Remarks on the lifetime of sterile neutrinos and the effect on detection of rare meson decays $M^{+}\to M^{\prime -}{\ell }^{+}{\ell }^{+}$

We review the detection of the $\mathrm{\Delta }L=2$ semileptonic meson decays of the form $M^{+}\to M^{\prime -}{\ell }^{+}{\ell }^{+}$ that we studied in a previous work, mediated by a Majorana neutrino with a mass in the range between the masses of the initial and final mesons. For such a range of masses, the Majorana neutrino will go on its mass shell, and the two charged leptons will appear at displaced vertices or, if the lifetime is long enough, most secondary vertices will fall outside the detector. We study the consequences of this effect on the experimental searches and limits that can be extracted.

Figure 1

The $t$-type and $s$-type weak amplitudes at the quark level that enter in the process $M^{+}\to {M^{\prime }}^{-}{\ell }_1^{+}{\ell }_2^{+}$ (plus the same diagrams with leptons exchanged if they are identical). $M^{+}$ and ${M^{\prime }}^{-}$ are pseudoscalar mesons, such as $B_c$, $B$, $D_s$, $K$, and $\pi$.

Figure 2

The normalized $N$ neutrino lifetime ${\tilde{\tau }}_N$ (using lepton mixing $|U_{\ell N}{|}^2=1$) vs its mass $m_N$. The actual lifetime is ${\tau }_N={\tilde{\tau }}_N/|U_{\ell N}{|}^2$. ${\tilde{\tau }}_N$ is estimated as inclusive decay into all allowed standard quarks and leptons and assumes equal $|U_{\ell N}{|}^2$ for all three flavors $\ell =e$, $\mu$, $\tau$. Current bounds are $|U_{\ell N}{|}^2<10^{-9}$ for $m_N\lesssim 500\mathrm{MeV}$ and $|U_{\ell N}{|}^2<10^{-7}$ for $m_N\gtrsim 1\mathrm{GeV}$ [8].