A. Zee, Quantum Field Theory in a Nutshell · Problem VIII.3.6

Field Theory Redux

Physics — QFT

Problem

Use the effective field theory approach to show that the scattering cross section of light on an electrically neutral spin 12\frac{1}{2} particle (such as the neutron) goes like σ∝ω2\sigma \propto \omega^2 to leading order, not ω4\omega^4. Argue further that the constant of proportionality can be fixed in terms of the magnetic moment μ\mu of the particle.
Intuitive explanation

In effective field theory, you treat the neutral particle's internal structure as a black box and expand interactions in powers of the photon energy ω\omega. The key insight is that a neutral particle has no electric dipole moment, so the leading electromagnetic coupling comes from the magnetic dipole interaction—not the electric quadrupole, which would give ω4\omega^4. The magnetic moment μ\mu couples to the photon field with strength proportional to ω2\omega^2, setting both the scaling and the overall coefficient of the cross section.

Intuitive explanation

In effective field theory, you treat the neutral particle's internal structure as a black box and expand interactions in powers of the photon energy ω\omega. The key insight is that a neutral particle has no electric dipole moment, so the leading electromagnetic coupling comes from the magnetic dipole interaction—not the electric quadrupole, which would give ω4\omega^4. The magnetic moment μ\mu couples to the photon field with strength proportional to ω2\omega^2, setting both the scaling and the overall coefficient of the cross section.

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