Off-Forward quark-quark amplitude in momentum space

[Diracobama2181]

TL;DR

I am attempting to find the off-forward quark-quark amplitude in momentum space.

I am having difficulty writing out

$\bra{p',\lambda}\psi^{\dagger}(-\frac{z^-}{2})\gamma^0\gamma^+\psi\frac{z^-}{2})\ket{p,\lambda}$ in momentum space.
Here, I am working in light-cone coordinates, where I am defining $z^-=z^0-z^3$, $r'=r=(0,z^{-},z^1,z^2)$.
My attempt at this would be
$$\bra{p',\lambda}\psi^{\dagger}(-\frac{z^{-}}{2})\gamma^0\gamma^{+}\psi\frac{z^{-}}{2})\ket{p,\lambda}=\Sigma_{r,r'}\bra{p',\lambda}(\ket{r'}\bra{r'})\psi^{\dagger}(-\frac{z^{-}}{2})\gamma^0\gamma^{+}\psi\frac{z^{-}}{2}(\ket{r}\bra{r})\ket{p,\lambda}\\\\
=\int d^3r exp[i(p'-p)\cdot z^{-}]\psi^{\dagger}(-\frac{z^-}{2})\gamma^0\gamma^+\psi(\frac{z^-}{2})$$.
From here, I can substitute in
$\psi(t,\vec{r})=\int\frac{d^3\vec{k}}{(2\pi)^3}exp[-i(k^0t-\vec{k}\cdot \vec{r})]\phi(\vec{k})$

Is this attempt correct so far, or am I overlooking something? Any comments are appreciated.

 

[azntoon]

Your attempt looks correct so far. The next step would be to substitute in the definition of $\psi$ and evaluate the integral. This should give you a result in momentum space.