Bounds on the Diagonal Degree of the Biadjoint Scalar Amplitude Matrix

Biadjoint scalar theory partial amplitudes, $m_n(α,β)$, define an $(n-1)!/2\times (n-1)!/2$ 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 $n\to \infty$. More explicitly, we establish the bounds $\exp(-1.5n)\leqΔ_n/(n-3)!\leq \exp \left[ -0.01n+O(1)\right]$.