Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2025 / 3

Start with the formally gauge-redundant path integral $Z=\mathrm{vol}(\mathcal{G}{)}^{-1}\int \mathcal{D}A{e}^{iS[A]}$. Use $A_{\mu }^{\theta }=A_{\mu }+{\partial }_{\mu }\theta$, and the generalized Lorenz gauge condition $F_{\omega }[A]={\partial }_{\mu }{A}^{\mu }-\omega$. In a perturbative neighborhood where residual zero modes are removed by boundary conditions, insert the Faddeev-Popov gauge-orbit identityThus ${\Delta }_{\mathrm{FP}}=\det \square$. Gauge invariance lets the $\theta$ integration factor out and cancel $\mathrm{vol}(\mathcal{G})$. Average over $\omega$ with the Gaussian integral weight $\exp [-i\int {\omega }^2/(2\xi )]$; the delta functional fixes $\omega =\partial \cdot A$ and givesRepresent the functional determinant by the independent Grassmann fields $\bar{c},c$. A field-independent determinant sign can be absorbed in normalization, so a convenient convention isThereforeThe Faddeev-Popov ghosts are free and decouple because $\square$ is independent of $A$. For a Non-Abelian gauge theory, an infinitesimal gauge transformation gives $\delta A_{\mu }^a=D_{\mu }^{ab}{\theta }^b$, so the Faddeev-Popov operator is ${\partial }^{\mu }{D}_{\mu }^{ab}[A]$ andIt contains a ghost-gluon vertex: these Faddeev-Popov ghosts do not decouple in covariant gauge fixing of Lorenz gauge type.

Choose the gauge covariant derivative $D_{\mu }={\partial }_{\mu }+ie{A}_{\mu }$. Write $\rho =f+h$, $r=1+h/f$ and $B_{\mu }={\partial }_{\mu }\pi /f+e{A}_{\mu }$. The polar parametrization of the Abelian Higgs field givesHence the full Lagrangian density isAssume $f>0$ and $\lambda >0$, so the minimum has $|\varphi |=f/\sqrt{2}$. In this convention the local gauge symmetry acts asThe combination ${\partial }_{\mu }\pi +ef{A}_{\mu }$, the radial field $h$, and the gauge field strength are each invariant, proving invariance of the displayed Lagrangian density.

This parametrization follows the circle of classical minima: $h$ measures distance away from the vacuum manifold, whereas $\pi$ moves along a gauge orbit. It separates the massive radial Higgs mode from the would-be Goldstone boson and makes its removal by the Higgs mechanism transparent. At quadratic order,The polar parametrization of the Abelian Higgs field is a local description near the chosen nonzero vacuum expectation value; it is singular at $\varphi =0$ and is not a global coordinate system for configurations with zeros or winding.

Unitary gauge fixes the scalar phase to zero. Locally choose $\alpha =-\pi /f$ in part b; then ${\pi }^{\prime }=0$ and $A_{\mu }^{\prime }=A_{\mu }+{\partial }_{\mu }\pi /(ef)$. The remaining fields are the radial Higgs mode and a massive vector. Its quadratic Lagrangian density is $-F^2/4+m_A^2{A}^2/2$ and its gauge-boson propagator isFor $k^2\ne 0$, introduce the transverse and longitudinal orthogonal projectors $P_{\mu \nu }^T=g_{\mu \nu }-k_{\mu }{k}_{\nu }/k^2$ and $P_{\mu \nu }^L=k_{\mu }{k}_{\nu }/k^2$. Omitting the infinitesimal in this algebraic decomposition,At large momentum in Euclidean space, the transverse coefficient falls as $k^{-2}$ but the longitudinal coefficient approaches a constant of order $m_A^{-2}$. Thus power counting applied directly to unitary gauge Feynman diagrams suggests increasingly severe ultraviolet divergences. This hides the renormalizability of physical scattering amplitudes; it does not disprove it. A finite-$\xi$ R-xi gauge exposes the better propagator scaling while preserving the fields needed to track the cancellations.

Let $m_A=ef$, $r=1+h/f$, and $H=\partial \cdot A-\xi m_A{r}^2\pi$, so the given $G=H/\sqrt{\xi }$ contributes ${\mathcal{L}}_{\mathrm{gf}}=-H^2/(2\xi )$. Its quadratic cross term is $+m_A\pi \partial \cdot A$, which cancels $m_{A}A\cdot \partial \pi$ from the scalar kinetic term after integration by parts. The free vector and would-be Goldstone boson propagators are therefore diagonal; $\pi$ has gauge-dependent mass squared $\xi m_A^2$.

The field-dependent factor $r^2$ cancels more than just the constant-coefficient mixing, but a residual derivative interaction remains. Indeed,Thus the nonlinear R-xi gauge in polar Higgs variables removes quadratic mixing; it does not remove every interaction containing $A$ and $\pi$.

Use the gauge transformation in part b. Since $h$ is invariant,The factor $-1/(e\sqrt{\xi })$ in $\delta G/\delta \alpha$ is field independent and is absorbed into the partition function normalization. Exponentiating the remaining Faddeev-Popov determinant with the sign convention of part a givesThe ghost coupling in the Abelian Higgs model therefore survives even though the gauge group is Abelian: these Faddeev-Popov ghosts interact with $h$ and do not decouple for generic $\xi$. The distinction from part a is the scalar dependence of the gauge fixing function. In the $\xi \to 0$ Landau gauge limit these particular ghost interactions vanish.

After the quadratic mixing cancellation, the vector inverse quadratic operator isInverting its two orthogonal projectors and multiplying by $i$ givesEquivalently $D^{(\xi )}=-i{P}^T/(k^2-m_A^2+i0)-i\xi P^L/(k^2-\xi m_A^2+i0)$. At fixed finite $\xi$, both sectors fall as $k^{-2}$ at large momentum in Euclidean space; $\xi =1$ gives $-i{g}_{\mu \nu }/(k^2-m_A^2+i0)$, and taking $\xi \to \infty$ at fixed $k$ recovers unitary gauge. The latter limit is not uniform in ultraviolet momentum.

The expected power counting conclusion is that this $k^{-2}$ behaviour, together with polynomial interactions of mass dimension at most four in Cartesian scalar variables and the gauge symmetry identities, permits renormalization of spontaneously broken gauge theories using a finite set of physical parameters. The propagator alone is not sufficient to establish that claim in the polar variables of this question. In fact, part b contains $h(\partial \pi {)}^2/f$ and $h^2(\partial \pi {)}^2/(2f^2)$, of dimensions five and six, and the nonlinear gauge fixing term contains $-\xi e^2{h}^4{\pi }^2/(2f^2)$, also of dimension six. Ordinary power counting is consequently not manifest in these variables.

For a representation where it is manifest, write $\varphi =(f+H+i\chi )/\sqrt{2}$ and choose the usual polynomial R-xi gauge function $\partial \cdot A-\xi ef\chi$. It yields the same free vector propagator, cancels the quadratic mixing, and leaves only dimension-at-most-four interactions, including its Faddeev-Popov ghosts. This supplies the missing vertex condition in the source's proposed inference; the equivalence theorem for field redefinitions relates physical predictions near the vacuum. Good propagator falloff is the necessary ingredient illustrated by the requested formula, rather than a complete proof for arbitrary parametrizations or nonlinear choices of gauge fixing.