Srednicki — Quantum Field Theory · Problem 73.2

The beta function captures how the gauge coupling $g$ runs with energy scale—it's determined by counting all the quantum loops that screen or anti-screen the force. In Yang–Mills theory, fermions and scalars contribute differently to this running because they couple to gluons with different strengths and loop topologies. Your job is to assemble the one-loop beta function by identifying the Casimir invariants $C_R$ for each representation and applying the standard formula, which sums fermion and scalar contributions with their respective numerical coefficients.

The beta function captures how the gauge coupling $g$ runs with energy scale—it's determined by counting all the quantum loops that screen or anti-screen the force. In Yang–Mills theory, fermions and scalars contribute differently to this running because they couple to gluons with different strengths and loop topologies. Your job is to assemble the one-loop beta function by identifying the Casimir invariants $C_R$ for each representation and applying the standard formula, which sums fermion and scalar contributions with their respective numerical coefficients.