= One-loop ghost kinetic counterterm in Feynman gauge
{title2=$\delta Z_c=g^2C/[16\pi^2(4-d)]$}
For canonically normalized fields with action $-\int[\tfrac14F^2+\tfrac12(\partial\cdot A)^2+\partial\bar c\cdot Dc]$, use signature $(-,+,+,+)$ and incoming Fourier modes $e^{ipx}$. The <ghost-gluon vertex> is $gp_\mu f_{abc}$ and the <ghost propagator> is $-i/(p^2-i0)$. The open ghost-line <one-particle-irreducible two-point vertex> at one loop is $ig^2C I_d(p)$, where
$$
I_d(p)=\frac1i\int\frac{d^dk}{(2\pi)^d}\frac{p\cdot k}{[(p-k)^2-i0][k^2-i0]},\qquad I_d(p)\big|_{\mathrm{pole}}=\frac{p^2}{16\pi^2(4-d)}.
$$
A <counterterm> $\mathcal L_{\mathrm{ct}}=-\delta Z_c\,\partial\bar c\cdot\partial c$ inserts $-i\delta Z_c p^2$ and cancels this pole. Equivalently, writing $d=4-2\epsilon$ gives $\delta Z_c=g^2C/(32\pi^2\epsilon)$. There is no closed ghost-loop sign for this open-line graph. The <Feynman i-epsilon prescription>, Fourier convention and antighost sign must be kept consistent throughout.
Back to article page
