Stringy scaling of multi-tensor hard string scattering amplitudes and the K-identities

We calculate n-point hard string scattering amplitudes (HSSA) with n − 2 tachyons and 2 tensor states at arbitrary mass levels. We discover the stringy scaling behavior of these HSSA. It is found that for HSSA with more than 2 transverse directions, the degree of stringy scaling dim\(\mathcal{M}\)₂ decreases comparing to the degree of stringy scaling dim\(\mathcal{M}\)₁ of the n − 1 tachyons and 1 tensor HSSA calculated previously. Moreover, we propose a set of K-identities which is the key to demonstrate the stringy scaling behavior of HSSA. We explicitly prove both the diagonal and off-diagonal K-identities for the 4-point HSSA and give numerical proofs of these K-identities for some higher point HSSA.