= Solution
{c}
An ordinary bosonic <Gaussian functional integral> produces an inverse determinant. An integral over independent <Grassmann fields> $c,\bar c$ instead produces a determinant, allowing a local action to represent the <Faddeev-Popov determinant>. With one convenient antighost sign convention,
$$
\det\mathcal M\ \propto\ \int\mathcal D\bar c\,\mathcal Dc\,\exp\left(i\int d^4x\,\mathcal L_{\mathrm{gh}}\right),\qquad
\boxed{\mathcal L_{\mathrm{gh}}=-\bar c(\partial^2+2\lambda A^\mu\partial_\mu)c.}
$$
Multiplying the operator by the field-independent factor $-i$ changes only the normalization of its determinant. After <integration by parts>, an equivalent density is
$$
\mathcal L_{\mathrm{gh}}=(\partial_\mu\bar c)(\partial^\mu c)-2\lambda\bar c A^\mu\partial_\mu c.
$$
Thus the <Faddeev-Popov ghost fields> have a massless kinetic term and a photon-ghost-antighost interaction proportional to $\lambda$. They are auxiliary anticommuting scalar fields, not physical external <particles>. Closed <ghost loops> carry the minus sign of <Grassmann variables>. Replacing $\bar c$ by $-\bar c$ reverses both displayed signs without changing any observable; both the kinetic and interaction conventions must be changed together.
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