A theoretical preprint places the familiar scattering equations at the lowest level of a much larger ladder of mathematical relations called generalized K-identities. In the paper's stated aim, the authors seek to prove the proposed general K-identities for one-tensor hard string scattering amplitudes, or HSSA, and introduce a G function that generates an infinite set of them. The analysis is built from symbolic equations and saddle-point expressions, with a general n-point treatment and worked examples.
The central result is a linear link between a quantity called I_k and the terms E_i that define the saddle-point equations. For a fixed vertex k, I_k is a weighted sum of the E_i terms, with weights built from differences between the associated worldsheet coordinates and a factor of 2. At a saddle point, where the E_i terms vanish, the paper states that I_k vanishes as well.
Building a hierarchy from saddle points
For cases with five or more points, the authors avoid solving the higher-order algebraic saddle-point equations and instead use the saddle-point equations directly. Their derivation also applies a partial-fraction decomposition to each pair of terms with one index greater than the other. Together, these steps let the authors handle the general relations without first finding explicit solutions for every saddle point.
The next step is to package the saddle-point terms into G(z), an auxiliary function whose terms have denominators formed from the difference between an auxiliary coordinate and each worldsheet coordinate, with 2E_i in the numerator. The paper uses G(z) as an off-shell generator of generalized K-identities, meaning the construction can be discussed away from the saddle point. It is intended to produce an infinite hierarchy rather than a single relation.
Read in order, the hierarchy gives the paper's main conceptual identification: the lowest-order generalized K-identities are the scattering equations, while the next level reproduces the K-identities used elsewhere in the analysis. The result puts both sets of relations into one formal sequence.
The paper also rederives a K-identity through string theory by matching two HSSA ratio expressions. That matching is reported to recover the identity for the second tensor vertex, labelled j = 2. In the related zero-norm-state-decoupling discussion, the ratios are stated to remain valid at all string loop orders.
A six-point illustration
To show how the framework bears on stringy scaling, the authors work through a six-point case with three transverse directions, five tachyon states and one tensor state. In that configuration, the relevant ratios depend on two kinematic variables instead of eight. The reported stringy-scaling deficit, written as dimM1, is therefore six.
For general one-tensor HSSA, the paper gives the reported stringy-scaling degree as , where n is the number of external string-scattering points and r is the number of transverse directions. The expression is presented for the specified one-tensor setting and extends the pattern illustrated by the six-point calculation.
What the calculation does not settle
The analysis also connects the identities with the CHY formalism. In that discussion, the paper states that only n minus 3 of the n scattering equations are independent, and that the equations have a number of solutions equal to the factorial of n minus 3. This gives the proposed hierarchy a place within a broader formulation of scattering equations.
The result remains tied to the hard scattering limit used in the derivation. The authors expect higher-order generalized K-identity sets to help calculate other HSSA, but those applications remain an expectation rather than a demonstrated calculation in the supplied work. The six-point scaling numbers likewise belong to the particular configuration described in the paper.
The document is identified as arXiv:2608.28311v2 and dated 12 September 2026. Its main contribution is a formal derivation of the identity framework; whether the higher-order members can deliver calculations beyond the cases treated remains an open question.
Paper data and sources
Original title: Scattering Equations as the lowest order K-identities in the calculation of Stringy Scaling of Hard String Scattering Amplitudes
Authors: Sheng-Hong Lai, Jen-Chi Lee, Yi Yang
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
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