A student draws all possible Feynman diagrams for a process at a given order but gets the wrong amplitude. They used the correct propagators and vertex factors. What is the most likely systematic error?
AThey forgot to include the symmetry factor, which accounts for the number of ways the diagram can be drawn from the same set of contractions
BThey forgot to include the phase factor from Lorentz transformations
CThey used the wrong metric signature
DThey forgot to sum over all possible orderings of the external particles
Symmetry factors are the most common source of error in Feynman diagram calculations. A symmetry factor S arises when a diagram has internal symmetries — permutations of internal lines and vertices that leave the diagram unchanged. The amplitude must be divided by S to avoid overcounting. For example, a self-energy loop with two identical propagators has S = 2. The vacuum bubble diagram in phi^4 theory (a single vertex with two loops) has S = 8. Forgetting symmetry factors gives amplitudes that are too large by a factor of S.
Question 2 Multiple Choice
In QED, the vertex factor is -ie gamma^mu. Each QED Feynman diagram with n vertices therefore contains a factor of e^n = (sqrt{4 pi alpha})^n. Why does this mean higher-order diagrams give smaller corrections?
ABecause gamma matrices become smaller at higher powers
BBecause alpha = e^2/(4 pi) is approximately 1/137, so each additional vertex introduces a suppression factor of roughly 1/137
CBecause momentum conservation at each vertex reduces the available phase space
DBecause higher-order diagrams have more internal lines which suppress the amplitude
Each QED vertex contributes a factor of e, and each loop introduces an additional factor of alpha = e^2/(4 pi) approximately equal to 1/137. A diagram with L loops is suppressed by alpha^L relative to the tree-level diagram. This is why perturbation theory works so well for QED: each successive order in alpha gives a correction roughly 137 times smaller than the previous one. The spectacular agreement between QED predictions and experiment (the electron g-2 is verified to 12 significant figures) is a direct consequence of alpha being small.
Question 3 True / False
Disconnected Feynman diagrams (diagrams with pieces not connected to any external line) contribute to physical scattering amplitudes.
TTrue
FFalse
Answer: False
Disconnected diagrams factorize into a connected part (involving the external particles) times vacuum bubble diagrams (closed loops with no external lines). The vacuum bubbles contribute an overall phase factor e^{iW} to the S-matrix, where W is the sum of all vacuum diagrams. This phase cancels when you compute physical quantities like cross sections and decay rates, which depend on |S-matrix element|^2. The linked cluster theorem guarantees that only connected diagrams contribute to the physically relevant connected S-matrix elements.
Question 4 Short Answer
State the complete set of Feynman rules for scalar QED (a complex scalar field coupled to the electromagnetic field) at tree level, and explain what each rule represents physically.
Think about your answer, then reveal below.
Model answer: External lines: each incoming/outgoing scalar contributes a factor of 1 (for the standard normalization); each external photon contributes a polarization vector epsilon^mu(k). Internal lines (propagators): scalar propagator i/(p^2 - m^2 + i epsilon) for each internal scalar line; photon propagator -i g_{mu nu}/(k^2 + i epsilon) in Feynman gauge for each internal photon line. Vertices: the scalar-scalar-photon vertex gives -ie(p + p')^mu where p and p' are the momenta of the two scalar lines; the scalar-scalar-photon-photon (seagull) vertex gives 2ie^2 g^{mu nu}. At each vertex, impose momentum conservation. For each internal momentum not fixed by conservation, integrate d^4p/(2pi)^4. Divide by the symmetry factor.
Each rule has a direct physical origin. Propagators describe free-particle propagation between interactions. Vertex factors encode the strength and structure of the interaction — they come from the interaction terms in the Lagrangian. Momentum conservation at vertices reflects translational invariance. The integration over undetermined momenta sums over all possible virtual particle momenta. The Feynman rules are a precise algorithm for turning the physical content of the Lagrangian into numerical predictions.