# How to calculate relative branching fractions of the $Z$ boson to specific pairs of “neutral lepton and anti-lepton”?

The PDG is listing values of "$Z$ couplings to neutral leptons" as

$$\begin{eqnarray} g^{\nu_{\ell}} & = & 0.5008 \, \pm \, 0.0008 \\ g^{\nu_{e}} & = & 0.53 \, \pm \, 0.09 \\ g^{\nu_{\mu}} & = & 0.502 \, \pm \, 0.017 \end{eqnarray}$$

where $\nu_e$ and $\nu_{\mu}$ denote neutrino species or neutrino fields "participating in the weak interaction", appearing (presumably) explicitly for instance in the decays

$$W^+ \rightarrow e^+ \, \nu_e$$ and $$W^+ \rightarrow \mu^+ \, \nu_{\mu},$$

and $\nu_{\ell}$ refers to an "average of neutrino species" which involves species $\nu_{\tau}$ as well.

Also, the value of the "invisible" relative branching fraction of $Z$ is given as

$$\Gamma( Z \rightarrow \text{invisible} ) / \Gamma^Z_{total} := \Gamma_6 / \Gamma^Z_{total} = (2.000 \, \pm \, 0.006) \times 10^{-1} .$$

Assuming that all "invisible" decays of the $Z$ boson proceed to the final state $\nu \, \overline{\nu}$ ("neutral lepton and anti-lepton") which all contribute to the "average of neutrino species, $\nu_{\ell}$", is it possible to calculate (or at least to express in terms of suitable parameters)

(1) the relative branching fraction of $Z$ e.g. to final state $\nu_e \, \overline{\nu_e}$,

(2) the relative branching fraction of $Z$ to any final state $\nu_{final} \, \overline{\nu_{final}}$ which is completely devoid of any $\nu_e$ or $\overline{\nu_e}$ contribution, i.e. such that

$$\langle \nu_{final} \, \overline{\nu_{final}} | \nu_e \, \overline{\nu_e} \rangle = 0,$$

(3) the relative branching fraction of $Z$ e.g. to final state $\nu_3 \, \overline{\nu_3}$, i.e. to a pair of "neutral lepton and anti-lepton" of one particular neutrino mass eigenstate, $\nu_3$ (and where the "suitable parameters" may involve coefficients of the PMNS matrix), and

(4) the relative branching fraction of $Z$ to any final state $\nu_{final} \, \overline{\nu_{final}}$ which is completely devoid of any $\nu_3$ or $\overline{\nu_3}$ contribution, i.e. such that

$$\langle \nu_{final} \, \overline{\nu_{final}} | \nu_3 \, \overline{\nu_3} \rangle = 0,$$

• Off the top of my head the only reason I see for a species-to-species difference would be phase-space differences due to the masses of the pairs. That being the case if the $Z$ is anywhere near the mass shell the differences are going to be very, very small. – dmckee Apr 25 '14 at 23:54