QED has three types of divergent diagrams: the electron self-energy, the vacuum polarization, and the vertex correction. A student asks: 'What if I compute a two-loop diagram and find a new type of divergence not related to these three?' Why can't this happen?
ABecause two-loop diagrams are always finite in QED
BBecause any divergent subdiagram in a higher-order graph must itself be one of these three types — higher-order divergences are nested combinations of the same three primitive divergences, and the counterterms that cancel them at one loop also cancel them at all loops
CBecause QED only has one coupling constant
DBecause gauge invariance forbids any new divergent structures
This is the content of renormalizability. In QED, power counting shows that only three superficially divergent Green's functions exist: the fermion two-point function (self-energy), the photon two-point function (vacuum polarization), and the fermion-photon three-point function (vertex). Any higher-order diagram either is convergent or contains one of these three as a divergent subdiagram. The counterterms (delta_m for mass, delta_Z for field renormalization, delta_e for charge) introduced at one loop absorb the corresponding divergences at every order. This is proven rigorously by the BPHZ theorem.
Question 2 True / False
The bare electric charge e_0 in the QED Lagrangian is infinite, and only the renormalized charge e_R (defined at a specific momentum scale) is finite and measurable.
TTrue
FFalse
Answer: True
The bare charge e_0 absorbs the ultraviolet divergence from the vacuum polarization. It is formally infinite (or cutoff-dependent if you use a regulator), and it is not directly measurable. The physical charge e_R is defined by a renormalization condition — typically the value of the photon-fermion vertex at a specific momentum transfer. The relation is e_0 = Z_3^{-1/2} e_R, where Z_3 contains the divergent vacuum polarization contribution. All physical predictions depend only on e_R (which equals the measured value of the fine structure constant at the chosen scale), not on e_0.
Question 3 Multiple Choice
The electron's anomalous magnetic moment a_e = (g-2)/2 is one of the most precisely tested predictions in physics. The leading QED correction (Schwinger's result) gives a_e = alpha/(2 pi). What makes this prediction so remarkable?
AIt is the only prediction that does not require renormalization
BIt is a finite, unambiguous prediction from a single one-loop diagram (the vertex correction) that agrees with experiment to extraordinary precision — and the agreement improves as higher-order corrections are included, validating the entire renormalization program
CIt was the first prediction of quantum mechanics
DIt shows that the electron is a point particle
The anomalous magnetic moment is computed from the vertex correction diagram. After renormalization, the result alpha/(2pi) approximately 0.00116 is finite and parameter-free (it depends only on the already-measured value of alpha). The current theoretical value includes contributions through five-loop order (tenth order in alpha) and agrees with the experimental measurement to better than one part in 10^12. This agreement validates not just the one-loop calculation but the entire perturbative framework and renormalization procedure at extraordinary precision.
Question 4 Short Answer
Explain the physical meaning of wave function renormalization Z_2 for the electron field, and why it is necessary even though it is not directly observable.
Think about your answer, then reveal below.
Model answer: Z_2 is the field strength renormalization factor: psi_0 = sqrt(Z_2) psi_R, where psi_0 is the bare field and psi_R is the renormalized field. Physically, Z_2 accounts for the fact that a bare electron continuously emits and reabsorbs virtual photons — the 'physical electron' is dressed by a cloud of virtual photons. The probability of finding the bare electron inside the physical electron is Z_2 (which is less than 1 and divergent in perturbation theory). Z_2 is not directly observable because it cancels between the LSZ reduction formula and the vertex renormalization (this cancellation is guaranteed by the Ward identity Z_1 = Z_2). However, it must be included for intermediate calculations to be consistent.
The Ward identity Z_1 = Z_2 is a consequence of gauge invariance and is one of the most important relations in QED. It ensures that the electric charge is renormalized only by the vacuum polarization (Z_3), not by the vertex or self-energy corrections. This is why all charged particles (regardless of spin or mass) have their charges renormalized by the same factor — the universality of charge renormalization.