Analytic structure of the high-energy gravitational amplitude. Multi-h diagrams and classical 5pm logarithms

We investigate the high-energy, small-angle limit of two-body gravitational scattering. Using power counting arguments and dispersion relations in an effective field theory for the Regge regime, we derive the general loop expansion that determines how the leading Regge logarithms and their complex structure arise as a power series in t/s. Focusing on the tower of multi-H diagrams that govern the leading logarithmic behaviour, we compute the leading double logarithm at four loops (or equivalently fifth post-Minkowskian order, 5PM) using both effective field theory methods and the multi-Regge expansion, finding complete agreement. Finally, using the aforementioned dispersion relations, we extract the single logarithmic contribution to the imaginary part of the eikonal phase at 5PM in the Regge limit.