Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2018 / 2 / b

Choose an infinitesimal gauge transformation $\delta A_{\mu }^a=D_{\mu }^{ab}{\omega }^b$, where $D_{\mu }^{ab}={\delta }^{ab}{\partial }_{\mu }+g{f}^{acb}{A}_{\mu }^c$. Varying the axial gauge function at fixed $\sigma$ givesThis is the Faddeev-Popov operator. The Faddeev-Popov determinant has a Grassmann representation $\det M\propto \int \mathcal{D}\bar{c}\mathcal{D}c{e}^{i\int \bar{c}Mc}$. Choosing this normalization for the antighost givesA simultaneous antighost rephasing changes the overall ghost-action sign and vertex conventions, but not the determinant's field-dependent content. The essential facts are the kinetic operator $n\cdot \partial$ and a ghost-gluon vertex proportional to $n^{\mu }$.

Now average the fixed-$\sigma$ constraint with the stated Gaussian:up to a field-independent normalization. ConsequentlyThe Faddeev-Popov ghost determinant is not generally field independent before imposing the strict axial condition: its interaction term contains $n\cdot A$. In the $\xi \to 0$ gauge this quantity vanishes on the integration slice, giving the decoupling established next. Defining the inverse of $n\cdot \partial$ still requires boundary conditions for residual gauge transformations.

Contract the numerator of the quoted light cone gauge propagator with $n_{\mu }$:Thus every internal gluon emerging from a ghost-gluon vertex contributes a factor $n_{\mu }{\tilde{\Delta }}^{\mu \nu }=0$. Every ghost-dependent contribution to a physical gluon or matter amplitude vanishes. Equivalently, the strict condition $n\cdot A=0$ removes the Faddeev-Popov ghost interaction in i, leaving the field-independent determinant $\det (n\cdot \partial )$, which cancels from normalized correlators.

This is ghost decoupling in strict axial gauge, not a statement that the free Faddeev-Popov ghost propagator itself is zero. The algebra assumes a consistent axial-gauge pole prescription for $n\cdot p=0$ and treatment of residual gauge zero modes. The overall sign assigned to the displayed vector propagator does not enter the contraction proof. With the mostly-minus convention and the usual $-F^2/4$ action, its conventional prefactor is $-i/(p^2+i0)$; the source prints $+i$. Either prefactor gives the same zero contraction, so no sign correction is needed to obtain the requested ghost-decoupling result.