  We study the $O(3)$ critical model and the free theory of a scalar triplet in
the presence of a magnetic impurity. We use analytic bootstrap techniques to
extract results in the $\varepsilon$-expansion. First, we extend by one order
in perturbation theory the computation of the beta function for the defect
coupling in the free theory. Then, we analyze in detail the low-lying spectrum
of defect operators, focusing on their perturbative realization when the defect
is constructed as a path-ordered exponential. After this, we consider two
different bulk two-point functions and we compute them using the defect
dispersion relation. For a free bulk theory, we are able to fix the form of the
correlator at all orders in $\varepsilon$. In particular, taking
$\varepsilon\to1$, we can show that in $d=3$ one does not have a consistent and
non-trivial defect CFT. For an interacting bulk, we compute the correlator up
to second order in $\varepsilon$. Expanding these results in the bulk and
defect block expansions, we are able to extract an infinite set of defect CFT
data. We discuss low-spin ambiguities that affect every result computed through
the dispersion relation and we use a combination of consistency conditions and
explicit diagrammatic calculations to fix this ambiguity.
