A one-loop Feynman diagram has one undetermined internal momentum that must be integrated over all values. The electron self-energy diagram in QED integrates over a loop momentum k from 0 to infinity. Why does this integral diverge?
ABecause the integrand oscillates without damping
BAt large k, the integrand falls off as 1/k^2 while the four-dimensional integration measure grows as k^3 dk — the integral diverges logarithmically because 1/k^2 times k^3 = k grows without bound
CBecause the electron propagator in the loop has a pole on the real axis
DBecause the photon is massless, creating an infrared divergence
In four spacetime dimensions, the loop integration measure is d^4k ~ k^3 dk (in spherical coordinates). The electron self-energy has a fermion propagator (~1/k) and a photon propagator (~1/k^2) in the loop, giving an integrand that falls as 1/k^3 at large k. Combined with the measure k^3 dk, the integral goes as dk/k ~ ln(k), which diverges logarithmically as k -> infinity. This is an ultraviolet divergence — it comes from arbitrarily high momenta (equivalently, arbitrarily short distances). Different diagrams have different power-counting behavior; some diverge quadratically, some logarithmically, and some are finite.
Question 2 True / False
Ultraviolet divergences indicate that QFT is mathematically inconsistent and must be replaced by a fundamentally different theory at high energies.
TTrue
FFalse
Answer: False
UV divergences do not mean the theory is inconsistent. They mean that the bare parameters in the Lagrangian (bare mass, bare charge) are not the physical parameters. Renormalization absorbs the divergences into redefinitions of these parameters, yielding finite predictions for all physical observables. The renormalized theory is perfectly well-defined and makes extraordinarily precise predictions (QED's electron g-2 agrees with experiment to 12 digits). What UV divergences may indicate is that the theory is an effective field theory — valid up to some energy scale but potentially replaced by a more complete theory at higher energies. But as a computational framework, renormalizable QFT is entirely self-consistent.
Question 3 Multiple Choice
The vacuum polarization diagram (a fermion loop inserted into a photon propagator) modifies the photon propagator. What is the physical effect of this modification?
AIt gives the photon a mass
BIt screens the bare electric charge at long distances — the effective charge is smaller at low energies and increases at higher energies (shorter distances), because virtual electron-positron pairs polarize the vacuum like a dielectric
CIt causes the photon to decay into electron-positron pairs
DIt violates gauge invariance
Virtual electron-positron pairs in the vacuum act as electric dipoles that partially screen the bare charge. At large distances (low energies), the screening is maximal and the measured charge is the familiar alpha ~ 1/137. At shorter distances (higher energies), you probe inside the screening cloud and see a larger effective charge. This is the running of the QED coupling constant. The photon does not acquire a mass — gauge invariance (enforced by the Ward identity) guarantees that the vacuum polarization tensor is transverse, which protects the photon's masslessness.
Question 4 Short Answer
Explain the difference between ultraviolet and infrared divergences, and give a physical example of each in QED.
Think about your answer, then reveal below.
Model answer: Ultraviolet divergences arise from loop momenta going to infinity (equivalently, distances going to zero) and indicate sensitivity to short-distance physics. Example: the electron self-energy, where a virtual photon is emitted and reabsorbed — the integral over the photon's momentum diverges logarithmically at high momentum. Infrared divergences arise from loop momenta going to zero (long distances) and are related to the emission of very soft (low-energy) photons. Example: the vertex correction in QED diverges logarithmically as the photon momentum goes to zero. IR divergences cancel when you include the corresponding real emission process (Bremsstrahlung) — the Bloch-Nordsieck theorem guarantees this cancellation for any process where you sum over all possible soft photon emissions.
UV and IR divergences have completely different origins and resolutions. UV divergences are handled by renormalization (redefining bare parameters). IR divergences cancel between virtual corrections and real emission when you compute physically measurable (inclusive) cross sections. The KLN (Kinoshita-Lee-Nauenberg) theorem generalizes this: all IR divergences cancel in sufficiently inclusive observables.