Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2024 / 3 / a / Solution

Choose $D_{\mu }={\partial }_{\mu }-ig{A}_{\mu }^a{t}^a$ on the Dirac field, $t^a={\sigma }^a/2$ in the fundamental representation of SU(2), and $[t^a,t^b]=i{ϵ}^{abc}{t}^c$. Define the gauge field strength and the Adjoint representation gauge covariant derivative byThe required gauge-fixed Yang-Mills Lagrangian in a covariant gauge, including the matter term, isHere $c^a$ and ${\bar{c}}^a$ are independent Grassmann fields in the Adjoint representation: the Faddeev-Popov ghost fields and antighost fields. Under an infinitesimal gauge transformation, $\delta A_{\mu }^a=(D_{\mu }\omega {)}^a$; hence the Lorenz gauge function $G^a={\partial }^{\mu }{A}_{\mu }^a$ varies by ${\partial }^{\mu }{D}_{\mu }^{ab}{\omega }^b$. Its Faddeev-Popov determinant is represented by the ghost term, with an irrelevant field-independent determinant sign absorbed into normalization. Integration by parts gives ${\mathcal{L}}_{\mathrm{gh}}=({\partial }^{\mu }{\bar{c}}^a)(D_{\mu }c{)}^a$, explicitly showing the ghost-gluon vertex.

There is no scalar vacuum expectation value here, so R-xi gauge reduces to the linear covariant gauge displayed above, with no Goldstone boson term. At $\xi =1$ the Feynman gauge gauge boson propagator is $-i{\delta }^{ab}{g}_{\mu \nu }/(k^2+i0)$. With Fourier convention $e^{-ikx}$ and all three boson four-momenta incoming, this choice gives exactly the source's $+g{ϵ}^{abc}$ three-boson Feynman rule, and the matter interaction vertex is $+ig{\gamma }^{\mu }{t}^a$. The sign of $D_{\mu }$ has been explicitly chosen here; the QED convention in question 2 instead has vertex $-ie{\gamma }^{\mu }$.