CKM Parameters Research Articles

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This provides the requisite large value of the CKM CP-violating phase while the strong CP phase θ¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\bar{\ heta }}$$\\end{document} remains zero or is tiny. Within the considered framework we discuss phenomenologically viable quark mass matrices with three types of texture zeros, which are realized by assigning both the left-handed and right-handed quark fields to A4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_4$$\\end{document} singlets 1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{1}$$\\end{document}, 1′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbf{1'}}$$\\end{document} and 1′′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbf{1''}$$\\end{document} with appropriate weights. The VEV of τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document} is restricted to reproduce the observed CKM parameters. We discuss cases in which the modulus VEV is close to the fixed points i, ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} and i∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$i\\infty $$\\end{document}. In particular, we focus on the VEV of τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}, which gives the absolute minima of the supergravity-motivated modular- and CP-invariant potentials for the modulus τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}, so called, modulus stabilisation. We present a successful model, which is consistent with the modulus stabilisation close to τ=ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au =\\omega $$\\end{document}.

Revisiting Representations of Quark Mixing Matrix

Abstract Using unitarity, unlike the approaches available in the literature, we construct 9 independent representations of the Cabibbo–Kobayashi–Maskawa) CKM matrix starting with each of the 9 elements of the matrix. The relationship of these independently constructed representations with those already available in the literature is compared and discussed. Further, the implications of these representations are explored for some of the CKM parameters such as $\delta$, J and $\epsilon _k$. Interestingly, we find that the PDG representation which is equivalent to our first representation seems to be most appropriate to incorporate the hierarchy of the elements of the CKM matrix as well as to describe the related phenomenology.

Prospects for the Belle II Experiment to Further Elucidate the KM Mechanism and Beyond

Abstract We present prospects for the Belle II experiment to further elucidate the KM mechanism and beyond. Recent measurements of the CKM parameter, and the radiative and electroweak penguin decays are also reported.

Kaon physics without new physics in varepsilon _K

Despite the observation of significant suppressions of brightarrow smu ^+mu ^- branching ratios no clear sign of New Physics (NP) has been identified in Delta F=2 observables Delta M_{d,s}, varepsilon _K and the mixing induced CP asymmetries S_{psi K_S} and S_{psi phi }. 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}| and |V_{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 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, significant NP contributions to K^+rightarrow pi ^+nu {bar{nu }}, K_{L}rightarrow pi ^0nu {bar{nu }}, K_Srightarrow mu ^+mu ^-, K_Lrightarrow pi ^0ell ^+ell ^-, varepsilon '/varepsilon and 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 coupling g^L_{sd}. 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 couplings to q{bar{q}}, nu {bar{nu }}, and mu ^+mu ^-. A characteristic feature of this scenario is a strict correlation between K^+rightarrow pi ^+nu {bar{nu }} and K_{L}rightarrow pi ^0nu {bar{nu }} branching ratios on a branch parallel to the Grossman-Nir bound. Moreover, 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_Lrightarrow mu ^+mu ^- which otherwise would bound NP effects in K^+rightarrow pi ^+nu {bar{nu }}. Of particular interest are the correlations of K^+rightarrow pi ^+nu {bar{nu }} and K_{L}rightarrow pi ^0nu {bar{nu }} branching ratios and of Delta M_K with the ratio varepsilon '/varepsilon . We investigate the impact of renormalization group effects in the context of the SMEFT on this simple scenario.

New physics in B q 0– mixing: present challenges, prospects, and implications for

The phenomenon of B0 q – mixing (q = d, s) provides a sensitive probe for physics beyond the Standard Model (SM). We explore the corresponding space for New Physics left through the current data, having a careful look at analyses of the Unitarity Triangle that are needed for the SM predictions of the B q mixing parameters. In particular, we explore the impact of tensions between inclusive and exclusive determinations of the CKM matrix elements ∣V ub ∣ and ∣V cb ∣. Moreover, we focus on the angle γ of the Unitarity Triangle, comparing measurements from B → DK and B → π π, ρ π, ρ ρ decays, where the latter are typically interpreted in terms of the angle α. We discuss various scenarios and present the corresponding state-of-the-art constraints on the New Physics parameters of B q 0– mixing. We point out that these results have an interesting application in the analysis of rare decays, allowing us to minimise the impact of CKM parameters in the search for New Physics. In view of the high-precision era, we make future projections. Interestingly, we find that for the extraction of the New Physics parameters in the B d system the determination of the apex of the Unitarity Triangle results in a key limiting factor. By contrast, the corresponding impact is negligible for the B s system, making it a promising candidate to reveal sources of New Physics.

