It is proved that a two-body reaction amplitude involving scalar particles and satisfying Mandelstam's representation is bounded by expressions of the form $\mathrm{Cs}{\ln }^{2}s$ at the forward and backward angles, and $C{s}^{\frac{3}{4}}{\ln }^{\frac{3}{2}}s$ at any other fixed angle in the physical region, $C$ being a constant, $s$ being the total squared c.m. energy. This corresponds to cross sections increasing at most like ${\ln }^{2}s$. These restrictions limit the freedom of choice of the subtraction terms to six arbitrary single spectral functions and one subtraction constant.

• Received 27 March 1961

DOI:https://doi.org/10.1103/PhysRev.123.1053