$SU(2)_L$ Gauge theory and particle-antiparticle annihilation
I have a problem understanding how to reconcile the particle antiparticle annihilation vertex with the $SU(2)$ gauge theory, in the context of the weak interaction. Let me explain better :
Invoking $SU(2)$ gauge invariance we deduce there must be three gauge bosons, associated to the three Pauli matrices. We take, as usual, the linear combination yielding $\sigma_+, \sigma_-$ and $\sigma_z$ that are respectively associated to $W^+, W^-$ and $Z$. I am aware that I should be considering $U(1)_Y\times SU(2)_L$, but in the context of this question I believe it is irrelevant.
Now consider the SU(2) doublets,$\begin{pmatrix}l^+\ l^- \end{pmatrix}$, where $l^+$ has weak isospin $1/2$ and $l^-$ has isospin $-1/2$. Let's take $\begin{pmatrix}v_e\ e^- \end{pmatrix}$, we find that the weak current by coupling to the $Z$ boson is: $$j^{\mu}_Z \propto \begin{pmatrix}\overline{v}_e & \overline{e}^- \end{pmatrix}\gamma^{\mu}\sigma_z \begin{pmatrix}v_e\ e^- \end{pmatrix}$$ Where $\overline{u} = u^{\dagger}\gamma^0$. 656
Expanding this, we find that : $$j^\mu_Z=\frac{1}{2}\overline{v}_e\gamma^{\mu}v_e-\frac{1}{2}\overline{e}^-\gamma^{\mu}e^-$$
Where, $v_e$ and $\overline{v_e}$ stands for the spinors of the neutrino, and likewise for the electron. As we can see from this, it seems that the Z-boson couples particles of same weak isospin. However, we can have an annihilation vertex where $e^-$ and $e^+$ annihilate into a Z boson, despite the fact that $e^-$ has $I_w^{(3)} = -1/2$ while $e^+$ has $I_w^{(3)} = 1/2$. How can this reconciled with the representation of Z as $\sigma_z$ ?
I know that there is some problem with my current, since obviously an $e^-$ cannot annihilate with an $e^-$, in a vertex such as : , but only in a vertex such as : .
However, in my derivation, there does not seem to be a distinction in which one of these vertex I'm considering, so I'm confident that there lies my mistake, but I am unable to figure it out. I think somehow, in an annihilation vertex, particles of opposite weak isospin should interact while in a scattering vertex particle of same weak isospin should interact. This is also consistent with conservation of weak isospin, but I am unable to understand how to make this distinction in the currents using $\sigma_Z$ as the Z boson coupling. 818