Scattering Equations: Algebra, Geometry & Applications

• Scattering equations are algebraic constraints that relate the kinematics of massless particles to the moduli space of punctured Riemann spheres.
• They underpin the CHY representation for tree-level S-matrix amplitudes by enabling efficient integrand reduction via companion matrix and elimination techniques.
• The equations reveal deep connections between positive geometries, string theory, and combinatorial algebraic structures, advancing our understanding of field theory amplitudes.

Scattering equations are a universal system of algebraic constraints that encode the relation between the kinematic data of massless particles and the geometry of the moduli space of $n$ marked points on the Riemann sphere. They serve as the backbone of the Cachazo–He–Yuan (CHY) representation for tree-level S-matrix amplitudes in a wide class of field theories. Beyond their computational utility, the scattering equations expose deep connections between positive geometries, string theory, symplectic geometry, and combinatorial algebraic geometry. Their explicit form, algebraic structure, and the broad spectrum of solution techniques reveal a rich interplay between physics and mathematics.

1. Foundational Formulation and Algebraic Structure

The core formulation of the scattering equations for $n$ massless particles is: $$E_a(\sigma) = \sum_{b\neq a} \frac{s_{ab}}{\sigma_a-\sigma_b} = 0,\quad a = 1,\ldots,n,$$ where $s_{ab}=(k_a+k_b)^2=2k_a\cdot k_b$ are Mandelstam invariants, and $\sigma_a$ are puncture coordinates on $\mathbb{CP}^1$, defined modulo $SL(2,\mathbb{C})$ transformations. Only $n-3$ equations are independent due to projective invariance and total momentum conservation $\sum_{a=1}^n k_a=0$. The system can also be formulated as a zero-dimensional ideal $\mathcal{I}=\langle E_1,\ldots,E_n\rangle$ of degree $(n-3)!$ in an appropriate coordinate ring (Lukowski et al., 2022).

Algebraically, the equations can be recast as multilinear homogeneous polynomials: $$h_m(z) = \sum_{|S|=m} (k_S)^2\prod_{a\in S} z_a = 0, \quad m=2,\ldots,n-2,$$ facilitating elimination theory and explicit degree counting via Bézout's theorem (Dolan et al., 2014, Dolan et al., 2015). This multilinear structure induces an H-basis (in the sense of Macaulay) for the ideal, enabling systematic integrand reduction and residue evaluation (Bosma et al., 2016).

2. Solution Counting, Geometric Interpretation, and Positive Geometries

For generic kinematics in $D$ dimensions, the number of inequivalent solutions is exactly $(n-3)!$ (Cachazo et al., 2013, Dolan et al., 2014). Geometrically, each solution to the scattering equations corresponds to a point in the moduli space $\mathcal{M}_{0,n}$, or equivalently to a configuration of $n$ marked points on the Riemann sphere fixed up to projective automorphisms.

Recent developments relate the scattering equations to positive geometries via pushforwards: canonical forms for kinematic polytopes such as the associahedron can be obtained as the image of the canonical form on worldsheet moduli space under the map defined by the scattering equations (Lukowski et al., 2022). The pushforward is defined as

$$\mathcal{I}_*(\omega) = \sum_{\xi\in V(\mathcal{I})} \xi^*(\omega),$$

where $\omega$ is a rational differential form, and the pushforward can be computed via chain rule and global residue theorems. For instance, in the biadjoint $\phi^3$ theory, the kinematic associahedron arises naturally from the worldsheet associahedron via this construction.

The scattering equations are further realized as the critical points of the master function

$\Phi(z) = \prod_{i<j}(z_i-z_j)^{s_{ij}}$

on the complement of the hyperplane arrangement in $\mathbb{C}^n$, providing a bridge between combinatorial algebraic geometry and physics (Betti et al., 4 Oct 2024).

3. Computational Methods and Elimination Techniques

Solving the scattering equations for large $n$ is nontrivial due to factorial scaling. Several algebraic and computational techniques have been developed:

• Companion Matrix Method: The zero-dimensional quotient ring $Q=\mathbb{C}(s)[\sigma]/\mathcal{I}$ of dimension $(n-3)!$ admits companion matrices $T_{i}$ for variables $\sigma_{i}$, with roots obtained as their joint eigenvalues. For any rational function $r(\sigma)$,

$$\sum_{\mathcal{I}=0} r(\sigma) = \mathrm{Tr}[r(T_1,\ldots,T_n)],$$

replacing explicit root-finding by linear algebra (Huang et al., 2015, Lukowski et al., 2022).

• Sylvester/Bézout Elimination: The system can be algebraically reduced to a single univariate resultant of degree $(n-3)!$ via Sylvester-type block Toeplitz matrices or Bézoutians. Their determinants yield the minimal polynomial for a chosen coordinate, and recursive constructions link the degrees and structures of all minors. Expansion in Plücker coordinates reveals underlying Grassmannian geometry (Cardona et al., 2015).
• Global Duality of Residues and Bezoutian Matrix: The sum over all solutions (global residue) can be efficiently computed by constructing the Bezoutian matrix, whose determinant encodes the required contractions. Notably, if the dual basis contains a constant polynomial, the global residue reduces to a single monomial coefficient extraction (Sogaard et al., 2015, Bosma et al., 2016).
• H-Basis for Integrand Reduction: The H-basis property allows any integrand polynomial to be reduced to degree at most $d^*=(n-3)(n-4)/2$, drastically simplifying residue computations. The global residue is determined solely by the coefficient of a single top-degree monomial in the remainder (Bosma et al., 2016).
• Numerical Algebraic Geometry Approaches: Homotopy continuation (both algebraic and "physical" in kinematic space), degeneration-based tracking, and Monte Carlo root-finding enable high-point or sector-decomposed explicit solution enumeration (Liu et al., 2018, Betti et al., 4 Oct 2024, Duhr et al., 2018, Farrow, 2018).

