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A note on the implications of gauge invariance in QCD

We compare and contrast the implications of gauge invariance for the structure of scattering amplitudes in QED and QCD. We derive the most general analogue for QCD of the famous QED rule that the scattering amplitude must be invariant if the polarization vectors $ε^{μ_j}(k_j)$ of any number of photons undergo the replacement $ ε^{μ_j}(k_j) \rightarrow ε^{μ_j}(k_j) + c \, k_j^{μ_j} $, where $c$ is an arbitrary constant.