Rare kaon decay to missing energy: Implications of the NA62 result for a $Z^{\prime }$ model

Meson decays offer a good opportunity to probe new physics. The rare kaon decay $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ is one of the cleanest of them and, for this reason, is rather sensitive to new physics, in particular, vector mediators. The NA62 Collaboration, running a fixed-target experiment at CERN, recently reported an unprecedented sensitivity to this decay, namely a branching fraction of $\mathrm{BR}(K^{+}\to {\pi }^{+}\nu \overline{\nu })=(11_{-3.5}^{+4.0})\times {10}^{-11}$ at 68% C.L. Vector mediators that couple to neutrinos may yield a sizeable contribution to this decay. Motivated by the new measurement, we interpret this result in the context of a concrete $Z^{\prime }$ model, and put our findings into perspective with the correlated $K_L\to {\pi }^0\nu \overline{\nu }$ decay measured by the KOTO Collaboration, current, and future colliders, namely the High-Luminosity and High-Energy Large Hadron Collider.

Figure 1

Feynman diagrams that illustrate how vector mediators can contribute to the $K^{+}\to {\pi }^{+}\nu \overline{\nu }$ decay. The first diagram requires $Z^{\prime }$ coupling to neutrinos, the second further requires couplings to the top quark, whereas in the third case the $Z^{\prime }$ boson mediates FCNC at tree level, which is the main focus of this paper.

Figure 2

$\mathrm{BR}(K_L\to {\pi }^0+\overline{\nu }\nu )$ versus $\mathrm{BR}(K^{+}\to {\pi }^{+}+\overline{\nu }\nu )$; the blue dots were generated varying the $Z^{\prime }$ mass, the module $|V_{L32}^{d*}{V}_{L31}^d|$, and the $\delta$ phase for the 3-3-1 r.h.n. model. We combined three different limits, the theoretical Grossman-Nir limit (green region), the NA62 bounds (blue region), and the exclusion region from KOTO (gray region).

Figure 3

Branching ratio versus $Z^{\prime }$ mass for kaon long (left panel), $\mathrm{BR}(K_L\to {\pi }^0+\overline{\nu }\nu )$, and for the $K^{+}$, $\mathrm{BR}(K^{+}\to {\pi }^{+}+\overline{\nu }\nu )$ (right panel). The green band provides the excluded region by the KOTO experiment, and the blue region provides that allowed by the NA62 experiment. The colored bar represents the variation of the $\delta$ phase; for details see the text.

Figure 4

Excluded parameter space regions in the 3-3-1 r.h.n model, in the plane $|V_{L32}^{d*}{V}_{L31}^d|$ versus $m_{Z^{\prime }}$, for $\delta =0$ (left) and $\delta =\pi /2$ (right). The colored regions represent the limits from NA62 (cyan), KOTO (blue) and LHC current (gray), and prospects (dashed lines).