Upper Bounds for the Scattering Amplitude at High Energy

Constraints imposed by unitarity and analyticity on the high-energy behavior of scattering amplitudes are studied in detail. The Froissart bound at nonforward (and nonbackward) angle is improved by two different methods: One is based on the analyticity in complex $cos\theta$ plane, and the other on the analyticity of the partial-wave amplitude in the complex angular-momentum plane. It is also shown that the Froissart bound at $\theta =0\mathrm{} or \pi$ cannot be improved insofar as one does not impose restrictions on the scattering amplitude other than unitarity and the analyticity in the $cos\theta$ plane. The scattering of particles with spin is discussed briefly.