Some Constraints on Partial Waves of Helicity Amplitudes which Follow from Analyticity, Unitarity, and Crossing Symmetry

Some inequalities involving finite numbers of partial-wave helicity amplitudes are derived for the elastic scattering process $\mathrm{ab}\ensuremath{\rightarrow}\mathrm{ab}$ (arbitrary spins and masses). One set of inequalities involves algebraic combinations of $t$-channel ($a\overline{a}\ensuremath{\rightarrow}b\overline{b}$) partial-wave helicity amplitudes and holds for any value of $t$ between 0 and $4{\ensuremath{\mu}}^{2}$ ($\ensuremath{\mu}$ is the lesser mass of the two particles involved in the scattering). A second set places restrictions on integrals over $s$-channel ($\mathrm{ab}\ensuremath{\rightarrow}\mathrm{ab}$) partial-wave helicity amplitudes. Finally, the above relations are applied to the particular case of ${\ensuremath{\pi}}^{\ensuremath{\circ}}$-nucleon elastic scattering, where inequalities among partial-wave helicity amplitudes are obtained.