K → μ+μ− as a third kaon golden mode

Recent progress has demonstrated that the K → μ + μ − decay carries clean short-distance information, attainable from a measurement of time-dependence sensitive to KL − KS interference effects. We review the ingredients that go into this proposed extraction, and discuss the sensitivity to the CKM parameter as well as to various NP scenarios.

Standard Model predictions for rare K and B decays without new physics infection

The Standard Model (SM) does not contain by definition any new physics (NP) contributions to any observable but contains four CKM parameters which are not predicted by this model. We point out that if these four parameters are determined in a global fit which includes processes that are infected by NP and therefore by sources outside the SM, the resulting so-called SM contributions to rare decay branching ratios cannot be considered as genuine SM contributions to the latter. On the other hand genuine SM predictions, that are free from the CKM dependence, can be obtained for suitable ratios of the K and B rare decay branching ratios to Delta M_s, Delta M_d and |varepsilon _K|, all calculated within the SM. These three observables contain by now only small hadronic uncertainties and are already well measured so that rather precise SM predictions for the ratios in question can be obtained. In this context the rapid test of NP infection in the Delta F=2 sector is provided by a |V_{cb}|-gamma plot that involves Delta M_s, Delta M_d, |varepsilon _K|, and the mixing induced CP-asymmetry S_{psi K_S}. As with the present hadronic matrix elements this test turns out to be negative, assuming negligible NP infection in the Delta F=2 sector and setting the values of these four observables to the experimental ones, allows to obtain SM predictions for all K and B rare decay branching ratios that are most accurate to date and as a byproduct to obtain the full CKM matrix on the basis of Delta F=2 transitions alone. Using this strategy we obtain SM predictions for 26 branching ratios for rare semileptonic and leptonic K and B decays with the mu ^+mu ^- pair or the nu {bar{nu }} pair in the final state. Most interesting turn out to be the anomalies in the low q^2 bin in B^+rightarrow K^+mu ^+mu ^- (5.1sigma ) and B_srightarrow phi mu ^+mu ^- (4.8sigma ).

On the superiority of the |V_{cb}|-gamma plots over the unitarity triangle plots in the 2020s

The unitarity triangle (UT) plots played already for three decades an important role in the tests of the Standard Model (SM) and the determination of the CKM parameters. As of 2022, among the four CKM parameters, |V_{us}| and beta are already measured with respectable precision, while this is not the case of |V_{cb}| and gamma . In the case of |V_{cb}| the main obstacle are the significant tensions between its inclusive and exclusive determinations from tree-level decays and it could still take some years before a unique value of this parameter will be known. The present uncertainty in gamma of 4^circ from tree-level decays will be reduced to 1^circ by the LHCb and Belle II collaborations in the coming years. Unfortunately in the common UT plots |V_{cb}| is not seen and the experimental improvements in the determination of gamma from tree-level decays at the level of a few degrees are difficult to appreciate. In view of these deficiencies of the UT plots with respect to |V_{cb}| and gamma and the central role these two CKM parameters will play in this decade, the recently proposed plots of |V_{cb}| versus gamma extracted from various processes appear to be superior to the UT plots in the flavour phenomenology of the 2020s. We illustrate this idea with Delta F=2 observables Delta M_s, Delta M_d, varepsilon _K and with rare decays B_srightarrow mu ^+mu ^-, B_drightarrow mu ^+mu ^-, K^+rightarrow pi ^+nu bar{nu } and K_Lrightarrow pi ^0nu bar{nu }. In particular the power of varepsilon _K, mathcal {B}(K^+rightarrow pi ^+nu bar{nu }) and mathcal {B}(K_Lrightarrow pi ^0nu bar{nu }) in the determination of |V_{cb}|, due to their strong dependence on |V_{cb}|, is transparently exhibited in this manner. Combined with future reduced errors on gamma and |V_{cb}| from tree-level decays such plots can better exhibit possible inconsistencies between various determinations of these two parameters, caused by new physics, than it is possible with the UT plots. This can already be illustrated on the example of the recently found 2.7sigma anomaly in B_srightarrow mu ^+mu ^-.

