# Finding the interaction Lagrangian from a Scattering Amplitude

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I am reading the book Scattering Amplitudes in Gauge Theory and Gravity from Elvang and Huang. In section 2.6 they seem to suggest that the mass-dimension of the kinematic part of the amplitude is in 1-to-1 correspondence to the number of derivatives in the interaction term in the Lagrangian. For example, for the 3-gluon amplitude
$$A_3(g_1^- g_2^- g_3^+) = g \frac{\langle 12\rangle^3}{\langle13\rangle\langle 23\rangle}$$
has a kinetic part with mass-dimension 1, which means it is compatible with the $AA\partial A$ interaction term in $\mathrm{Tr}F_{\mu\nu}F^{\mu\nu}$. Similarly they argue that the amplitude
$$A_3(g_1^-g_2^-g_3^+) = g' \frac{[13][23]}{[12]^3}$$
has mass-dimension -1, and thus must come from $g' AA \frac{\partial}{\Box} A$ (and is thus nonlocal and unphysical). Later on a similar claim is made about the amplitude
$$A_3(g_1^-g_2^-g_3^-) = a \langle12\rangle\langle13\rangle\langle23\rangle$$
coming from an interaction term with three derivatives ($\partial A \partial A \partial A$)

This seems intuitive, but it also seems to contradict an earlier result from QED with massless fermions discussed in section 2.4. In that section we derive the 3-particle amplitude
$$A_3(f^- \bar{f}^+ \gamma^-) = \tilde{e} \frac{\langle 13\rangle^2}{\langle12\rangle}.$$
By the same logic you would expect this to come from a Lagrangian interaction term with one derivative, but in fact it seems to arise from the $\gamma^\mu\bar{\Psi} A_\mu \Psi$ interaction term, which does not have any derivatives.

Can someone clarify this for me?

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