Final answer:
The zero contribution of the loop-level three-photon vertex in QED is a result of gauge invariance which would change if the invariance was broken. Breaking gauge invariance could lead to massive photons and could permit new processes, potentially impacting the predictions of Quantum Electrodynamics as well as other theories like GUTs.
Step-by-step explanation:
The cancellation of loop-level three-photon vertex contributions in QED is indeed tied to its gauge invariance. If gauge invariance were broken—for example, in scenarios with spontaneous symmetry breaking—the photon could acquire mass. In that case, not only would its mass modify the interactions but the inclusion of longitudinal polarizations could also potentially allow for processes that otherwise vanish due to gauge invariance. However, in Quantum Electrodynamics (QED), which is the quantum field theory of electromagnetism, photons are massless and therefore only have two transverse polarization states.
In theories such as the Grand Unified Theories (GUTs), at very high energies (around 100 GeV and above), the electromagnetic and weak forces unify, and the gauge bosons (like W and Z bosons) act like massless particles such as photons and gluons. These considerations tie into the range of different forces; massive carrier particles like the W and Z bosons, due to their mass, mediate forces with very short range compared to the long-range electromagnetic force, which is mediated by massless photons. If we were to break the gauge invariance in QED, we might expect to see new types of contributions to various processes—including those to a three-photon vertex—if it were allowed by the new symmetries of the theory.
In grand unified theories, breaking gauge invariance could have profound implications, such as the violation of lepton and baryon number conservation—and potentially observable effects like proton decay with extremely long lifetimes. While we do not typically see this due to the energies required being beyond our experimental reach, these theories predict phenomena that are fundamentally different from what is observed at lower energies where QED operates.