Oscillation of heavy sterile neutrino in decay of $B\to \mu e\pi$

In a scenario with two almost mass degenerate heavy sterile Majorana neutrinos with a mass $\sim 1\mathrm{GeV}$, we present the semileptonic decay widths of heavy charged pseudoscalars such as $B$ mesons, either lepton number violating ($B^{\pm }\to {\mu }^{\pm }{e}^{\pm }{\pi }^{\mp }$), or lepton number conserving ($B^{\pm }\to {\mu }^{\pm }{e}^{\mp }{\pi }^{\pm }$), mediated by such on-shell neutrinos. It had been explained in the literature that such decays may be eventually detected, and that they can present even detectable $CP$ violation effects. In this work we point out that, in addition, such decays may present detectable effects of heavy neutrino oscillation, allowing us to extract the oscillation length and thus the heavy neutrino mass difference $\mathrm{\Delta }{M}_N$, as well as a $CP$-violating Majorana phase.

Figure 1

The LNV decay $B^{+}\to {\mu }^{+}{e}^{+}{\pi }^{-}$ via exchange of an on-shell neutrino $N_j$ ($j=1,2$).

Figure 2

The LNC decay $B^{+}\to {\mu }^{+}{e}^{-}{\pi }^{+}$ via exchange of an on-shell neutrino $N_j$ ($j=1,2$).

Figure 3

The suppression factors $\eta (y)/y$ and $\delta (y)$ as a function of $y\equiv \mathrm{\Delta }{M}_N/{\mathrm{\Gamma }}_N$: (a) for $0<y<1$ (on the linear $y$ scale); (b) for $1<y<10$ (on the logarithmic $y$ scale). Figures were taken from Refs. [13, 15]. For $y<0$ we have $\delta (y)=\delta (-y)$ and $\eta (y)=\eta (-y)$.

Figure 4

The canonical probability ${\overline{P}}_N$, as defined in Eq. (23), as a function of the neutrino mass $M_N$, with the Lorentz lab time dilation factor chosen to be ${\gamma }_N[\equiv (1-{\beta }_N^2{)}^{-1/2}]=2$.

Figure 5

Graphical representation of the hierarchy equation (35) of the lengths $\mathrm{\Delta }{L}_{12}$, $\mathrm{\Delta }{L}_{\text{wp}}$ and the detector length $L$. The two interaction vertices (the production and the decay vertex of the neutrino $N$) are denoted as $v_1$ and $v_2$, respectively. Note that at the production vertex ($v_1$) the wave packets of $N_1$ and $N_2$ are not mutually displaced, unlike in the decay vertex ($v_2$).

Figure 6

Graphical schematical representation of the differential decay rate $d{\mathrm{\Gamma }}_{\text{eff}}^{(\text{osc})}(B\to \mu e\pi ;L)/dL$; cf. Eqs. (42) and (45).

Figure 7

The LNC decay $B^{+}\to {\mu }^{+}{e}^{-}{\pi }^{+}$: at the production vertex, the $|{\mathcal{N}}_1⟩$ state is produced; at the decay vertex, the $|{\mathcal{N}}_2⟩$ state is absorbed.