It is well-known that interacting QFTs with conformal symmetry preserved at the quantum level have a vanishing $\beta$-function. Another common statement is that mass terms break conformal invariance.
The free massless scalar field is conformal, and adding a mass term breaks the conformal invariance. Although it is still commonly called "free", I think you could view the mass term $m^2 \phi \phi$ as an interaction term between two scalar fields. After all, what is the difference between $m^2 \phi^2$ and $\lambda \phi^4$, apart from the number of fields being coupled in the vertex?
So is there some sort of a $\beta$-function related to the mass? If yes, how would it look like? Or does the statement above only make sense when there is a coupling constant for $n \geq 3$ fields for some reason?