One-loop massive scattering amplitudes and ward identities in string theory

We calculate bosonic open string one-loop massive scattering amplitudes for some low-lying string states. By using the periodicity relations of Jacobi theta functions, we explicitly prove an infinite number Of one-loop) type I stringy Ward identities derived from type 1. zero-norm states in the old covariant first quantized (OCFQ) spectrum of open bosonic string. The subtlety in the proofs of one-loop type 11 stringy Ward identities is discussed by comparing them with those of string-tree cases. High-energy limit of these stringy Ward identities can be used to Fix the proportionality constants between one-loop massive high-energy scattering amplitudes of different string states with the same momenta, These proportionality constants cannot be calculated directly from sample calculation as we did previously in the cases of string-tree scattering amplitudes.