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Scattering Amplitudes from Feynman Diagrams (Spinor Helicity Formalism)

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$\require{cancel}$
I am trying to do an exercise from Scattering Amplitudes By Elvang (Exercise 2.9) which states:

Show that $A_5(f^-\bar{f}^-\phi\phi\phi) = g^3\frac{[12][34]^2}{[13][14][23][24]} + 3\leftrightarrow 5 + 4\leftrightarrow 5$ in Yukawa theory

So, I draw the feynman diagram, which I think looks something like this (the interaction term is $L_i = g\phi\psi\bar{\psi}$):

Is this diagram correct? Using the Feynman rules for Yukawa theory (in the Massless Spinor Helicity formalism) I evaluate this to be:
$$
A_5(f^-\bar{f}^-\phi\phi\phi) = g^3\langle2|\frac{(\cancel{p_1} + \cancel{p_2})}{(({p_1} + p_2)^2}\frac{(\cancel{p_1} + \cancel{p_2} + \cancel{p_3})}{(p_1 + p_2 + p_3)^2}|5\rangle \\~~~\\+ ~1\leftrightarrow 3 + ~1\leftrightarrow 4 + ~3\leftrightarrow 4
$$

My strategy thus far has been calculate the first term then simply do the permutations at the very end. In general, is this a good strategy to take with diagrams like this?

Doing this, I end up with the following for the first term:
$$
A_5^{(1)} = g^3\langle2|\frac{s_{13}}{s_{12}(s_{12} + s_{13} + s_{23})}|5\rangle
$$

Where $s_{ij} = -(p_i + p_j)^2 = 2p_i\cdot p_j$ and I have used the Weyl equation $\langle 2|p_2 = 0$.

I can go further, using the fact that $s_{ij} = \langle ij\rangle[ij]$, to end up with:
$$
A_5^{(1)} = g^3\langle2|\frac{\langle 13\rangle[13]}{\langle 12\rangle[12](\langle 12\rangle[12] + \langle 13\rangle[13] + \langle 23\rangle[23])}|5\rangle
$$

I can't seem to simplify this further. Am I going the right away about solving this? Are there any tricks I am missing?

asked Oct 11, 2015 in Theoretical Physics by Akoben (15 points) [ no revision ]

I am not sure if this helps, but I suggest you give a look at:

http://arxiv.org/pdf/hep-ph/9601359v2.pdf

Although these calculations are in QCD you might find some other sources from the same author.

Good luck!

Thanks @MathematicalPhysicist will take a look

I believe you have mislabeled your diagram. The final answer ought to be symmetric under exchange of the scalars but the way you have constructed your answer is clearly not. I don't happen to have my copy of Elvang with me currently but I may write a more detailed response to this later.

I think I've found the problem: your initial expression for the amplitude using the Feynman rules is correct, but your second step is wrong. \slashed{p} should still have some matrix structure that would allow you to contract the external spinors with something. It looks like you took traces over the product of gamma matrices. Instead use Elvang's equation 2.15. That and momentum conservation (equation 2.39) should allow you to simplify the amplitude.

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