Past exam of the advanced quantum field theory for particle physics course of the University of Oxford / 2017 / 3 / a / ii / Solution

For the ghost coupling in a linear Cartesian Abelian Higgs gauge, the common factor in the scalar parametrization cancels from the gauge transformation. Separating real and imaginary parts givesConsequentlyFor the variables and gauge function actually printed, the ghost mass and Higgs-ghost vertex are thereforeFor the usual nonnegative gauge parameter, $m_c=\sqrt{\xi }g|v|$. The momentum-space ghost propagator is $-i/(k^2+m_c^2-i0)$ in this kinetic convention. There is no direct ghost-photon vertex or ghost-$b$ vertex. At $\xi =0$ the ghost becomes massless and decouples.

The scalar prefactor $1/4$ is incompatible with the stated canonical broken-vacuum interpretation. At $h=b=0$ it gives $|\varphi {|}^2=v^2/16$, whereas the potential is minimized at $|\varphi {|}^2=v^2/2$. In fact the printed parametrization givesIts $h,b$ kinetic terms have coefficient $-1/16$, not $-1/2$, and its vector mass parameter is $g^2{v}^2/8$. The given gauge function consequently does not cancel the vector-scalar mixing in those literal variables. The ghost operator computed above nevertheless follows algebraically from that gauge function without changing the printed fields.

With a canonical Abelian Higgs shift $\varphi =(v+h+ib)/\sqrt{2}$, the expansion is about a true minimum, the vector mass is $m_A=g|v|$, the gauge fixing cancels the mixing, and the same ghost formulas acquire the usual interpretation $m_c^2=\xi m_A^2$. For the literal $1/4$ parametrization, a canonical fluctuation is $h_{\mathrm{can}}=h/\sqrt{8}$, so its ghost vertex would instead be $-i\sqrt{8}\xi g^{2}v$; the non-vacuum background problem still remains. These two interpretations must not be silently mixed.