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

Take metric signature $(+---)$ and the Fourier transform convention $e^{-ipx}$. Introduce an even source $J$ for the gauge field and odd sources $\eta ,\bar{\eta }$ for the Faddeev-Popov ghost field:At tree level the two-point functions come from the quadratic action. Integrating the gauge kinetic term by parts gives its momentum-space kernelIts inverse follows from transverse and longitudinal projectors: $K=-p^2(P_T+{\zeta }^{-1}{P}_L)$, so $K^{-1}=-p^{-2}(P_T+\zeta P_L)$. Completing the bosonic Gaussian square gives $Z_0[J]=\exp [-iJ{K}_F^{-1}J/2]$, with the Feynman i-epsilon prescription in the inverse. Two source derivatives and their insertion factors give the gauge-boson propagator $D_F=i{K}_F^{-1}$. HenceFor $\zeta =1$ this reduces to the Feynman-gauge adjoint propagator. The longitudinal double pole is understood with the compatible Feynman boundary prescription.

The ghost kinetic term as printed is $-{\partial }^{\mu }{\bar{c}}^a{\partial }_{\mu }{c}^a$. Integration by parts gives ${\bar{c}}^a\square c^a$, so its Fourier-space quadratic kernel is $K_c^{ab}(p)=-p^2{\delta }^{ab}$. Completing the Grassmann square in the same generating functional givesKeep the source order fixed, using a left derivative for $\bar{\eta }$ and a right derivative for $\eta$. Including the two insertion factors produces $⟨Tc\bar{c}⟩=i{K}_{c,F}^{-1}$, just as in the Gaussian generating functional for a Dirac field. ThereforeThe minus sign follows from the particular ghost Lagrangian in this paper. A convention with the opposite ghost kinetic sign has the opposite $c\bar{c}$ propagator; redefining the Faddeev-Popov antighost field changes its interaction and source signs at the same time. Mixing propagator and vertex conventions would give an erroneous loop sign. This is the ghost propagator for a negative derivative kinetic term.

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.