I Weak Interaction Hamiltonian V-A Structure Derivation

[deuteron]

TL;DR Summary

the Hamiltonian I end up with is very different than the usual V-A structure, and I don't know why

Hi! I am trying to come to the V-A structure of the weak interaction Hamiltonian, but I am having some issues with it.
In Feynman & Gell-Mann 1958 paper, they argue that each particle field must be projected onto its left chiral component, which means:

$$H=\displaystyle\sum_{i=S,V,T,A,P} C_i (\overline{\psi_n } O_i \psi_p )(\overline{\psi_\nu} O_i \psi_e) \to \displaystyle\sum_i C_i ([\overline{P_L\psi_n}] O_i [P_L\psi_p]) ([\overline{P_L\psi_\nu}] O_i[P_L\psi_e])$$

and then, they argue that since $P_L^2 = P_L$, and $P_RP_L=0$,
$$S:\quad P_RP_L=0,
\quad P: \quad P_R\gamma_5 P_L = -P_RP_L = 0,
\quad T:\quad P_R\sigma_{\mu\nu} P_L = \sigma_{\mu\nu} P_RP_L = 0,
\quad V:\quad P_R\gamma_\mu P_L = \gamma_\mu P_LP_L=\gamma_\mu P_L,
\quad A:\quad P_R\gamma_\mu \gamma_5 P_L = \gamma_\mu P_L\gamma_5P_L = -\gamma_\mu P_LP_L=-\gamma_\mu P_L$$
so, only the V and A contributions must survive.

And this is where I start to have a problem, because when I only take the V and A contributions, I get:
$$H =C_V ( \overline{\psi_p}\gamma_\mu P_L \psi_n)(\overline{\psi_\nu} \gamma_\mu P_L \psi_e) + C_A(\overline{\psi_p}(-\gamma_\mu P_L) \psi_n)(\overline{\psi_\nu} (-\gamma_\mu P_L) \psi_e)
\\ = C_V ( \overline{\psi_p}\gamma_\mu P_L \psi_n)(\overline{\psi_\nu} \gamma_\mu P_L \psi_e) + C_A(\overline{\psi_p}\gamma_\mu P_L \psi_n)(\overline{\psi_\nu} \gamma_\mu P_L \psi_e)
\\ = (C_V+C_A) (\overline{\psi_p}\gamma_\mu P_L \psi_n)(\overline{\psi_\nu} \gamma_\mu P_L \psi_e)
\\ = \frac {C_V+C_A}4 ( \overline{\psi_p} \gamma_\mu (1-\gamma_5) \psi_n)(\overline{\psi_\nu} \gamma_\mu (1-\gamma_5) \psi_e)
\\ = \frac{C_V+C_A}4 [ \overline{\psi_p} \gamma_\mu \psi_n - \overline{\psi_p}\gamma_\mu \gamma_5 \psi_n ] [ \overline{\psi_\nu} \gamma_\mu \psi_e - \overline{\psi_\nu} \gamma_\mu\gamma_5 \psi_e]
\\ = \frac{C_V+C_A}4 [(\overline{\psi_p}\gamma_\mu \psi_n )(\overline{\psi_\nu} \gamma_\mu \psi_e ) -(\overline{\psi_p}\gamma_\mu \psi_n ) ( \overline{\psi_\nu} \gamma_\mu\gamma_5 \psi_e) + (\overline{\psi_p}\gamma_\mu \gamma_5 \psi_n) ( \overline{\psi_\nu} \gamma_\mu \gamma_5 \psi_e)-(\overline{\psi_p} \gamma_\mu \gamma_5 \psi_n )(\overline{\psi_\nu} \gamma_\mu \psi_e) ]$$

And this does not have the V-A structure, or the term with the $\lambda$ whatsoever. My question is: at which step am I doing something wrong?

Thanks for reading so far :)