Anomalous mass dimensions and Schwinger-Dyson equations boundary condition

Theories with large mass anomalous dimensions ($\gamma_m$) have been extensively studied because of their deep consequences for models where the scalar bosons are composite. Large $\gamma_m$ values may appear when a non-Abelian gauge theory has a large number of fermions or is affected by four-fermion interactions. In this note we provide a simple explanation how $\gamma_m$ can be directly read out from the IR and UV boundary conditions derived from the gap equation, and verify that moderate $\gamma_m$ values appear when the theory possess a large number of fermions, but large $\gamma_m$ values are obtained only when four-fermion interactions are added to the theory. We also verify how the critical line separating the different chiral phases emerge from these conditions.