  Scalar-fermion models, such as the Gross-Neveu-Yukawa model, admit natural
$1d$ defects given by the exponential of a scalar field integrated along a
straight line. In $4-\varepsilon$ dimensions the defect coupling is weakly
relevant and the setup defines a non-trivial interacting defect CFT. In this
work we study correlation functions on these defect CFTs to order
$\varepsilon$. We focus on $1d$ correlators constrained to the line, which
include canonical operators like the displacement and the one-dimensional
analog of the spin field. These results give access to perturbative CFT data
that can be used as input in the numerical bootstrap. We also consider local
operators outside the line, in particular two-point functions of scalars whose
dynamics are non-trivial due to the presence of the defect.
