Discussion

Contributors: G. Isidori and G. D'Ambrosio.

Kaon physics is one of the richest sources of information about the standard model of electroweak interactions (SM). The properties of kaons and their decay patterns provide stringent constraints on any extension of the SM[1] (e.g. $\epsilon_K$, $\Delta M_K$, $\Gamma(K_L\to\mu^+\mu^-)$, etc...).

Discussion centered on decay widths of modes dominated by short distance physics. These observables are of great interest because, even in absence of direct new physics contributions, their measurement would provide valuable information about the flavor structure of the SM. The best examples of such observables are the widths of the neutrino modes $K_L\to\pi^0\bar{\nu}\nu$ and $K^+\to \pi^+\bar{\nu}\nu$ [2]. These occur at second order in electroweak interactions and are virtually free from long-distance contamination. Top-quark loops make the dominant contribution to these decays because of the GIM mechanism and so the decays are sensitive to mixing between the third generation and the light s and d quarks.

The ratio $\Gamma(K_L\to\pi^0\bar{\nu}\nu)/\vert{\rm Im}(V_{td}V_{ts}^*)\vert^2$can be calculated to better than 1$\%$(a precision currently unobtainable in calculations of B-meson decay modes), with uncertainty in the top mass increasing the uncertainty to 3$\%$[3]. Moreover, a combined measurement of $\Gamma(K_L\to\pi^0\bar{\nu}\nu)$ and $\Gamma(K^+\to \pi^+\bar{\nu}\nu)$could determine $\sin 2\beta$ complementarily and comparable to the cleanest CP asymmetry in B decays ( $B\to J/\Psi K_S$) [4]. Many new physics models predict different sensitivity to the value of $\sin 2\beta$ between rare kaon decays and B-meson decays [4,5]. The strong CKM suppression of $K\to\pi\bar{\nu}\nu$ amplitudes within the SM ( $V_{td}V_{ts}^* \sim \lambda^5$) implies that even small new physics effects could produce sizable modifications to these decays rates, while not affecting B-meson decay modes. Two popular examples where sizable effects can be found are provided by supersymmetric extensions of the SM [6] and models with a fourth generation of quarks [7], even if the masses of the new particles are $\sim~1~TeV$.

$\Gamma(K_L\to\pi^0 e^+ e^-)$ can be considered a short-distance observable if correlated with two other measurements: the rate of $K_S\to\pi^0 e^+ e^-$and a precise determination of the low diphoton invariant mass spectrum of $K_L\to\pi^0 \gamma \gamma$ [8]. Very interesting from a theoretical point of view but quite challenging is the measurement of the $K_{L,S}\to\pi^0 e^+ e^-$ interference term.

The high statistics required to observe $K_L\to\pi^0\bar{\nu}\nu$, $K^+\to \pi^+\bar{\nu}\nu$, and $K_{L}\to\pi^0 e^+ e^-$ decays will provide data in other channels as well. For example,

1.

Chiral tests in semileptonic kaon decays. The $\pi\pi$ phase shifts determined from Kl4would help constrain the size of the quark condensate in the chiral limit [9].

2.

$K_L\to\mu^+\mu^-$ can provide additional constraints on the CKM matrix and/or new physics. However, the long-distance contributions must be very well known. In principle, a better knowledge of these effects could be achieved by a precise measurement of the dilepton invariant mass spectrum in $K_L\to l^+ l^- \gamma$ and $K_L\to\mu ^+\mu ^- e^+ e^-$ [10].

3.

Other non-leptonic channels, like $K \to\pi \gamma \gamma$ and $K \to\pi \gamma ^*$, can establish the role of the vector and axial meson contributions in the weak sector; $K_S\to \gamma \gamma$ is a pure test of chiral loops; $K\to \pi\pi\gamma$, $K\to 3\pi\gamma$, and $K\to 3\pi$ decays could provide new CP-violating observables (even if not well established from a theoretical point of view) and several tests of CHPT (even in the $\Delta I=3/2$ sector) [11].

On the experimental front, an experiment on $K_L \rightarrow \pi^0 \nu\overline \nu$ has been approved at KEK [12] which should reach a sensitivity on the branching ratio of around 10-10. This will be very useful for excluding some new physics scenarios but is not enough to reach the SM value. Two proposals, one at FNAL[13] and one at BNL[14], are slated to reach sensitivities around 10-12. One $K^+ \rightarrow \pi^+ \nu \overline \nu$event has been observed by the Brookhaven experiment E787[15].

Next: Conclusion Up: Kaon Physics Previous: Speakers and Topics