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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

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

please ?

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  • $\begingroup$ 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. $\endgroup$ – dmckee Apr 25 '14 at 23:54
  • $\begingroup$ dmckee: "[...] differences are going to be very, very small." -- Thanks (interesting, perhaps not quite unexpected in itself). But please don't miss the conceptual purpose of my question. Analogy: If a car maker is known only to produce and sell cars painted red, green, or blue then what's the actual output (correspondingly) in terms of cars colored magenta, or cyan, or orange? (Are there any actually being made at all? If so, how many, in which proportions? ...) But also analogous: photon pairs from SPDC, and "what they do" ... $\endgroup$ – user12262 Apr 26 '14 at 6:59

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