Searching for New Physics in Rare \(K\) and \(B\) Decays without \(|V_{cb}|\) and \(|V_{ub}|\) Uncertainties

We reemphasize the strong dependence of the branching ratios $B(K^+\to\pi^+\nu\bar\nu)$ and $B(K_L\to\pi^0\nu\bar\nu)$ on $|V_{cb}|$ that is stronger than in rare $B$ decays, in particular for $K_L\to\pi^0\nu\bar\nu$. 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 $K_S\to\mu^+\mu^-$, $B_{s,d}\to\mu^+\mu^-$ 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 $B(K^+\to\pi^+\nu\bar\nu)$ and $B(B_s\to\mu^+\mu^-)$ 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 $B(B^+\to K^+\nu\bar\nu)$ and $B(B_s\to\mu^+\mu^-)$ 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 $\epsilon_K$, $\Delta M_d$ and $\Delta M_s$ finding tensions between these three determinations of $|V_{cb}|$ within the SM. From $\Delta M_s$ and $S_{\psi K_S}$ alone we find $|V_{cb}|=41.7(6)\times 10^{-3}$ and $|V_{ub}|=3.64(12)\times 10^{-3}$. We stress the importance of a precise measurement of $\gamma$.

Padé approximants to B→πℓνℓ and Bs→Kℓνℓ and determination of |Vub|

In light of the first observation of the semileptonic decay $B^{0}_{s}\to K^{-}\mu^{+}\nu_{\mu}$ by the LHCb collaboration, we revisit the determination of the CKM parameter $|V_{ub}|$ from exclusive semileptonic $B$-meson decays. A controlled theoretical input on the Standard Model $B\to\pi$ and $B_{s}\to K$ vector and scalar form factors from Lattice QCD in the large $q^{2}$ region, in combination with experimental measurements of the differential $B\to\pi\ell\nu_{\ell}$ and $B^{0}_{s}\to K^{-}\mu^{+}\nu_{\mu}$ branching ratio distributions, has allowed us determine $|V_{ub}|=3.86(11)\times10^{-3}$ and $|V_{ub}|=3.58(9)\times10^{-3}$ from the analyses of the individual decay channels, respectively, and $|V_{ub}|=3.68(5)\times10^{-3}$ from a simultaneous analysis of both decays, which is only a $1.4\%$ error and differs by $1.8\sigma$ with respect to the value from inclusive determinations $|V_{ub}|=4.25(12)^{+15}_{-14}(23)\times10^{-3}$. Our results are based on the use of Pad\'{e} approximants to the participating form factors, highlight the importance of the decay $B_{s}\to K\mu\nu_{\mu}$ in complementing the traditional $B\to\pi\ell\nu_{\ell}$ one in the exclusive determination of $|V_{ub}|$, and allow obtain, to the best of our knowledge, the first correlated results for the $B\to\pi$ and $B_{s}\to K$ vector and scalar form factors. We hope that our study strengthens the case for precise measurements of the differential $B_{s}\to K\ell\nu_{\ell}$ decay rate with a finer resolution of the $q^{2}$ bins, as it would definitely allow achieving more conclusive results for $|V_{ub}|$.

Implications of precision measurements and unitarity on CKM paradigm

In the context of precise measurements in some of the CKM elements and angles of the unitarity triangle coupled with latest theoretical improvements of lattice QCD calculations of “hadronic factors,” we have explored the implication of these and unitarity on some of the CKM parameters which are ridden with ambiguities. In particular, we have undertaken a thorough scrutiny of the implications of the precise measurements of the angles of the unitarity triangle along with the element [Formula: see text], on the [Formula: see text], [Formula: see text], [Formula: see text], as well as the CP violating parameter [Formula: see text]. Interestingly, we find a unitarity-based analysis while unambiguously favoring [Formula: see text] exclusive-over-inclusive value which is also in agreement with the present data pertaining to [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], etc. The present analysis also shows that the presence of NP in [Formula: see text] and [Formula: see text] systems, if any, would be less than a few percent only.

