Student Friendly Introduction to QFT · Problem 13.5

The key insight is recognizing that the photon self-energy $\Pi_c(k^2)$ contains a series expansion in powers of $k^2$, but physical photons satisfy the on-shell condition $k^2 = 0$. This means only the constant (zeroth-order) term in the expansion survives—the higher-order $k^2$ corrections vanish. By carefully extracting this leading term and applying the same renormalization logic used in (13-60), you can isolate the physical, finite contribution to the amplitude while the divergent parts cancel through charge renormalization.

The key insight is recognizing that the photon self-energy $\Pi_c(k^2)$ contains a series expansion in powers of $k^2$, but physical photons satisfy the on-shell condition $k^2 = 0$. This means only the constant (zeroth-order) term in the expansion survives—the higher-order $k^2$ corrections vanish. By carefully extracting this leading term and applying the same renormalization logic used in (13-60), you can isolate the physical, finite contribution to the amplitude while the divergent parts cancel through charge renormalization.