Write the position-dependent factor as $e^f$, with
$$
f=\alpha'\sum_{j<l}p_j\cdot p_l\log|z_j-z_l|.
$$
In the fixed-angle hard-scattering limit the integral is governed by stationary points. Differentiation gives the <scattering equation>[scattering equations]
$$
\boxed{\sum_{j\ne i}\frac{p_i\cdot p_j}{z_i-z_j}=0},
\qquad
\boxed{\sum_{j\ne i}\frac{p_i\cdot p_j}{\bar z_i-\bar z_j}=0}.
$$
Only $m-3$ complex equations are independent after quotienting by $SL(2,\mathbb C)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 306 / 3 / b / Solution

{wiki=CHY_formalism}

For null or hard external momenta, the scattering equations are
$$
\sum_{j\ne i}\frac{p_i\cdot p_j}{z_i-z_j}=0.
$$
They arise as stationary-point equations for the Koba-Nielsen exponent and are covariant under projective transformations of the insertion points.


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