The following table summarizes core solution techniques:

Method	Algebraic Object	Key Property
Companion matrix	Quotient ring $Q$	Trace reproduces sum over solutions
Sylvester/Bézout	Resultant determinant	Eliminates to univariate polynomial
Bezoutian/global residue	$(n-3)\times(n-3)$ matrix	Pairing/dual basis structure
H-basis reduction	Polynomial ideal generators	Fast integrand reduction/residue

4. Applications: Amplitudes and Factorization

The scattering equations localize CHY-type worldsheet integrals for amplitudes in scalar, Yang–Mills, gravity, Einstein–Yang–Mills, and mixed-spin theories. The $n$-point tree amplitude is generally

$$\Omega(k) = \int_{\Gamma} \omega(\sigma) \prod_{a=1}^n \delta(E_a(\sigma)),$$

with $\omega(\sigma)$ encoding theory-specific information (Parke–Taylor, reduced Pfaffians). The standard pushforward formula reads

$$\Omega(k) = \sum_{\sigma^*:E(\sigma^*)=0} \frac{\omega(\sigma^*)}{\det\left[\frac{\partial E}{\partial \sigma}\right]\big|_{\sigma^*}}.$$

For Möbius-invariant integrands with simple poles, universal combinatorial rules bypass the need for explicit solution enumeration, associating each compatible set of internal poles with Feynman-diagram propagator structures (Baadsgaard et al., 2015).

The orthogonality property with respect to the KLT kernel ensures that Parke–Taylor vectors on different solutions are mutually orthogonal, underpinning the double-copy structure of gravity amplitudes (Cachazo et al., 2013).

In specific kinematic regimes (multi-Regge, quasi-multi-Regge), the solutions to the scattering equations reorganize to reflect the rapidity hierarchy, and corresponding factorization theorems become manifest (Duhr et al., 2018).

5. Connections to String Theory, Ambitwistor Strings, and Degenerations

The scattering equations emerge as the worldsheet saddle-point equations in the high-energy (Gross–Mende) limit of string theory, connecting the localization of string amplitudes to their field-theory counterparts. The CHY formalism can be recast as a $\delta$-function insertion of the saddle equations, yielding compact, resummed dual models that interpolate between string and field theory amplitudes in $\alpha'$ (Bjerrum-Bohr et al., 2014).

Ambitwistor string theory generalizes this connection, providing a chiral, infinite-tension worldsheet model whose correlation functions are localized to scattering equation solutions for both gauge and gravity amplitudes. Extensions to loop level via higher-genus worldsheet theory yield loop-level "scattering equations" enforcing additional conditions on period integrals (Mason et al., 2013).

Degeneration-based algorithms have also been developed: by viewing the scattering equations as linear constraints on reciprocal linear spaces, Proudfoot–Speyer degenerations provide homotopy-based optimal tracking schemes for numerical and symbolic root-finding, tied to the combinatorics of hyperplane arrangements (Betti et al., 4 Oct 2024).

6. Generalizations, Diagrammatics, and Algebraic Structures

Scattering equations underlie several generalized frameworks:

• Diagrammatic Rules: Möbius-invariant integrands with only simple poles can be evaluated by summing over compatible graph coverings, matching Feynman diagram expansions directly (Baadsgaard et al., 2015).
• Factorization and Positive Geometry: Pushforwards via scattering equations relate worldsheet and kinematic positive geometries universally. In the biadjoint $\phi^3$ case, the worldsheet associahedron is mapped to the kinematic associahedron, and more generally, any CHY-integrable positive geometry in moduli space can be transferred to kinematic space (Lukowski et al., 2022).
• Kinematic Lie Algebras and BCJ Duality: The structure constants $X_{a,b}=s_{ab}/(\sigma_a-\sigma_b)$ and $\bar X_{a,b}=(\sigma_a-\sigma_b)h_a h_b$ equip the solutions to the scattering equations with a pair of infinite-dimensional Lie algebra structures. These underpin the color-kinematic duality (BCJ) and the double-copy construction of gravity amplitudes, as reflected in decompositions of the CHY Jacobian and amplitude numerator structures (Monteiro et al., 2013).

7. Outlook and Open Problems

The universality of the scattering equations spans field theories, string theory, and enumerative algebraic geometry. Outstanding problems include rigorous proofs of factorization properties in nontrivial kinematic limits, extensions to loop-level and massive or higher-spin cases, and deeper exploration of associated geometric structures such as positive geometries, reciprocal linear spaces, and scattering varieties (Calabi–Yau, Fano, or K3 geometries) (He et al., 2014, Betti et al., 4 Oct 2024).

Recent algorithmic advances and the recognition of the scattering equations as an H-basis, as well as their realization via degeneration and quotient ring techniques, are likely to continue expanding the computational and conceptual tools available for amplitudes and their mathematical underpinnings (Bosma et al., 2016, Liu et al., 2018, Lukowski et al., 2022).