# Anomalous mass dimensions and Schwinger-Dyson equations boundary condition

## TL;DR

This paper explains how to determine the mass anomalous dimension from boundary conditions in gap equations, showing large values arise with four-fermion interactions, impacting models with composite scalars.

## Contribution

It provides a straightforward method to extract the anomalous dimension from IR and UV boundary conditions and clarifies the conditions leading to large anomalous dimensions.

## Key findings

- Large $\gamma_m$ occurs with many fermions.
- Four-fermion interactions induce large $\gamma_m$.
- Critical line separates chiral phases.

## Abstract

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.