In quantum electrodynamics, Furry's theorem states that if a Feynman diagram consists of a closed loop of fermion lines with an odd number of vertices, its contribution to the amplitude vanishes.
The theorem was first derived by Wendell H. Furry in 1937,[1] as a direct consequence of the conservation of energy and charge conjugation symmetry.
photon operators, this argument shows that under charge conjugation this picks up a factor of
, Furry's theorem states that the correlation function of any odd number of on-shell or off-shell photon fields and/or currents must vanish in quantum electrodynamics.
[3] At leading order this means that any fermion loop with an odd number of vertices must have a vanishing contribution to the amplitude.
The vanishing of the three vertex loop can also be seen as a consequence of the renormalizability of quantum electrodynamics since the bare Lagrangian does not have any counterterms involving three photons.
[4] Furry's theorem allows for the simplification of a number of amplitude calculations in quantum electrodynamics.
[5] In particular, since the result also holds when photons are off-shell, all Feynman diagrams which have at least one internal fermion loops with an odd number of vertices have a vanishing contribution to the amplitude and can be ignored.
[6] It also does not apply in the presence of a strong background magnetic field where photon splitting interactions
are allowed, a process that may be detected in astrophysical settings such as around neutron stars.
In particular, the non-vanishing of the triangle diagram with Weyl fermions gives rise to the chiral anomaly, with the sum of these having to cancel for a quantum theory to be consistent.
While the theorem has been formulated in quantum electrodynamics, a version of it holds more generally.
For example, while the Standard Model is not charge conjugation invariant due to weak interactions, the fermion loop diagrams with an odd number of photons attached will still vanish since these are equivalent to a purely quantum electrodynamical diagram.
It is however no longer true that all odd number photon diagrams need to vanish.
For example, relaxing the requirement of charge conjugation and parity invariance of quantum electrodynamics, as occurs when weak interactions are included, allows for a three-photon vertex term.
[9] In non-Abelian Yang–Mills theories, Furry's theorem does not hold since these involve noncommuting color charges.
For example, the quark triangle diagrams with three external gluons are proportional to two different generator traces
[10][11] However, charge conjugation arguments can still be applied in limited cases such as to deduce that the triangle diagram