Bounds on the Diagonal Degree of the Biadjoint Scalar Amplitude Matrix
Abstract
Biadjoint scalar theory partial amplitudes, mn(α,β), define an (n−1)!/2× (n−1)!/2 matrix, Mn, with rows and columns labeled by orderings. The CHY construction of Mn implies that its rank is (n−3)!. Invertible submatrices of maximal rank enter in the KLT construction relating graviton amplitudes to gluon amplitudes. Motivated by finding the KLT formula with the smallest number of terms, one of the authors posed the question of whether there is a critical nc such that for all n>nc there exists a diagonal submatrix of Mn of rank (n−3)!. In this work we obtain bounds on the size of the largest non-singular diagonal submatrix of Mn, which we call its diagonal degree Δn, and we answer the question in the negative. We prove that the diagonal degree is exponentially suppressed relative to the CHY rank: Δn/(n−3)! = exp(−Θ(n)) as n→ ∞. More explicitly, we establish the bounds exp(−1.5n)≤Δn/(n−3)!≤ exp [ −0.01n+O(1)].
Related
BibTeX
@misc{cachazo2026bounds,
title = {{Bounds on the Diagonal Degree of the Biadjoint Scalar Amplitude Matrix}},
author = {Freddy Cachazo and Rajbir Longia},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.40355},
eprint = {2610.40355},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.40355},
}
RIS
TY - UNPB
TI - Bounds on the Diagonal Degree of the Biadjoint Scalar Amplitude Matrix
AU - Freddy Cachazo
AU - Rajbir Longia
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.40355
UR - https://arxiv.org/abs/2610.40355
AB - Biadjoint scalar theory partial amplitudes, m_n(α,β), define an (n−1)!/2× (n−1)!/2 matrix, M_n, with rows and columns labeled by orderings. The CHY construction of M_n implies that its rank is (n−3)!. Invertible submatrices of maximal rank enter in the KLT construction relating graviton amplitudes to gluon amplitudes. Motivated by finding the KLT formula with the smallest number of terms, one of the authors posed the question of whether there is a critical n_c such that for all n>n_c there exists a diagonal submatrix of M_n of rank (n−3)!. In this work we obtain bounds on the size of the largest non-singular diagonal submatrix of M_n, which we call its diagonal degree Δ_n, and we answer the question in the negative. We prove that the diagonal degree is exponentially suppressed relative to the CHY rank: Δ_n/(n−3)! = exp(−Θ(n)) as n→ ∞. More explicitly, we establish the bounds exp(−1.5n)≤Δ_n/(n−3)!≤ exp [ −0.01n+O(1)].
ER -