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

TL;DR

This paper advances the understanding of hard string scattering by computing $n$-point amplitudes with $n-2$ tachyons and two tensor states, uncovering a stringy scaling pattern in which the number of independent kinematic variables decreases with increasing transverse directions. Using saddle-point methods augmented by novel $K$-identities (diagonal and off-diagonal), the authors derive explicit scaling relations for four-point amplitudes and extend the framework to general $n$-point cases, confirming consistency with zero-norm state decoupling. They show that the degree of stringy scaling, $\dim\mathcal{M}_{k}$, shifts from $\dim\mathcal{M}_{1}$ to $\dim\mathcal{M}_{2}$ according to the transverse dimension $r$, with $\dim\mathcal{M}_{2}=\dim\mathcal{M}_{1}$ for $r=1,2$ and $\dim\mathcal{M}_{2}=\dim\mathcal{M}_{1}-r+2$ for $r\geq3$. The $K$-identities are validated analytically for $4$-point amplitudes and numerically for higher points, and are argued to reflect momentum conservation at the saddle point, offering a sturdy tool for probing high-energy string dynamics and its universal scaling properties.

Abstract

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 dimM2 decreases comparing to the degree of stringy scaling dimM1 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.