One-Loop Massive Scattering Amplitudes and Ward Identities in String Theory

We calculate bosonic open string one-loop massive scattering amplitudes for some lowlying 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 I zeronorm states in the old covariant first quantized (OCFQ) spectrum of open bosonic string. The subtlety in the proofs of one-loop type II 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.