A New Approach to a Global Fit of the CKM Matrix

TL;DR

This paper introduces the Rfit frequentist framework for a global CKM fit, emphasizing a clear separation between metrology and model testing and a cautious treatment of theoretical uncertainties. It develops a detailed likelihood-based approach combining experimental data and theoretical inputs, compares Rfit with Bayesian and 95% CL scan methods, and provides comprehensive CKM parameter constraints, UT analyses, and rare-decays predictions. The work demonstrates SM consistency within quantified confidence levels, explores a minimal SUSY extension, and discusses future experimental prospects to sharpen CKM tests. It also provides methodological insights and a public tool (CkmFitter) for reproducing and extending these global CKM analyses.

Abstract

We report on a global CKM matrix analysis taking into account most recent experimental and theoretical results. The statistical framework (Rfit) developed in this paper advocates formal frequentist statistics. Other approaches, such as Bayesian statistics or the 95% CL scan method are also discussed. We emphasize the distinction of a model testing and a model dependent, metrological phase in which the various parameters of the theory are determined. Measurements and theoretical parameters entering the global fit are thoroughly discussed, in particular with respect to their theoretical uncertainties. Graphical results for confidence levels are drawn in various one and two-dimensional parameter spaces. Numerical results are provided for all relevant CKM parameterizations, the CKM elements and theoretical input parameters. Predictions for branching ratios of rare K and B meson decays are obtained. A simple, predictive SUSY extension of the Standard Model is discussed.

Figures (18)

• Figure 1: The rescaled Unitarity Triangle in the Wolfenstein parameterization.
• Figure 2: Constraints in the $(\bar{\rho},\,\bar{\eta})$ plane for the most relevant observables. The theoretical parameters used correspond to some "standard" set chosen to reproduce compatibility. The dashed lines indicate the rectangle on which we concentrate in the following for the global fit.
• Figure 3: The $\kappa$ parameter as a function of $\zeta$ (see text).
• Figure 4: The left hand plot shows the Hat functions ($\bar{x_0}=0$ and $\sigma_o=1$) used for the Rfit scheme, the ERfit scheme and the Gaussian treatment. The right hand plot shows the combined likelihood $_{\rm exp}{\cal L}_{\rm syst}$ (with $\bar{x_0}=0$ and $\sigma_{{\rm exp}}=\sigma_o=1$) obtained from Eq. (\ref{['digest']}) for the Rfit scheme, the ERfit scheme, a convolution of a Gaussian with a uniform distribution (hence taken as a PDF, following the Bayesian approach) and a convolution of two Gaussians (see Appendix \ref{['TheBayesianMethodcaughtunder-conservative']}).
• Figure 7: Confidence levels in the $(\bar{\rho},\,\bar{\eta})$ plane for the individual constraints. The upper right hand plot shows in addition to $\Delta m_d$ the improved constraint from $\Delta m_s$ via $\xi$ on $\Delta m_d$. The shaded areas indicate the regions of $\ge90\%$, $\ge32\%$ and $\ge5\%$ CLs, respectively.