Discovering sterile neutrinos lighter than $M_W$ at the LHC

We study the purely leptonic $W$ decays $W^{+}\to e^{+}{\mu }^{-}{e}^{+}{\nu }_e$ and $W^{+}\to e^{+}{e}^{+}{\mu }^{-}{\overline{\nu }}_{\mu }$ (or their charge conjugates) produced at the LHC, induced by sterile neutrinos with mass below $M_W$ in the intermediate state. While the first mode is induced by both Dirac or Majorana neutrinos, the second mode is induced only by Majorana neutrinos, as it violates lepton number. We find that, even when the final (anti-)neutrino goes undetected, one could distinguish between these two processes, thus distinguishing the Dirac or Majorana character of the sterile neutrinos, by studying the muon spectrum in the decays.

Figure 1

The diagram for the lepton number violating (LNV) decay $W^{+}\to e^{+}{e}^{+}{\mu }^{-}{\overline{\nu }}_{\mu }$, which is mediated by a Majorana neutrino.

Figure 2

The diagram for the lepton number conserving (LNC) decay $W^{+}\to e^{+}{\mu }^{-}{e}^{+}{\nu }_e$, which is mediated by either a Dirac or Majorana neutrino.

Figure 3

Normalized muon energy spectrum, $(1/\mathrm{\Gamma })d\mathrm{\Gamma }/d{E}_{\mu }$, given in the $N$ rest frame, for the LNV process $W^{+}\to e^{+}{e}^{+}{\mu }^{-}{\overline{\nu }}_{\mu }$ [cf. Eq. (2)] (solid line), and for the LNC process $W^{+}\to e^{+}{e}^{+}{\mu }^{-}{\nu }_e$ [cf. Eq. (5)] (dashed line). In each curve the normalizing factor $\mathrm{\Gamma }$ is the rate itself, so that the integral over either normalized spectrum is unity.

Figure 4

Normalized muon energy spectrum for the LNV + LNC decays $d\mathrm{\Gamma }/d{E}_{\mu }(W^{+}\to e^{+}{e}^{+}{\mu }^{-}{\overline{\nu }}_{\mu })+d\mathrm{\Gamma }/d{E}_{\mu }(W^{+}\to e^{+}{e}^{+}{\mu }^{-}{\nu }_e)$, normalized by the sum of the two rates. The rates are proportional to $|U_{Ne}{|}^2$ and $|U_{N\mu }{|}^2$, respectively. The curves correspond to $|U_{Ne}{|}^2/|U_{N\mu }{|}^2=10$, 1, and $1/10$ (solid, dashed and dotted lines, respectively). Similar shapes correspond to the spectrum of $d\mathrm{\Gamma }/d{E}_{\mu }(W^{+}\to {\mu }^{+}{\mu }^{+}{e}^{-}{\overline{\nu }}_e)+d\mathrm{\Gamma }/d{E}_{\mu }(W^{+}\to {\mu }^{+}{\mu }^{+}{e}^{-}{\nu }_{\mu })$ if one exchanges the roles $|U_{Ne}{|}^2\leftrightarrow |U_{N\mu }{|}^2$.

Figure 5

The reduced branching ratio $\overline{\mathrm{Br}}$ as a function of the intermediate neutrino mass $m_N$ [as in Eqs. (3) and (6)], where the lepton mixing factors have been removed. The actual branching ratios are obtained as $\mathrm{Br}=\overline{\mathrm{Br}}\times |U_{Ne}{|}^4/(\sum _{\ell }|U_{N\ell }{|}^2)$ and $\mathrm{Br}=\overline{\mathrm{Br}}\times |U_{Ne}{U}_{N\mu }{|}^2/(\sum _{\ell }|U_{N\ell }{|}^2)$ for the LNV and LNC decays, respectively.

Figure 6

Decay length of a heavy sterile neutrino vs its mass, $m_N$, for a Lorentz factor $\gamma =1$ and a mixing parameter $|U_{N\ell }{|}^2$ at its current upper bound of $10^{-7}$ (solid line), and at lower value $10^{-9}$ (dashed line). For other values of the parameters, in general ${\lambda }_N$ grows proportional to $\gamma /|U_{N\ell }{|}^2$. For the expressions see Eq. (7) and the Appendix.