Charged current $e^-e^+$ weak interaction Feynman diagram

Question

I am new to the weak interactions and in particular to the interactions given by charged current interaction lagrangian $$ \mathcal{L}_{int}^{CC} = \frac{g}{2\sqrt{2}}\left[J^{\mu}W^{+}_{\mu}+J^{\mu \dagger} W^{-}_{\mu}\right] $$ where currents are given as $J_{\mu}=\bar{\psi}_{\nu_e}\gamma_{\mu}(1-\gamma_5)\psi_e$ and $J^\dagger_{\mu}=\bar{\psi}_e\gamma_{\mu}(1-\gamma_5)\psi_{\nu_e}$, $W^{\pm}$ is vector boson.

I would like to draw Feynman diagram and build the matrix element for the process $e^-e^+\rightarrow W^+W^-$. My question is why in the Feynman diagram $W^-$ shares vertex with $e^-$ instead of $e^+$ (and the same for the other vertex)?

In the lagrangian $W^-_{\mu}$ stands with the current $J^{\mu \dagger} $, which contains wave function $\bar{\psi}_e$ that destroys antiparticle — positron, so shouldn't $W^-$ share vertex with $e^+$ and $W^+$ with $e^-$ instead of what it is in the diagram? Following this logic, $W^+$ and $W^-$ on the right-hand side of the diagram should be interchanged, but every source I found has the above diagram for this process, so the error is on my side. Can someone kindly explain, why we'd obtain the above diagram for this process?

Answer 1

Observe the fields in your lagrangian density $$ \mathcal{L}_{int}^{CC} = \frac{g}{2\sqrt{2}}\left[ \bar{\psi}_{\nu_e}\gamma^{\mu}(1-\gamma_5)\psi_e W^{+}_{\mu}+J^{\mu \dagger}\bar{\psi}_e\gamma^{\mu}(1-\gamma_5)\psi_{\nu_e} W^{-}_{\mu}\right] $$ conserve charge in every term. The first term destroys an electron and (destroys a $W^+$, whence it) creates a $W^-$.

Conversely, the second term destroys a positron and creates a $W^+$.

So at every term/vertex, the charged lepton passes its charge on to the vector boson.