= Solution
{c}
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 $\mathcal L_{\rm gf}=-H^2/(2\xi)$. Its quadratic cross term is $+m_A\pi\partial\cdot A$, which cancels $m_AA\cdot\partial\pi$ 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,
$$
\begin{aligned}
\mathcal L_{\rm mix}+\mathcal L_{\rm gf,cross}
&=m_A r^2(A\cdot\partial\pi+\pi\partial\cdot A)\\
&=\partial_\mu(m_A r^2A^\mu\pi)-m_A\pi A^\mu\partial_\mu r^2\\
&\simeq-2e\left(1+\frac hf\right)\pi A^\mu\partial_\mu h.
\end{aligned}
$$
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 <3/b>. Since $h$ is invariant,
$$
\delta_\alpha H=-\frac1e\Box\alpha-\xi m_A r^2f\alpha
=-\frac1e[\Box+\xi m_A^2r^2]\alpha.
$$
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 <3/a> gives
$$
\boxed{\mathcal L_{\rm gh}=-\bar c\left[\Box+\xi e^2(f+h)^2\right]c
=-\bar c(\Box+\xi m_A^2)c
-2\xi e^2fh\bar cc-\xi e^2h^2\bar cc.}
$$
The <ghost coupling in the Abelian Higgs model> therefore survives even though the <gauge group> is Abelian: these <Faddeev-Popov ghosts> interact with $h$ and \b[do not decouple for generic $\xi$]. The distinction from part <3/a> is the scalar dependence of the <gauge fixing> function. In the $\xi\to0$ <Landau gauge> limit these particular ghost interactions vanish.
Back to article page
