$\nu_e \bar{\nu_e} \rightarrow e^- e^+$ confusion

Question

Interacting part of the $SU(2)_L$describing the Higgs and fermionic sector with one family

$$\mathcal{L}= \bar{l}_Li\not Dl_L+\bar{e}_Ri\not\partial e_R+ \bar{\nu}_Ri\not\partial\nu_R - (y_{\nu}\bar{l}_L\tilde{\Phi}\nu_R - y_{e}\bar{l}_L{\Phi}e_R+h.c)$$

Where $l_L=(\nu_L, e_L)^T$ and $D_{\mu}=\partial_{\mu}+ig\hat{W_{\mu}}$ , $\tilde{\Phi}=i\sigma_{2}\Phi^{*}$ , $\Phi = \Big(\frac{v+h}{\sqrt{2}}\Big)e^{i\Pi/v}(0,1)^T$

From the interaction terms in the Lagrangian only 3-point interactions are:

$$\mathcal{L_{int}}= -\frac{g}{2}\gamma^{\mu}[\bar{\nu_L} W^{3}_{\mu} \nu_L + \sqrt{2}\bar{\nu_L}W^{+}_{\mu} e_L + \sqrt{2}\bar{e_L}W^{-}_{\mu} \nu_L -\bar{e_L} W^{3}_{\mu} e_L] -y_{\nu}\frac{1}{\sqrt{2}}\bar{\nu}_L h \nu_R + iy_{\nu}\frac{1}{2\sqrt{2}}\bar{\nu}_L \Pi^{3} \nu_R + iy_{\nu}\frac{1}{2}\bar{e}_L \Pi^{-} \nu_R + \frac{y_e}{\sqrt{2}}\bar{e_L}h e_R + \frac{iy_e}{2}\bar{\nu_L}\Pi^{+}e_R - \frac{iy_e}{2\sqrt{2}}\bar{e_L}\Pi^3 e_R$$

And from these, Feynman graphic rules are as follows:

Since I want to calculate the processes contributing $\nu_e \bar{\nu_e} \rightarrow e^- e^+$ . I would expect to get the following scattering diagrams:

The way I proceeded is:

I matched the diagrams with the same propagators (and with their h.c. in case of charged propagators). However I don't know how to interpret $\nu_R$ and $\bar{\nu}_L$ in vertex diagrams.

Aren't they right-handed neutrino and left-handed anti-neutrino? Why do they even appear?

Just by looking at the vertex with Z boson it seems like something is off since L.H. neutrino annihilates with L.H. anti-neutrino (yet L.H. neutrino should annihilate with R.H. anti-neutrino). Same mismatch appears in the vertex with Higgs however this time there is a R.H. neutrino and L.H. anti-neutrino which is even more confusing.

Edit

At this point, if anyone could show how to correctly glue the vertex diagrams to get neutrino pair annihilation to electron positron pair, I will accept the answer.

Answer 1

I'm really not sure what your trouble or question are. When in doubt, check your expressions versus (10.2) of the PDG review. In your expression, you need to intercalate the γ matrices in-between the two spinors.

For neutrinos, the neutral current is particularly (uniquely among fermions!) clean. Only left-chiral neutrinos and right-chiral antineutrinos couple to the Z bosons, since the neutrino fails to couple to the photon, and only couples to the Z. So right-chiral neutrinos and left-chiral antineutrinos do not couple to it: $$ {\sqrt{g^2+g'^2}\over 2} \overline {\nu_L}Z_\mu \gamma^\mu \nu_L. $$ This term destroys a Z and supplants it with a created pair of L neutrino and R antineutrino; or, vice versa, creates a Z out of a destroyed pair of L neutrino and R antineutrino. (It can also represent an L neutrino emitting a virtual Z in the t-channel and continuing on as an L neutrino.)

By contrast, the Higgs coupling, $\propto y_\nu h \bar\nu \nu= y_\nu h \bar\nu (L+R)\nu= y_\nu h (\overline{\nu_R} \nu_L+\overline{\nu_L} \nu_R )$, lacking a chirality flipping γ, creates an L neutrino and L antineutrino when a higgs is destroyed, and, in the same breath, with the same strength, a R neutrino and R antineutrino. But with a ridiculously small strength, as $$ y_\nu \sim m_v/v < 10^{-12}. $$ It of course also creates a higgs out of a destruction of a LL or RR neutrino-antineutrino pair. Recall $\overline {\nu_L}=\bar\nu R$.