It's been a time since you posted the question. I met a derivation recently which I think is kind of similar to your question. As for ##F(\bar{k}_{A}^{z})## you wrote, if we have constraints ##\bar{k}_{A}^{\perp}=k_{A}^{\perp}## and ##\bar{k}_{B}^{\perp}=k_{B}^{\perp}##, I don't think there is...