Gluon self-energy (${\Pi }^{\mu \nu ,ab}(k)$)

The proper two-point correction to the gluon propagator in QCD. At one loop it receives quark, gluon, Faddeev-Popov ghost field and four-gluon tadpole diagram contributions. The Slavnov-Taylor identity makes the total tensor transverse, $k_{\mu }{\Pi }^{\mu \nu ,ab}=0$, although the individual gluon and ghost loop contributions need not be transverse. The sign of the symbol must be tied to whether the amputated insertion is defined as $i\Pi$ or $-i\Pi$.

With insertion convention $i\Pi$, ghost propagator $i/p^2$, and oriented ghost-gluon Feynman rule $g{f}^{abc}{p}^{\mu }$ with $p$ pointing away along the ghost arrow, the ghost loop is $i{\Pi }_{\mathrm{gh}}=-g^2{C}_A{\delta }^{ab}\int d^{D}l(2\pi {)}^{-D}(l+k{)}^{\mu }{l}^{\nu }/[l^2(l+k{)}^2]$. In dimensional regularization with $D=4-ϵ$, a Feynman parameter and a shift give the displayed ultraviolet pole. Its contraction is $g^2{C}_A{\delta }^{ab}{k}^2{k}^{\nu }/(32{\pi }^2ϵ)$; it cancels the nontransverse part of the two-three-gluon-vertex contribution. A different insertion or ghost-vertex convention must be translated before comparing signs.