Permutation symmetry of the scattering equations
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Bibliographic record
Abstract
Closed formulas for tree amplitudes of $n$-particle scatterings of gluon, graviton, and massless scalar particles have been proposed by Cachazo, He, and Yuan. They depend on $(n\ensuremath{-}3)$ quantities ${\ensuremath{\sigma}}_{\ensuremath{\alpha}}$ which satisfy a set of coupled scattering equations, with momentum dot products as input coefficients. These equations are known to have $(n\ensuremath{-}3)!$ solutions; hence, each ${\ensuremath{\sigma}}_{\ensuremath{\alpha}}$ is believed to satisfy a single polynomial equation of degree $(n\ensuremath{-}3)!$. In this article, we derive the transformation properties of ${\ensuremath{\sigma}}_{\ensuremath{\alpha}}$ under momentum permutation and verify them with known solutions at low $n$, and with exact solutions at any $n$ for special momentum configurations. For momentum configurations not invariant under a certain momentum permutation, new solutions can be obtained for the permuted configuration from these symmetry relations. These symmetry relations for ${\ensuremath{\sigma}}_{\ensuremath{\alpha}}$ lead to symmetry relations for the $(n\ensuremath{-}3)!+1$ coefficients of the single-variable polynomials, whose correctness are checked with the known cases at low $n$. The extent to which the coefficient symmetry relations can determine the coefficients is discussed.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it