Spinor Helicity Formalism: Reference Spinors $q$ in Compton Scattering

My question is rather straight forward, but the setup in order to pose the question is a little lengthy; please bear with me!

I am trying to calculate the average over initial states and sum over final states for Compton Scattering, using spinor helicity techniques. I’m having some trouble getting the amplitudes independent of the arbitrary reference spinors $q$ we get from the polarization vectors.

I use the conventions of Elvang and Huang, where all external particles are taken to be outgoing, and we use the spacetime metric (-+++). We have four terms to calculate:

begin{equation}
frac{1}{4} sum_{h_{A}, h_{B}, h_{1}, h_{2}=+,-}left|mathcal{M}left(A^{h_{A}} B^{h_{B}} 1^{h_{1}} 2^{h_{2}}right)right|^{2} = frac{1}{4}left[left|mathcal{M}left(A^{+} B^{-} 1^{+} 2^{-}right)right|^{2}+left|mathcal{M}left(A^{-} B^{-} 1^{+} 2^{+}right)right|^{2}+left|mathcal{M}left(A^{+} B^{+} 1^{-} 2^{-}right)right|^{2}+left|mathcal{M}left(A^{-} B^{+} 1^{-} 2^{+}right)right|^{2}right].
end{equation}

For the first amplitude, we use the usual QED Feynman rules to obtain

begin{equation}
frac{1}{e^{2}} mathcal{M}left(A^{+} B^{-} 1^{+} 2^{-}right) = bar{u}_{1}^{+} not varepsilon_{2}^{*-} frac{(not A+not B)}{(A+B)^{2}} not varepsilon_{A}^{*+} v_{B}^{-} + bar{u}_{1}^{+} not varepsilon_{A}^{*+} frac{(not 2+not B)}{(2+B)^{2}} not varepsilon^{*-}_{2} v_{B}^{-},
end{equation}

since the $s$– and $t$-channel contribute to the Compton scattering process.

After translating this to the angle and square brackets of the spinor helicity formalism, we simplify and see that

begin{equation}
frac{1}{e^{2}} mathcal{M}left(A^{+} B^{-} 1^{+} 2^{-}right) = frac{2[12]leftlangle A q_{2}rightrangle}{leftlangle 2 q_{2}rightrangle [A B]} +frac{2[12]leftlangle B q_{2}rightrangleleft[B q_{A}right]}{leftlangle 2 q_{2}rightrangleleft[A q_{A}right][A B]}+frac{2left[1 q_{A}right]langle A Brangleleftlangle q_{2} Brightrangle}{left[A q_{A}right]leftlangle 2 q_{2}rightranglelangle 2 Brangle}.
end{equation}

Now, my understanding is that this expression does not depend on our specific choice of reference spinors $q_A$ and $q_2$. We should therefore be able to fiddle with this expression to get rid of the $q$‘s.

If we were considering a process with only three external particles, and we had some expression like

begin{equation}
frac{langle 13rangle[2 q]}{[3 q]},
end{equation}

we could multiply by $1=langle 12rangle /langle 12rangle$ and use energy-momentum conservation to eliminate the $q$:

begin{equation}
frac{langle 13rangle[2 q]}{[3 q]}
=frac{langle 13rangle[2 q]langle 12rangle}{[3 q]langle 12rangle}
=frac{langle 13rangle(-[3 q]langle 13rangle)}{[3 q]langle 12rangle} = -frac{langle 13 rangle ^2}{langle 12 rangle},
end{equation}

which is independent of $q$.

However, stepping back to Compton scattering, if I multiply

begin{equation}
frac{2[12]leftlangle A q_{2}rightrangle}{leftlangle 2 q_{2}rightrangle[A B]}
end{equation}

by $1= [A1] / [A1]$, energy-momentum conservation gives $leftlangle A q_{2}rightrangle[A1] = -langle2q_2rangle [21] – langle Bq_2 rangle [B1]$ in the numerator. We see that the $leftlangle 2 q_{2}rightrangle$ in the denominator is cancelled, but we end up with an additional term that still depends on reference spinors.

I don’t know how to treat this additional term in order to remove all $q$ dependence, and I’m having the same issue with the other two terms in $frac{1}{e^{2}} mathcal{M}left(A^{+} B^{-} 1^{+} 2^{-}right)$.

Any pointers would be much appreciated!