High-energy unitarity of gravitation and strings

It is known that the behavior of a four-point string amplitude at large center-of-mass energy {}s and fixed momentum transfer q= {}-t is not perturbative. We study this region of phase space by summing multiple Reggeized graviton exchange in the eikonal approximation in D space-time dimensions. It is argued that the eikonal sum is at least representative of the summation of the leading powers of s in a string theory. The masslessness and high spin of the (Reggeized) graviton determine the character of the result. For κ²{sq}^{D{{-}}4}1 (κ is the gravitational coupling), the eikonal amplitude is dominated by single Reggeized graviton exchange. The amplitude in the region{{κ}}²sq^D-4{}1 is quite nonperturbative in character: simple Regge behavior and the Froissart bound are violated, and the amplitude does not satisfy a fixed-momentum-transfer dispersion relation. Although order by order the amplitude exhibits in q² the exponential decrease of Regge behavior, the final amplitude has only power-law falloff dependent on the number of space-time dimensions but independent of the Regge slope. The unitarity of the partial-wave projections of the eikonal amplitude is also studied. It is demonstrated that for D{≥}4 noncompact dimensions, the partial-wave amplitudes are bounded as s{→}{∞} only for large values of angular momentum, l{}x₀ {}s , where x₀ is the dominant value of the impact parameter. A heuristic argument is presented that the eikonal approximation is successful in unitarizing Reggeized graviton exchange as t/s{→}0 in four dimensions but not in higher dimensions.