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 ${\kappa }^2$ ${\mathrm{sq}}^{D\mathrm{-}4}$≲1 (κ is the gravitational coupling), the eikonal amplitude is dominated by single Reggeized graviton exchange. The amplitude in the region ${\kappa }^2$ ${\mathrm{sq}}^{D\mathrm{-}4}$≫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 $q^2$ 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≳$x_0$ √s , where $x_0$ 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.