Factorization, Kinematic Singularities, and Conspiracies

Factorization of residues of poles of the $S$ matrix is derived from the requirements of unitarity for partialwave helicity amplitudes. Careful attention is given to questions of spin and the kinematic singularities of the relevant amplitudes, especially at $t=0\mathrm{}$. Residues $\beta$ of a pole in the full partial-wave amplitude satisfy factorization in the simple form ${{\beta }_{\mathrm{ba}}}^2={\beta }_{\mathrm{aa}}{\beta }_{\mathrm{bb}}$. In general, $\beta$ can be written as $\beta =K\gamma$, where $K$ contains the standard kinematic singularities of the Hara-Wang type, plus threshold behavior, and $\gamma$ is a reduced residue. The $K\text{'}\mathrm{s}$ for various mass classes are exhibited in a compact and consistent form, and the corresponding factorization statements for the reduced residues are derived. These factorization relations are of the form $t^x{{\gamma }_{\mathrm{ba}}}^2={\gamma }_{\mathrm{aa}}{\gamma }_{\mathrm{bb}}$, where $x$ is an integer. The reduced residues are analytic in the neighborhood of thresholds and pseudothresholds, but may, in the case of conspiracies, contain poles at $t=0\mathrm{}$. Various examples are presented to illustrate the use of factorization. These include LeBellac's argument on the behavior of the pion residue at $t=0\mathrm{}$ and its circumvention with a type-II conspiracy. mandelstam's treatment of Adler's self-consistency condition and the hypothesis of partially conserved axial-vector current using an $M=1\mathrm{}$ pion is discussed from the viewpoint of factorization. It is shown that factorization for an $M=1\mathrm{}$ pion seems to imply smallness of both soft-pion and hard-pion amplitudes. The smallness of the latter casts some doubt on the $M=1\mathrm{}$ assignment for the pion. The nature of the relations between amplitudes and the behavior of the reduced residues at $t=0\mathrm{}$ for conspiracies with unequal masses is also considered.