Analyticity Constraints on Unequal-Mass Regge Formulas

A Regge-pole formula is derived for the elastic scattering of two unequal-mass particles that combines desirable $l$-plane analytic properties (i.e., a simple pole at $l=\alpha$ in the right-half $l$ plane) and Mandelstam analyticity. It is verified that such a formula possesses the standard asymptotic Regge behavior $u^{\alpha \mathrm{}\left(s\right)\mathrm{}}$ even in regions where the cosine of the scattering angle of the relevant crossed reaction may be bounded. The simultaneous requirements of $l$-plane and Mandelstam analyticity enforce important constraints, and the consistency of these constraints is studied. These considerations lead to the appearance of a "background" term proportional asymptotically to $u^{\alpha \mathrm{}\left(0\right)-1\mathrm{}}$ which has no analog in the equal-mass problem. We also conclude that a necessary condition for consistency is $\alpha \left(\infty \right)<\mathrm{}0\mathrm{}$.