Complete path integration of a field Consider a theory of two fields $\phi$ and $\chi$ admitting an action $S[\phi, \chi]$ which has both self-interations and mixed interactions. Now, suppose we were to perform the path integral over $\chi$ exactly $$Z[\phi] = \int [\mathcal{D}\chi] e^{-iS[\phi, \chi]}$$
What is then the physical interpretation of $Z[\phi]$? Can we consider $\ln Z[\phi]$ to be some action for $\phi$ field? How is this different from the action obtained by setting $\chi =0$ in the original action $S[\phi, \chi]$? How is the knowledge of $\ln Z[\phi]$ helpful?
 A: What you are describing here is a text book example of ''integrating out fields''.
$\mathcal{Z}[\phi]$ can be considered as a partition function for $\phi$. But really $\mathcal{Z}[\phi]$ is describing a theory where the $\chi$ contributions have been included to all orders.
For example, consider the interaction term in the Lagrangian:
$$\frac{\lambda}{4} \phi^2 \chi^2.$$
This term can contribute at $\mathcal{O}(\lambda^2)$ to the $\phi$ 4-point vertex of the theory with 2 $\chi$ internal lines forming a loop. Integrating out the $\chi$ fields in the path integral would produce a genuine $\phi^4$ term (and others!) in $\mathcal{Z}[\phi]$ whose coefficient take into account all possible contributions from the previous $\chi$ fields.
Looking at the above example interaction term you can also see how this is different from setting $\chi=0$ in the action; It's not that we're erasing $\chi$ from the physics, we're repackaging the physics into a new $\mathcal{Z}[\phi]$.
For a more detailed discussion I'd highly recomment looking at David Skinner's AQFT notes, section 2.4.2 on p.27.