Fermion masses and mixings and some phenomenological aspects of a 3-3-1 model with linear seesaw mechanism

We propose a viable theory based on the $SU(3)_C\times SU(3)_L\times U(1)_X$ gauge group supplemented by the $S_4$ discrete group together with other various symmetries, whose spontaneous breaking gives rise to the current SM fermion mass and mixing hierarchy. In the proposed theory the small light active neutrino masses are generated from a linear seesaw mechanism mediated by three Majorana neutrinos. The model is capable of reproducing the experimental values of the physical observables of both quark and lepton sectors. Our model is predictive in the quark sector having 9 effective parameters that allow to successfully reproduce the four CKM parameters and the six Standard Model (SM) quark masses. In the SM quark sector, there is particular scenario, motivated by naturalness arguments, which allows a good fit for its ten observables, with only six effective parameters. We also study the single heavy scalar production via gluon fusion mechanism at proton-proton collider. Our model is also consistent with the experimental constraints arising from the Higgs diphoton decay rate.

Disentangling weak and strong interactions in Brightarrow K^{*}(rightarrow Kpi )pi Dalitz-plot analyses

Dalitz-plot analyses of Brightarrow Kpi pi decays provide direct access to decay amplitudes, and thereby weak and strong phases can be disentangled by resolving the interference patterns in phase space between intermediate resonant states. A phenomenological isospin analysis of Brightarrow K^*(rightarrow Kpi )pi decay amplitudes is presented exploiting available amplitude analyses performed at the BaBar, Belle and LHCb experiments. A first application consists in constraining the CKM parameters thanks to an external hadronic input. A method, proposed some time ago by two different groups and relying on a bound on the electroweak penguin contribution, is shown to lack the desired robustness and accuracy, and we propose a more alluring alternative using a bound on the annihilation contribution. A second application consists in extracting information on hadronic amplitudes assuming the values of the CKM parameters from a global fit to quark flavour data. The current data yields several solutions, which do not fully support the hierarchy of hadronic amplitudes usually expected from theoretical arguments (colour suppression, suppression of electroweak penguins), as illustrated from computations within QCD factorisation. Some prospects concerning the impact of future measurements at LHCb and Belle II are also presented. Results are obtained with the CKMfitter analysis package, featuring the frequentist statistical approach and using the Rfit scheme to handle theoretical uncertainties.

Flavor physics scenario for the 750 GeV diphoton anomaly

A simple variant of a realistic flavour symmetry scheme for fermion masses and mixings provides a possible interpretation of the diphoton anomaly as an electroweak singlet "flavon". The existence of TeV scale vector-like T-quarks required to provide adequate values for CKM parameters can also naturally account for the diphoton anomaly. Correlations between $V_{ub}$ and $V_{cb}$ with the vector-like T-quark mass can be predicted. Should the diphoton anomaly survive in a future Run, our proposed interpretation can also be tested in upcoming B and LHC studies.

Quark flavour observables in the Littlest Higgs model with T-parity after LHC Run 1.

