A. Zee, Quantum Field Theory in a Nutshell · Problem III.6.4

Field Theory Redux

Physics — QFT

Problem

In chapter II.6, when discussing electron-proton scattering, we ignored the strong interaction that the proton participates in. Argue that the effects of the strong interaction could be included phenomenologically by replacing the vertex uˉ(P,S)γμu(p,s)\bar{u}(P, S)\gamma^\mu u(p, s) in (II.6.1) by ⟨P,S∣Jμ(0)∣p,s⟩=uˉ(P,S)[γμF1(q2)+iσμνqν2mF2(q2)]u(p,s)\langle P, S| J^\mu(0)|p, s\rangle = \bar{u}(P, S)\left[\gamma^\mu F_1(q^2) + \frac{i\sigma^{\mu\nu}q_\nu}{2m} F_2(q^2)\right] u(p, s)
Intuitive explanation

The key insight is that the proton isn't a point particle—it's a composite object with internal structure governed by the strong force. Rather than calculate those complicated strong interactions explicitly, you can absorb all their effects into two form factors, F1F_1 and F2F_2, that depend only on the momentum transfer q2q^2. The first term (F1F_1) modifies the familiar electromagnetic vertex, while the second (F2F_2) represents an anomalous magnetic moment contribution. This phenomenological approach lets you use experimental data to determine the form factors, then apply them to any electromagnetic probe without solving QCD.

Intuitive explanation

The key insight is that the proton isn't a point particle—it's a composite object with internal structure governed by the strong force. Rather than calculate those complicated strong interactions explicitly, you can absorb all their effects into two form factors, F1F_1 and F2F_2, that depend only on the momentum transfer q2q^2. The first term (F1F_1) modifies the familiar electromagnetic vertex, while the second (F2F_2) represents an anomalous magnetic moment contribution. This phenomenological approach lets you use experimental data to determine the form factors, then apply them to any electromagnetic probe without solving QCD.

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