$Z$-boson decays into Majorana or Dirac heavy neutrinos

We computed the kinematics of $Z$-boson decay into a heavy–light neutrino pair when the $Z$-boson is produced at rest in $e^{+}{e}^{-}$ collisions, including the subsequent decay of the heavy neutrino into a visible final state containing a charged-lepton. We concentrated on heavy-neutrino masses of order dozens of GeV and the issue of addressing the nature of the neutrinos—Dirac fermions or Majorana fermions. We find that while it is not possible to tell the nature of the heavy and light neutrinos on an event-by-event basis, the nature of the neutrinos can nonetheless be inferred given a large-enough sample of heavy–light neutrino pairs. We identify two observables sensitive to the nature of neutrinos. One is the forward-backward asymmetry of the daughter-charged-leptons. This asymmetry is exactly zero if the neutrinos are Majorana fermions and is nonzero (and opposite) for positively- and negatively-charged daughter-leptons if the neutrinos are Dirac fermions. The other observable is the polarization of the heavy neutrino, imprinted in the laboratory-frame energy distribution of the daughter-charged-leptons. Dirac neutrinos and antineutrinos produced in $e^{+}{e}^{-}$ collisions at the $Z$-pole are strongly polarized while Majorana neutrinos are at most as polarized as the $Z$-bosons.

Figure 1

Normalized differential cross-section for $e^{+}{e}^{-}\to Z\to {\nu }_4{\overline{\nu }}_{\text{light}}$ (left) and $e^{+}{e}^{-}\to Z\to {\overline{\nu }}_4{\nu }_{\text{light}}$ (right) as a function of the direction of the heavy (anti)neutrino $\cos \theta$, for different values of the heavy neutrino mass $m_4$. The neutrinos are assumed to be Dirac fermions.

Figure 2

The forward-backward asymmetry $A_{\mathrm{FB}}^D$ of heavy neutrino or antineutrino production in $e^{+}{e}^{-}\to Z\to {\nu }_4{\overline{\nu }}_{\text{light}}$ or $e^{+}{e}^{-}\to Z\to {\overline{\nu }}_4{\nu }_{\text{light}}$ as a function of the heavy neutrino mass. The neutrinos are assumed to be Dirac fermions.

Figure 3

The polarizarion $P_D$ of heavy neutrinos (dashed lines) or antineutrinos (solid lines) produced in $e^{+}{e}^{-}\to Z\to {\nu }_4{\overline{\nu }}_{\text{light}}$ or $e^{+}{e}^{-}\to Z\to {\overline{\nu }}_4{\nu }_{\text{light}}$ as a function of the direction of the heavy (anti)neutrino $\cos \theta$, for different values of the heavy neutrino mass $m_4$. The neutrinos are assumed to be Dirac fermions.

Figure 4

Normalized differential cross-section for $e^{+}{e}^{-}\to Z\to {\nu }_4{\nu }_{\text{light}}$ as a function of the direction of the heavy neutrino, $\cos \theta$, for different values of the heavy neutrino mass $m_4$. The neutrinos are assumed to be Majorana fermions.

Figure 5

The polarizarion $P_M$ of heavy neutrinos produced in $e^{+}{e}^{-}\to Z\to {\nu }_4{\nu }_{\text{light}}$ as a function of the direction of the heavy neutrino $\cos \theta$, for different values of the heavy neutrino mass $m_4$. The neutrinos are assumed to be Majorana fermions. The range of $P_M$ values here is much smaller than that of $P_D$ values in Fig. 3.

Figure 6

Normalized differential decay widths of ${\nu }_4\to {\ell }^{\pm }{\pi }^{\mp }$ as a function of the energy of the charged-lepton, for ${\nu }_4$ produced in $Z$-decay-at-rest. The different curves correspond to different values of ${\alpha }_{\pm }P\in [-1,1]$ and $m_4=30\mathrm{GeV}$. Se Eq. (4.5).

Figure 7

Averaged, normalized differential decay widths of ${\nu }_4\to {\ell }^{-}{\pi }^{+}$ as a function of the energy of the charged-lepton, averaged over the heavy-neutrino production angle, for ${\nu }_4$ produced in $Z$-decay-at-rest assuming the heavy neutrinos are Dirac (left) and Majorana (right) fermions. The different curves correspond to different values of $m_4$. The same curves apply, both in the left-hand and in the right-hand panels, to the ${\ell }^{+}{\pi }^{-}$ final-states.