Sterile neutrinos facing kaon physics experiments

We discuss weak kaon decays in a scenario in which the Standard Model is extended by massive sterile fermions. After revisiting the analytical expressions for leptonic and semileptonic decays we derive the expressions for decay rates with two neutrinos in the final state. By using a simple effective model with only one sterile neutrino, compatible with all current experimental bounds and general theoretical constraints, we conduct a thorough numerical analysis which reveals that the impact of the presence of massive sterile neutrinos on kaon weak decays is very small, less than 1% on decay rates. The only exception is $\mathcal{B}(K_L\to \nu \nu )$, which can go up to $\mathcal{O}({10}^{-10})$, thus possibly within the reach of the KOTO, NA62 and SHIP experiments. Plans have also been proposed to search for this decay at the NA64 experiment. In other words, if all the future measurements of weak kaon decays turn out to be compatible with the Standard Model predictions, this will not rule out the existence of massive light sterile neutrinos with non-negligible active-sterile mixing. Instead, for a sterile neutrino of mass below $m_K$, one might obtain a huge enhancement of $\mathcal{B}(K_L\to \nu \nu )$, otherwise negligibly small in the Standard Model.

Figure 1

Result of the scan in the scenario of three active and one (effective) sterile neutrino displayed in the plane $|U_{e4}{|}^2$ vs $m_4$. Perturbative unitarity cuts the parameter space for large $m_4$. The red points agree with all constraints while the blue ones are excluded by requiring the compatibility with experimental results for the leptonic $\tau$ decays.

Figure 2

As in Fig. 1 but for $m_4\le 1\mathrm{GeV}$, and for all $|U_{\ell 4}{|}^2$.

Figure 3

$|R_{Ke2}-1|$ as a function of the sterile neutrino mass, $m_4$ (left panel) and of $U_{e4}$ (right panel). The full ensemble of points (red and blue ones) correspond to $R_{Ke2}$ computed using Eq. (24) in the Standard Model and in our scenario, with an additional sterile neutrino, by using the parameters selected in the scan discussed in Sec. 3. Blue points are in conflict with $R_{Ke2}^{\exp }$ shown by the dashed line.

Figure 4

Predictions for $|R_{Ke3}-1|$ and $|R_{K\mu 3}-1|$ computed by using the parameters obtained in Sec. 3 are shown as functions of the mass of the sterile neutrino $m_4$. All the allowed points in red remain far below the experimental limit shown by the dashed line. Blue points correspond to those discarded by incompatibility with $R_{Ke2}^{\exp }$.

Figure 5

$|R_{K^{\pm }\to {\pi }^{\pm }\nu \nu }-1|$ and $|R_{K_L\to {\pi }^0\nu \nu }-1|$ remain within 1%, which means that $K\to \pi \nu \nu$ decays are not sensitive to the presence of an extra (massive) sterile neutrino once experimental constraints are applied.

Figure 6

$\mathcal{B}(K_L\to \nu \nu )$ as a function of mass of the sterile neutrino for $m_4\le m_{K^0}$. Notice that in the Standard Model this branching fraction is zero while the values close to the upper bound found here are possibly within the reach of the KOTO, NA62(-KLEVER) and SHIP experiments.