L}}\rightarrow \pi ^0\nu \bar \nu )_{\mathrm {SM}}=(2.94\pm 0.15)\times 10^{-11}\) with only CKM uncertainty coming from \(\beta \), that is already precisely known. These are the most precise determinations to date. Assuming no NP in \(\Delta M_{s,d}\) allows to obtain analogous results for all \(B\) decay branching ratios considered in our paper without any CKM uncertainties.

abstract

The definitions of the complete modulus–modulus synchronization (CMMS), modulus–modulus combination synchronization (MMCS), and modulus–modulus combination–combination synchronization (MMCCS) for chaotic complex systems are introduced. These types of synchronization may be considered as a generalization of many other types of synchronization in the literature. Based on the active control method, three schemes are stated to achieve: CMMS, MMCS, and MMCCS. Three theorems are presented and proved to provide us with analytical formulas for the control functions. We present examples to test the validity of the control functions to achieve CMMS, MMCS, and MMCCS. Using the Runge–Kutta of the order of 4 method, we got the numerical solutions of our systems which agree well with the analytical results. Based on the CMMS of two chaotic complex systems, the processes of encryption and decryption of images are introduced. The experimental results of image encryption and decryption as well as the information entropy and histograms are calculated. Similar studies using MMCS and MMCCS are also investigated.

abstract

Based on the relations from the meson–meson mass mixing matrix, the mixing angles of the isoscalar state have been re-evaluated via mass relations and the latest experimental results. The results in the present work are compared with the values from different theoretical models and the quarkonia content of the isoscalar state is presented. In order to check the validity of the analysis, some predictions on the decays of the isoscalar state are presented. These predictions may be useful for the phenomenological analysis for meson nonet in future experiments.