The Littlest Higgs model with T-parity (LHT) belongs to the simplest new physics scenarios with new sources of flavour and CP violation. The latter originate in the interactions of ordinary quarks and leptons with heavy mirror quarks and leptons that are mediated by new heavy gauge bosons. Also a heavy fermionic top partner is present in this model which communicates with the SM fermions by means of standard W^pm and Z^0 gauge bosons. We present a new analysis of quark flavour observables in the LHT model in view of the oncoming flavour precision era. We use all available information on the CKM parameters, lattice QCD input and experimental data on quark flavour observables and corresponding theoretical calculations, taking into account new lower bounds on the symmetry breaking scale and the mirror quark masses from the LHC. We investigate by how much the branching ratios for a number of rare K and B decays are still allowed to depart from their SM values. This includes K^+rightarrow pi ^+nu bar{nu }, K_{L}rightarrow pi ^0nu bar{nu }, K_Lrightarrow mu ^+mu ^-, Brightarrow X_sgamma , B_{s,d}rightarrow mu ^+mu ^-, Brightarrow K^{(*)}ell ^+ell ^-, Brightarrow K^{(*)}nu bar{nu }, and varepsilon '/varepsilon . Taking into account the constraints from Delta F=2 processes, significant departures from the SM predictions for K^+rightarrow pi ^+nu bar{nu } and K_{L}rightarrow pi ^0nu bar{nu } are possible, while the effects in B decays are much smaller. In particular, the LHT model favours mathcal {B}(B_{s}rightarrow mu ^+mu ^-) ge mathcal {B}(B_{s}rightarrow mu ^+mu ^-)_mathrm{SM}, which is not supported by the data, and the present anomalies in Brightarrow K^{(*)}ell ^+ell ^- decays cannot be explained in this model. With the recent lattice and large N input the imposition of the varepsilon '/varepsilon constraint implies a significant suppression of the branching ratio for K_{L}rightarrow pi ^0nu bar{nu } with respect to its SM value while allowing only for small modifications of K^+rightarrow pi ^+nu bar{nu }. Finally, we investigate how the LHT physics could be distinguished from other models by means of indirect measurements and discuss the consequences for quark flavour observables of not finding any LHT state in the coming years.

Occam's razor in quark mass matrices

On the standpoint of Occam's Razor approach, we consider the minimum number of parameters in the quark mass matrices needed for the successful CKM mixing and CP violation. We impose three zeros in the down-quark mass matrix with taking the diagonal up-quark mass matrix to reduce the number of free parameters. The three zeros are maximal zeros in order to have a CP violating phase in the quark mass matrix. Then, there remain six real parameters and one CP violating phase, which is the minimal number to reproduce the observed data of the down-quark masses and the CKM parameters. The twenty textures with three zeros are examined. Among them, the thirteen textures are viable for the down-quark mass matrix. As a representative of these textures, we discuss a texture $M_d^{(1)}$ in details. By using the experimental data on $\sin 2\beta$, $\theta_{13}$ and $\theta_{23}$ together with the observed quark masses, the Cabibbo angle is predicted to be close to the experimental data. It is found that this surprising result remains unchanged in all other viable textures. We also investigate the correlations among $|V_{ub}/V_{cb}|$, $\sin 2\beta$ and $J_{CP}$. For all textures, the maximal value of the ratio $|V_{ub}/V_{cb}|$ is $0.09$, which is smaller than the upper-bound of the experimental data, $0.094$. We hope that this prediction will be tested in future experiments.

K + → π + ν ν ¯ $$ {K}^{+}\to {\pi}^{+}\nu \overline{\nu} $$ and K L → π 0 ν ν ¯ $$ {K}_L\to {\pi}^0\nu \overline{\nu} $$ in the Standard Model: status and perspectives

In view of the recent start of the NA62 experiment at CERN that is expected to measure the $K^+\to\pi^+\nu\bar\nu$ branching ratio with a precision of 10%, we summarise the present status of this promising decay within the Standard Model (SM). We do likewise for the closely related $K_L\to\pi^0\nu\bar\nu$, which will be measured by the KOTO experiment around 2020. As the perturbative QCD and electroweak corrections in both decays are under full control, the dominant uncertainties within the SM presently originate from the CKM parameters $V_{cb}$, $V_{ub}$ and $\gamma$. We show this dependence with the help of analytic expressions as well as accurate interpolating formulae. Unfortunately a clarification of the discrepancies between inclusive and exclusive determinations of $V_{cb}$ and $V_{ub}$ from tree-level decays will likely require results from the Belle II experiment available at the end of this decade. Thus we investigate whether higher precision on both branching ratios is achievable by determining $V_{cb}$, $V_{ub}$ and $\gamma$ by means of other observables that are already precisely measured. In this context $\varepsilon_K$ and $\Delta M_{s,d}$, together with the expected progress in QCD lattice calculations will play a prominent role. We find $\mathcal{B}(K^+\to\pi^+\nu\bar\nu) = (9.11\pm 0.72) \times 10^{-11}$ and $\mathcal{B}(K_L\to\pi^0\nu\bar\nu) = (3.00\pm 0.30) \times 10^{-11}$, which is more precise than using averages of the present tree-level values of $V_{cb}$, $V_{ub}$ and $\gamma$. Furthermore, we point out the correlation between $\mathcal{B}(K^+\to\pi^+\nu\bar\nu)$, $\overline{\mathcal{B}}(B_s\to\mu^+\mu^-)$ and $\gamma$ within the SM, that is only very weakly dependent on other CKM parameters. Finally, we also update the ratio $\varepsilon'/\varepsilon$ in the SM and present its correlation with $K_L\to\pi^0\nu\bar\nu$.

