The Rare Decays $K\to\pi\nu\bar\nu$, $B\to X\nu\bar\nu$ and $B\to l^+l^-$ -- An Update

TL;DR

This paper updates Standard Model predictions for rare kaon and B-meson decays, providing new upper limits and discussing uncertainties, with implications for CKM matrix elements and future experimental measurements.

Contribution

It introduces a theoretically clean method to set upper limits on $B(K^+ o ext{ extpi}^+ uar u)$ using mixing ratios, avoiding large hadronic uncertainties, and reviews NLO QCD corrections with minor updates.

Findings

Upper limit on $B(K^+ o ext{ extpi}^+ uar u)$ is $1.67\times 10^{-10}$.

Method based on $\Delta M_d/\Delta M_s$ ratio reduces theoretical uncertainties.

NLO QCD corrections include a previously missed small term, with negligible phenomenological impact.

Abstract

We update the Standard Model predictions for the rare decays $K^+\to\pi^+\nu\bar\nu$ and $K_L\to\pi^0\nu\bar\nu$. In view of improved limits on $B_s$--$\bar B_s$ mixing we derive a stringent and theoretically clean Standard Model upper limit on $B(K^+\to\pi^+\nu\bar\nu)$, which is based on the ratio of $B_d$--$\bar B_d$ to $B_s$--$\bar B_s$ mixing, $\Delta M_d/\Delta M_s$, alone. This method avoids the large hadronic uncertainties present in the usual analysis of the CKM matrix. We find $B(K^+\to\pi^+\nu\bar\nu)< 1.67\cdot 10^{-10}$, which can be further improved in the future. In addition we consider the extraction of $|V_{td}|$ from a future measurement of $B(K^+\to\pi^+\nu\bar\nu)$, discussing the various sources of uncertainties involved. We also investigate theoretically clean constraints on $B(K_L\to\pi^0\nu\bar\nu)$. We take the opportunity to review the next-to-leading order (NLO) QCD corrections to $K\to\pi\nu\bar\nu$, $K_L\to\mu^+\mu^-$, $B\to X\nu\bar\nu$ and $B\to l^+l^-$, including a small additional term that had been missed in the original publications. The phenomenological impact of this change is negligible, the corresponding numerical shift being essentially within the small perturbative uncertainties at the NLO level.