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?
Best Answer
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.