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The Soft graviton theorem factorizes an amplitude with one low-energy graviton into the hard amplitude times a universal sum over external momenta. Gauge invariance of the soft factor requires energy-momentum conservation and universal gravitational coupling.

A supersymmetric gauge theory combines vector superfields with charged matter superfields so that gauge invariance and supersymmetry are both manifest.

An amplitude is ultraviolet finite if its regulated high-loop momentum contribution has a finite regulator-removal limit after whatever lower-order subtractions are specified. A zero superficial degree of divergence does not force divergence: gauge invariance can cancel the candidate local term. Conversely, superficial convergence alone does not exclude divergent subgraphs. Ultraviolet finiteness makes no claim about infrared divergences or physical threshold singularities.

For an on-shell amplitude $\mathcal{M}={ϵ}_{\mu }(k){\mathcal{M}}^{\mu }$ with an external photon, gauge invariance implies $k_{\mu }{\mathcal{M}}^{\mu }=0$.

In a gauge-preserving regularization and consistent renormalization convention, the proper-vertex Ward-Takahashi identity in QED equates the vertex and electron wave-function renormalization constants. With ${\psi }_0=Z_2^{1/2}\psi$ and $A_0=Z_3^{1/2}A$, the vertex coefficient gives $e_0={\mu }_R^{ϵ}{Z}_{1}e/(Z_2\sqrt{Z_3})$, proving the displayed charge relation in $d=4-2ϵ$. Gauge invariance restricts counterterms but does not forbid charge running.

A hyperon decay that changes quark flavour while emitting a photon. An example is ${\Xi }^0\to \Lambda \gamma$, which changes strangeness and cannot be purely electromagnetic. Its amplitude can contain short-distance loops and long-distance hadronic contributions. photon attachments to a weak quark-exchange graph must be completed consistently with electromagnetic gauge invariance. A quark topology identifies allowed flavour flow but does not by itself calculate a reliable hadronic rate.