Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 304 / 4 / d / ii / Solution

The anticommuting fields $c$ and $\bar{c}$ represent the field-dependent Faddeev-Popov determinant. They are Lorentz-scalar Grassmann fields, appear only on internal lines, and contribute a minus sign for each closed ghost loop. Their diagrams cancel unphysical gauge-polarization contributions and are required for gauge-independent, unitary amplitudes in a non-Abelian covariant gauge.