Improved anatomy of ε′/ε in the Standard Model

We present a new analysis of the ratio epsilon'/epsilon within the Standard Model (SM) using a formalism that is manifestly independent of the values of leading (V-A)x(V-A) QCD penguin, and EW penguin hadronic matrix elements of the operators Q_4, Q_9, and Q_10, and applies to the SM as well as extensions with the same operator structure. It is valid under the assumption that the SM exactly describes the data on CP-conserving K -> pi pi amplitudes. As a result of this and the high precision now available for CKM and quark mass parameters, to high accuracy epsilon'/epsilon depends only on two non-perturbative parameters, B_6^(1/2) and B_8^(3/2), and perturbatively calculable Wilson coefficients. Within the SM, we are separately able to determine the hadronic matrix element <Q_4>_0 from CP-conserving data, significantly more precisely than presently possible with lattice QCD. Employing B_6^(1/2) = 0.57+-0.19 and B_8^(3/2) = 0.76+-0.05, extracted from recent results by the RBC-UKQCD collaboration, we obtain epsilon'/epsilon = (1.9+-4.5) 10^-4, substantially more precise than the recent RBC-UKQCD prediction and 2.9 sigma below the experimental value (16.6+-2.3) 10^-4, with the error being fully dominated by that on B_6^(1/2). Even discarding lattice input completely, but employing the recently obtained bound B_6^(1/2) <= B_8^(3/2) <= 1 from the large-N approach, the SM value is found more than 2 sigma below the experimental value. At B_6^(1/2) = B_8^(3/2) = 1, varying all other parameters within one sigma, we find epsilon'/epsilon = (8.6+-3.2) 10^-4. We present a detailed anatomy of the various SM uncertainties, including all sub-leading hadronic matrix elements, briefly commenting on the possibility of underestimated SM contributions as well as on the impact of our results on new physics models.

Current status of the standard model CKM fit and constraints onΔF=2new physics

This letter summarises the status of the global fit of the CKM parameters within the Standard Model performed by the CKMfitter group. Special attention is paid to the inputs for the CKM angles $\alpha$ and $\gamma$ and the status of $B_s\to\mu\mu$ and $B_d\to \mu\mu$ decays. We illustrate the current situation for other unitarity triangles. We also discuss the constraints on generic $\Delta F=2$ New Physics. All results have been obtained with the CKMfitter analysis package, featuring the frequentist statistical approach and using Rfit to handle theoretical uncertainties.

Rates and asymmetries ofB→πℓ+ℓ−decays

The LHCb experiment observed ${B}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ decay with $1.0\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ data, which is the first measurement of a flavor changing neutral current $b\ensuremath{\rightarrow}d{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}$ decay ($\ensuremath{\ell}=e,\ensuremath{\mu}$). Based on QCD factorization, we give standard model predictions for the branching ratios, direct CP asymmetries, and isospin asymmetry for $B\ensuremath{\rightarrow}\ensuremath{\pi}{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}$ decays, in the kinematic region where the dilepton invariant mass is small. We find that the contribution from weak annihilation enhances the direct CP asymmetry for low ${\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}$ pair mass. Anticipating improved measurements, we assess the utility of ${B}^{+}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}$ observables, when combined with ${B}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{\ensuremath{-}}{\ensuremath{\ell}}^{+}\ensuremath{\nu}$ and ${B}^{+}\ensuremath{\rightarrow}{K}^{+}{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}$, for determining Cabibbo-Kobayashi-Maskawa matrix parameters in the future.