For $L[A^a]=\partial^\mu A_\mu^a$, eliminating the <Nakanishi-Lautrup field> gives the <covariant gauge> Lagrangian
$$
\mathcal L=\frac14F_{\mu\nu}^aF^{\mu\nu,a}
+\frac1{2\xi}(\partial^\mu A_\mu^a)^2
-\bar c^a\partial^\mu(D_\mu c)^a.
$$
The first term supplies the gluon kinetic term, three-gluon vertex, and four-gluon vertex. The second makes the quadratic gauge-field operator invertible. The last supplies the ghost propagator and ghost-antighost-gluon vertex.

The nonzero one-loop contributions to the <gluon propagator> are a gluon bubble with two three-gluon vertices and a closed ghost bubble with two ghost-antighost-gluon vertices. A four-gluon tadpole is also present with a cutoff regulator; for massless fields it is a scaleless integral and vanishes in <dimensional regularization>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 304 / 4 / d / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-304/4/d/i/solution