Neutrino Electron Scattering In The Standard Model Approach

[Radwan Parvez]

Previously I posted a problem concerning Electron-neutrino scattering, but as I couldn't describe the problem clearly, so I am trying to post the problem using latex codes to present it in the correct way.

For a couple of months, I am trying to calculate the invariant amplitude of the Neutrino electron scattering in the standard model (SM) approach where I am not considering any kind of approximation (like $q^2 << {M_W}^2$ , $m_e^2 \approx 0$ etc ).

So my invariant amplitude is $$\mathcal{M} = \mathcal{M}_W + \mathcal{M}_Z$$ Hence,
\begin{align*}
|\mathcal{M}|^2 & = {\mathcal{M}}^*\mathcal{M} \\
& = (\mathcal{M}_W + \mathcal{M}_Z)^* (\mathcal{M}_W + \mathcal{M}_Z) \\
& = |\mathcal{M}_W|^2 + |\mathcal{M}_Z|^2 + 2{\mathcal{M}_W}^*\mathcal{M}_Z
\end{align*}

And
$$ \mathcal{M}_W = \frac{-g^2}{8} [\overline{\nu_e} (k^\prime) \gamma^\mu (1-\gamma^5) e(p)] \frac{g_{\mu\nu} - \frac{q_\mu q_\nu}{M_W^2}}{q^2-M_W^2} [\overline{e} (p^\prime) \gamma^\nu (1-\gamma^5) \nu_e(k)]$$
$$ \mathcal{M}_Z = \frac{-g^2}{8} [\overline{e} (p^\prime) \gamma^\mu (1-\gamma^5) e(p)] \frac{g_{\mu\nu} - \frac{q_\mu q_\nu}{M_Z^2}}{q^2-M_Z^2} [\overline{\nu_e} (k^\prime) \gamma^\nu (C_V- C_A\gamma^5) \nu_e(k)]$$

I asked my teacher and he said that I have to use Fierz Rearrangement (FR) and applying FR, I could make the $\mathcal{M}_W$ term much similar to the $\mathcal{M}_Z$ term by the expense of a minus sign.

But, Fierz Rearrangement is $$ \overline{\psi_1}\gamma^\mu (1-\gamma^5)\psi_2 \overline{\psi_3}\gamma_\mu (1-\gamma^5)\psi_4 = - \overline{\psi_3}\gamma^\mu (1-\gamma^5)\psi_2 \overline{\psi_1}\gamma_\mu (1-\gamma^5)\psi_4$$

Notice that in FR the same Lorentz index $(\gamma^\mu and \gamma_\mu)$ is in upper and lower position, but for my case, as I'm not omitting the term $\frac{q_\mu q_\nu}{M_W^2}$, I will find one part of $\mathcal{M}_W$ to be with different Lorentz index $(\gamma^\mu and \gamma^\nu)$ both in the upper position, i.e,
\begin{align*}
\mathcal{M}_W = & -\frac{g^2}{8} [\overline{\nu_e} (k^\prime) \gamma^\mu (1-\gamma^5) e(p)] \frac{1}{q^2-M_W^2} [\overline{e} (p^\prime) \gamma_\mu (1-\gamma^5) \nu_e(k)] \\
& + \frac{g^2}{8} [\overline{\nu_e} (k^\prime) \gamma^\mu (1-\gamma^5) e(p)] \frac{q_\mu q_\nu}{M_W^2(q^2-M_W^2)} [\overline{e} (p^\prime) \gamma^\nu (1-\gamma^5) \nu_e(k)]
\end{align*}

So, How can I resolve this problem? I mean, How can I apply Fierz Rearrangement in this term?

*Nevertheless, though I can calculate the $|\mathcal{M}_W|^2$ and $|\mathcal{M}_Z|^2$ separately, I can not calculate the interference term $2\mathcal{M}_W^*\mathcal{M}_Z$ due to the same complication.

*Is there any Book or Paper on this problem? I found none. The renowned book by Gunti, "Fundamentals of Neutrino Physics and Astrophysics" also used the approximated approach where they approximate that $q^2 << {M_W}^2,{M_Z}^2$, for which the second part doesn't appear in their calculation.I hope I described my problem clearly.

 

[Radwan Parvez]

The $C_V$ and $C_A$ in the $\mathcal{M}_Z$ should be in the electron-electron current, not in the neutrino-neutrino current. I did an unwanted mistake. But that's not what the problem is about.