Waiting for Precise Measurements of K^+->pi^+ nu nu and K_L->pi^0 nu nu

TL;DR

The paper analyzes the theoretically clean rare kaon decays $K^+\rightarrow\pi^+\nu\bar{\nu}$ and $K_{\rm L}\rightarrow\pi^0\nu\bar{\nu}$ as precise probes of the CKM sector and CP violation. It compiles and derives the SM formulae, emphasizes the NNLO charm contribution and long-distance control, and explores MFV and scenarios with new CP-violating phases, deriving new relations that connect kaon observables to the unitarity triangle. A detailed numerical analysis assesses how future measurements, with improved inputs for $|V_{cb}|$, the top mass, and charm parameters, could determine $\sin 2\beta$, $\gamma$, $|V_{cb}|$, and $|V_{td}|$ with high precision, testing the SM and MFV against a landscape of new physics models (LHT, $Z'$, MSSM, UED, etc.). The work highlights golden relations that enable cross-checks between kaon and B-physics observables and demonstrates that kaon decays can provide uniquely clean access to CP-violating phases and potential new physics effects, complementing the B-system program. Overall, precise future measurements of the kaon rates could constrain or reveal new short-distance dynamics and CP-violating phases, with significant implications for flavor theory and beyond-Standard-Model physics.

Abstract

In view of future plans for accurate measurements of the theoretically clean branching ratios Br(K+ -> pi+ nu nu) and Br(KL -> pi0 nu nu), that should take place in the next decade, we collect the relevant formulae for quantities of interest and analyze their theoretical and parametric uncertainties. We point out that in addition to the angle beta in the unitarity triangle (UT) also the angle gamma can in principle be determined from these decays with respectable precision and emphasize in this context the importance of the recent NNLO QCD calculation of the charm contribution to K+ -> pi+ nu nu and of the improved estimate of the long distance contribution by means of chiral perturbation theory. In addition to known expressions we present several new ones that should allow transparent tests of the Standard Model (SM) and of its extensions. While our presentation is centered around the SM, we also discuss models with minimal flavour violation and scenarios with new complex phases in decay amplitudes and meson mixing. We give a brief review of existing results within specific extensions of the SM, in particular the Littlest Higgs Model with T-parity, Z' models, the MSSM and a model with one universal extra dimension. We derive a new "golden" relation between B and K systems that involves (beta,gamma) and Br(KL -> pi0 nu nu) and investigate the virtues of (R_t,beta), (R_b,gamma), (beta,gamma) and (etabar,gamma) strategies for the UT in the context of K -> pi nu nu decays with the goal of testing the SM and its extensions.

Figures (18)

• Figure 1: The penguin and box diagrams contributing to $K^+\rightarrow\pi^+\nu\bar{\nu}$. For $K_{\rm L}\rightarrow\pi^0\nu\bar{\nu}$ only the spectator quark is changed from $u$ to $d$.
• Figure 2: Unitarity Triangle.
• Figure 3: Unitarity triangle from $K\to\pi\nu\bar{\nu}$.
• Figure 4: $Br(K^+\rightarrow\pi^+\nu\bar{\nu})$ as a function of $\gamma$ for different values of $\beta$ and $|V_{cb}|$.
• Figure 5: The UT from $K\to\pi\nu\bar{\nu}$ in Scenario I of table \ref{['tabin']}. Also lines corresponding to several values of $Br(K^+\rightarrow\pi^+\nu\bar{\nu})$ and $Br(K_{\rm L}\rightarrow\pi^0\nu\bar{\nu})$ (in units of $10^{-11}$) are shown.