Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 46 / 4 / Solution

Expanding the covariant derivative isolates the ghost-gluon vertex interactionThe derivative acts on the antighost. With all momenta incoming, let its momentum be $k$, the gluon's Lorentz index be $\mu$, and their color indices be $(a,b,c)$ in the order $(\bar{c},A,c)$. The convention $e^{-ikx}$ replaces ${\partial }^{\mu }$ by $-i{k}^{\mu }$, and multiplication by $i$ from $e^{iS}$ givesMomentum conservation is implicit. The structure constants reflect the Adjoint representation, and the antighost momentum follows directly from its differentiated field. Reversing the momentum convention changes the displayed vertex sign consistently; exchanging the ghost and antighost source order also carries a fermionic sign.