  We investigate from a mathematical perspective how Feynman amplitudes appear
in the low-energy limit of string amplitudes. In this paper, we prove the
convergence of the integrands. We derive this from results describing the
asymptotic behavior of the height pairing between degree-zero divisors, as a
family of Riemann surfaces degenerates. These are obtained by means of the
nilpotent orbit theorem in Hodge theory.
