EW theory vertices

I'm trying to undertand the following vertex:

Initial state of up and anti-down quarks with finalk state made of $$W^+$$ boson. Does it go with left or right projector? I think that from Lagrangian it should go with left projector but the vertex with $$e^+$$ and $$W^-$$ in initial state and $$\bar{\nu}_e$$ in the final one goes with right projector and this is not read from Lagrangian.

Accordingly to my professor, the case R-positron with $$W^-$$ to give R-antineutrino has a vertex that goes with $$\gamma^\mu P_R$$, so what I want to know is how extract this vertex if the Lagrangian does not contain that term, just $$P_R\gamma^\mu P_L = \gamma^\mu P_L$$

It would be helpful if you just wrote the Lagrangian terms you are looking at. Destroying a Left-handed u and R antidown to create a W+ corresponds to the vertices $$W_\mu^- \overline {d} P_R\gamma^\mu P_L u + W_\mu^+ \overline {u} P_R\gamma^\mu P_L d ,$$ while destroying a R-positron and a W- to yield a R-antineutrino to $$W_\mu^-\overline {e} P_R\gamma^\mu P_L \nu + W_\mu^+ \overline{\nu} P_R\gamma^\mu P_L e ~~~,$$ where you focus on the first term in each line.
• Destroying a positron requires of $\bar{e}$ while destroying $W^-$ needs of $W^-$ field. Plus creating $\bar{\nu}$ suppose the necessity of $\nu$ field, so the corresponding Lagrangian term that leads to this proicess is the one that goes as $\gamma^\mu P_L$, not $\gamma^\mu P_R$ as books say. What am I misunderstanding? – Vicky Mar 16 at 1:20
• Yes, but I don't see how the vertex in $\gamma^\mu P_R$ appears since Lagrangian has not right projector. I've always been told that to get vertices you need to rib out fields of interaction Lagrangian and the remaining is the vertex. Nevertheless, there is no $P_R$ in Lagrangian, but there it is in vertex. I'm stuck – Vicky Mar 16 at 2:14
• You are telling me this: $\bar{\psi_L}\gamma^\mu = (P_L\psi)^\dagger\gamma^0\gamma^\mu = \bar{\psi}P_R\gamma^\mu = \bar{\psi}\gamma^\mu P_L$. So we don't get $P_R$ what does appear in vertex, this is why I don't understand the vertex with $P_R$, I don't see where this right projector comes – Vicky Mar 16 at 2:20