Tree-amplitude factorization tests

Let P be the sum of a consecutive subset of incoming momenta in an ordered tree. Near an isolated simple massless pole, multiply the amplitude by P squared and then take the limit. Terms regular in that channel vanish in the product. The remaining residue describes two lower-point processes joined by an on-shell state. This limit is more informative than substituting a small arbitrary number for P squared and comparing large values.

Use a convention in which the analytic partial amplitude retains its Feynman i factors and the internal scalar propagator is i divided by P squared. The internal momentum entering the left amplitude is minus P, while the right amplitude receives plus P. Helicity labels are opposite across the sewing, with crossing phases chosen consistently. Omitting the propagator i or independently rephasing only one lower amplitude can change a residue's sign or complex phase.

An ordered channel must first be allowed by cyclic adjacency. Next, each lower-point amplitude must exist for the indicated species and helicities. A nominal pole can have zero residue because a required subamplitude vanishes. For complex three-point corners, the relevant angle or square branch must match the channel solution. These checks prevent counting every denominator in a possible diagram as a nonzero singularity of every helicity amplitude.

Spurious denominators can appear in intermediate representations, especially after solving momentum constraints or reducing a basis. Their residues must cancel in the complete expression if no corresponding physical channel exists. Tests should be made at generic points of the chosen pole surface, away from simultaneous additional singularities. Otherwise a special zero in another factor can conceal a wrong residue or make a legitimate pole appear absent.

A local four-field interaction contributes a term regular in all exchange channels. Adding it changes scattering at generic kinematics while preserving every exchange residue. Factorization alone consequently cannot determine its Wilson coefficient. Derivative power counting, symmetry, large-complex-momentum behavior, matching to heavier physics, or independent measurements may supply that missing information. Which input is appropriate depends on whether the target is a renormalizable theory or an effective expansion.

Gauge invariance supplies another useful diagnostic: replacing an external polarization by its momentum should annihilate a properly assembled physical amplitude. A candidate can pass one numerical test and fail this structural one. At loops, poles coexist with cuts, ultraviolet renormalization, and infrared singularities; the simple meromorphic-tree analysis becomes only part of the problem. Passing all intended tree tests establishes consistency within their assumptions, rather than a general proof of a complete quantum theory.