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I was reading up on the history of $W/Z$ bosons today and I got a little puzzled. I always assumed that people measured $M_Z$ and $M_W$ and then derived the Weinberg angle. But it appears that they knew what the Weinberg angle was by the late 70's (but not with a lot of precision). How do you measure the Weinberg angle experimentally without using $W$ and $Z$ masses?

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The Weinberg angle was first estimated using the muon neutrino and anti-neutrino scattering data from the Gargamelle cloud chamber.

The estimation is based on the Paschos-Wolfenstein formula:

$\frac{\sigma^{\nu}_{NC} - \sigma^{\bar{\nu}}_{NC} }{\sigma^{\nu}_{CC} - \sigma^{\bar{\nu}}_{CC}} = \frac{1}{2}- sin^2\theta_W$

Where $\sigma_{NC}$ is the Neutral current processes (Mediated by the $Z$ boson) total cross section (for example $\nu_{\mu}+p \rightarrow \nu_{\mu}+p +\pi^+ + \pi^-$), and, $\sigma_{CC}$ is the charged current processes (Mediated by the $W$ bosons) total cross section (for example $\nu_{\mu}+n \rightarrow \mu^-+Hadrons$).

Please see the following article by Kretzer and Reno.

One of the earliest estimations was given by: C.H. Albright in 1974.

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Excuse me. Do you know why for using Paschos-Wolfenstein formula for experiment we need to subtract the contribution which is given by antineutrino? Without this substraction we will get something like $$ \frac{1}{2} - sin^{2}(\theta_{W}) + \frac{20}{27}sin^{4}(\theta_{W}), $$ but it doesn't satisfy the experiment data. –  Andrew McAddams Aug 6 at 1:06

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