A new theoretical study has derived a formal link between saddle-point equations, the conditions that define the calculation's saddle point, and K-identities in one-tensor hard string-scattering amplitudes. The paper finds an off-shell linear relation between them, meaning the relation remains valid even when a relevant variable has not been placed at its saddle point. At the saddle point, vanishing saddle-point equations imply that every K-identity expression in the system is zero. The result offers a formal route to these identities without solving every higher-point saddle point.
The study works with formal amplitudes and equation systems rather than empirical participant data. Its conclusions are limited to the hard-scattering and saddle-point framework described in the derivation.
How the proof avoids a harder calculation
For cases involving five or more external points, the proof uses the saddle-point equations directly instead of solving the higher-order algebraic system. It also applies partial-fraction decomposition to each ordered pair of indices in which the first index is greater than the second. These are algebraic steps in the proof, not statistical tests.
A connection to string theory
The authors also connect the identities back to string theory. They rederive the identities at the saddle point by identifying the saddle-point calculation with zero-norm-state decoupling, a string-theory procedure used here to obtain relations among amplitudes. Ratios at each fixed mass level are calculated through that same zero-norm-state decoupling method.
That connection puts the algebraic proof and the string-theory calculation in the same formal picture. The supplied analysis reports no independent numerical or empirical validation, so the finding remains a result within the stated theoretical framework.
The equations form a hierarchy
A separate generating construction, called the G function, organizes the equations into an infinite hierarchy of generalized K-identities. At the lowest order, the hierarchy contains the scattering equations used in the CHY formalism, a framework built around those equations. At the next-to-leading order, it contains the K-identities used for one-tensor hard string-scattering amplitudes.
The construction begins with a Laurent expansion of the function K around each point in the calculation. In ordinary language, the expansion examines the function's behavior near each point and sorts the result into terms of different order, allowing the authors to arrange the equations on the same ladder.
The hierarchy's upper levels are where the paper becomes cautious. The authors conjecture that higher-order K-identities may help calculate next-to-leading-order hard string-scattering amplitudes or amplitudes involving multiple tensors. The work does not demonstrate those calculations, and their practical value in concrete problems remains an open question.
One example, with a clear limit
The proposed reduction in kinematic-variable dependence is illustrated with a six-point example involving three transverse directions. In that configuration, ratios depend on two variables rather than eight. The reported stringy-scaling deficit is six, the difference between the eight starting variables and the two retained variables.
That example should be read narrowly. It is a configuration-specific theoretical illustration, not evidence that every possible scattering setup will have the same reduction. The supplied analysis also leaves open how far the formal results extend beyond the hard-scattering and saddle-point assumptions.
What remains untested
The work is a version-one arXiv preprint dated 28 August 2026. Its main result is an analytic route for relating saddle-point equations, K-identities and scattering equations. Whether higher-order members can support calculations of higher-order or multi-tensor hard string-scattering amplitudes remains to be established.
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
Original paper · Full text