An Introduction To Quantum Field Theory - M. Peskin, D. Schroeder (Perseus, 1995) · Problem 16.2

The key insight is that minimal coupling to a non-Abelian gauge field works exactly like QED, but with group generators $T^a$ replacing the single $U(1)$ charge. For part (a), the scalar propagator stays the same, but each vertex gains a factor of $igT^a$ (one for each attached gluon leg), just as QED has $ieQ$. Part (b) requires computing the one-loop box diagram where the scalar circulates in the loop—this contributes to the running coupling through the Casimir invariant $C(r)$ of the representation, which quantifies how strongly the scalar "feels" the gauge field. The $\beta$ function combines this scalar contribution with the pure-gauge contribution from gluon loops (proportional to $C_2(G)$), showing how different matter content shifts the theory's coupling evolution.

The key insight is that minimal coupling to a non-Abelian gauge field works exactly like QED, but with group generators $T^a$ replacing the single $U(1)$ charge. For part (a), the scalar propagator stays the same, but each vertex gains a factor of $igT^a$ (one for each attached gluon leg), just as QED has $ieQ$. Part (b) requires computing the one-loop box diagram where the scalar circulates in the loop—this contributes to the running coupling through the Casimir invariant $C(r)$ of the representation, which quantifies how strongly the scalar "feels" the gauge field. The $\beta$ function combines this scalar contribution with the pure-gauge contribution from gluon loops (proportional to $C_2(G)$), showing how different matter content shifts the theory's coupling evolution.