Linear relations and their breakdown in high energy massive string scatterings in compact spaces

We calculate high energy massive scattering amplitudes of closed bosonic string compactified on the torus. For each fixed mass level with given quantized and winding momenta (mR,12nR), we obtain infinite linear relations among high energy scattering amplitudes of different string states. For some kinematic regimes, we discover that linear relations with NR=NL break down and, simultaneously, the amplitudes enhance to power-law behavior instead of the usual exponential fall-off behavior at high energies. It is the space-time T-duality symmetry that plays a role here. This result is consistent with the coexistence of the linear relations and the softer exponential fall-off behavior of high energy string scattering amplitudes as we pointed out previously. It is also reminiscent of our previous work on the power-law behavior of high energy string/domain-wall scatterings.