Mass as a coupling and mass as a Casimir operator

In Poincare group, we consider mass as a Casimir of the group. Hence it is a constant in various frames (I do not mean old fashion Lorentz transformation). But, in the quantum field theory mass is the coefficient of the field square term, and could be considered as a coupling constant.
What is the relation between these two viewpoints? And if we consider the mass as a coupling then it could be run after renormalization. How this running one could be related to the Casimir operator?