  We study graviton-graviton scattering in partial-wave amplitudes after
unitarizing their Born terms. In order to apply $S$-matrix techniques, based on
unitarity and analyticity, we introduce an $S$-matrix associated to this
resummation that is free of infrared divergences. This is achieved by removing
the diverging phase factor calculated by Weinberg that multiplies the $S$
matrix, and that stems from the virtual infrared gravitons. A scalar
graviton-graviton resonance with vacuum quantum numbers is obtained as a pole
in the nonperturbative $S$-wave amplitude, which is called the graviball. Its
resonant effects along the physical real $s$-axis may peak at values
substantially lower than the UV cutoff squared of the theory, similarly to the
$\sigma$ resonance in QCD. These techniques are also applied to study
nonrelativistic Coulomb scattering up to next-to-leading order in the
unitarization program. A comparison with the exact known solution is very
illuminating.
