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The four-Fermi interaction in the form\begin{equation} \mathcal{L}_{int}=-\frac{G_F}{\sqrt{2}}[\overline{\psi}_{(e)}\gamma^\mu(c_V-c_A\gamma_5)\psi_{(e)})][\overline{\psi}_{(\nu)}\gamma_\mu(1-\gamma_5)\psi_{(\nu)})]. \end{equation} is used to calculate the elastic neutrino-electron scattering. The combination occurring in the neutrino bilinear involves left-chiral projections while the co-efficients in the electron bilinear involving vector and axial vector currents have coefficients $c_V$ and $c_A$ respectively.

Why are $c_V$ are $c_A$ left unspecified? Why does such an asymmetry exist between neutrinos and charged leptons?

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The four-Fermi interaction is an effective term for the exchange of a $Z$. Then a vertex $\nu\nu Z$ is proportional to $\gamma_\mu(1-\gamma_5)$ whereas a vertex $ee Z$ is proportional to $\gamma_\mu\big((-1+4\sin^2\theta_W)+\gamma_5\big)$.

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