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There is Paschos-Wolfenstein relation: $$ \tag 1 \frac{\sigma^{NC}_{\nu_{\mu}} - \sigma^{NC}_{\bar{\nu}_{\mu}}}{\sigma^{CC}_{\nu_{\mu}} - \sigma^{CC}_{\bar{\nu}_{\mu}}} = \frac{1}{2} - \sin^{2}(\theta_{W}). $$ In the most simple case the neutral current (NC) processes from $(1)$ are caused by $\nu q \to \nu q$, $\bar{\nu}q \to \bar{\nu}q$ reactions, while the charged current (CC) processes are caused by $\nu_{\mu} X \to \mu^{-}Y$, $\bar{\nu}_{\mu} X \to \mu^{+}Y$. Here $q, X, Y$ refer to $u$, $d$ quarks. It is not hard to derive $(1)$. Some historical experiments (like NuTeV) have used this relation for determination of $\sin (\theta_{W})$ value.

Here is my question: I don't understand how to arrange the experiment which would give the data which correspond $(1)$ (and don't correspond usual $\frac{\sigma_{\nu}^{NC}}{\sigma_{\nu}^{CC}}$). Maybe, I would understand why in the denominator of $(1)$ there is a minus sign: neutrinos make possible $\nu d \to \mu^{-}u$ reactions, while antineutrinos make possible $\bar{\nu} u \to \mu^{+}d$ reactions. So we detect the number of neutrinos-caused events minus antineutrinos-caused events. But I don't understand why there is minus sign into the nominator of $(1)$ (from the viewpoint of the experimental data).

I would be grateful for your help.

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