2. 2-trees

We begin by defining the class of graphs referred to in Theorem1.4. A 2-tree is a graph $T$ with $2n-5$ edges on the vertex set $[n-1] = \{1,2,\ldots ,n-1\}$ , which can be constructed inductively as follows. We start with the single edge graph $\{12\}$ , which is the unique 2-tree for $n-1=2$ . For $k=3,4,5,\ldots ,n-1$ , we proceed inductively as follows: we select an edge $ij$ whose vertices $i$ and $j$ are in $\{1,\ldots ,k-1\}$ , and we introduce the two new edges $ik$ and $jk$ . There are $2k-5$ choices at this stage, so our process leads to $\,(2n-7)!! = 1 \cdot 3 \cdot 5 \cdot 7 \cdots (2n-7)\,$ distinct 2-trees. Of course, many pairs of these 2-trees will be isomorphic as graphs.

The number of 2-trees up to isomorphism appears as the entry A054581 in the Online Encyclopedia of Integer Sequences (OEIS). That sequence begins with the counts

\begin{equation*} 1,1, \textbf {2}, 5, 12, 39, 136, 529, 2171, 9368, 41534,\ldots \quad \mbox{for } n = 4,5,\textbf{6},7,8,9,10,11,12,13, \ldots \end{equation*}

For instance, there are $\textbf { 2}$ distinct unlabeled 2-trees for $n=\textbf { 6}$ . Representatives are given by

(10) \begin{equation} T_1 = \{12,13,23,14,34,15,45\} \quad \textrm {and} \quad T_2 = \{12,13,23,24,34,25,35\}. \end{equation}

We identify each 2-tree $T$ with a subset of the set $S$ in (6), by removing the initial edge $12$ . This thus specifies a function (7). For example, the 2-tree $T_1$ in (10) is identified with $T$ in (8).

Theorem1.4 has two directions. First, (7) has only one critical point when $T$ is a 2-tree. Second, every maximal graph with this property is a 2-tree. We begin with the first direction.

Proof. Let $m=2n-6$ , and identify the coordinates on $({\mathbb{C}}^*)^m$ with the pairs in $T$ . We write $X_T$ for the $(n-3)$ -dimensional subvariety of $({\mathbb{C}}^*)^m$ that is parametrized by the $2 \times 2$ -minors $p_{ij}$ with $(i,j) \in T$ . Thus, $X_T$ is a very affine variety, defined as the complement of an arrangement of $m$ hyperplanes in ${\mathbb{C}}^{n-3}$ . The scattering potential (7) is the log-likelihood function for $X_T$ . The number of critical points, also known as the ML degree, equals the signed Euler characteristic of $X_T$ ; see [Reference Agostini, Brysiewicz, Fevola, Kühne, Sturmfels and TelenABFKST23, Reference Huh and SturmfelsHS14, Reference Sturmfels and TelenST21]. Therefore, our claim says that $|\chi (X_T)| = 1$ .

To prove this, we use the multiplicativity of the Euler characteristic. This states that, for any fibration $f: E \to B$ , with fiber $F$ , the following relation holds: $\chi (E)=\chi (F) \cdot \chi (B)$ .

We proceed by induction on $n$ . The base case is $n=4$ , where $T = \{(1,3),(2,3)\}$ . This set represents the unique $2$ -tree on $[3]$ , which is the triangle graph with vertices $1,2,3$ . Here, the very affine variety is the affine line ${\mathbb{C}}^1$ with two points removed. In symbols, we have

\begin{equation*} X_T = \{ \,(x_1,x_1-1) \in {\mathbb{C}}^2 \,: \, x_1 \not=0,1 \,\} = \{\,(p_{13},p_{23}) \in ({\mathbb{C}}^*)^2 \, : p_{13} - p_{23} = 1 \,\} \, \simeq \, \mathcal{M}_{0,4} . \end{equation*}

This punctured curve satisfies $\chi (X_T)=-1$ , so the base case of our induction is verified.

We now fix $n \geq 4$ , and we assume that $X_T \subset ({\mathbb{C}}^*)^{2n-6}$ has Euler characteristic $\pm 1$ for all 2-trees $T$ with $n-1$ vertices. Note that $\textrm {dim}(X_T) = n-3$ . Let $T^{\prime}$ be any 2-tree with $n$ vertices. The vertex $n$ is connected to exactly two vertices $i$ and $j$ . Suppose $i \lt j$ . The associated very affine variety $X_{T^{\prime}}$ lives in $({\mathbb{C}}^*)^{2n-4}$ . Note that $\textrm {dim}(X_{T^{\prime}}) = n-2$ .

We write the coordinates on $({\mathbb{C}}^*)^{2n-4}$ as $(\,p,\,p_{in}, p_{jn})$ , where $p \in ({\mathbb{C}}^*)^{2n-6}$ . Consider the the map $\,\pi \,:\,({\mathbb{C}}^*)^{2n-4} \rightarrow ({\mathbb{C}}^*)^{2n-6}, \,(\,p,\,p_{in}, p_{jn}) \mapsto p\,$ which deletes the last two coordinates.

Let $T$ be the 2-tree without the vertex $n$ and the two edges $in$ and $jn$ . Then $ij$ is an edge of $T$ , so $p_{ij}$ is among the coordinates of $p$ . The restriction of $\pi$ to $X_{T^{\prime}}$ defines a fibration

\begin{equation*} \pi \,:\, X_{T^{\prime}} \rightarrow X_T , \,\, \,(\,p,\,p_{in}, p_{jn}) \mapsto p . \end{equation*}

Indeed, the fiber over $p \in X_T$ equals

\begin{equation*} F = \pi ^{-1} (p) = \{ \, (p_{in}, p_{jn}) \in ({\mathbb{C}}^*)^2 \,:\, p_{jn} - p_{in} = c \,\}, \quad \hbox{ where $c = p_{ij} \not=0$.} \end{equation*}

This is the punctured line above, i.e. $F \simeq \mathcal{M}_{0,4}$ . We know that $\chi (F)=-1$ . Using the induction hypothesis, and the multiplicativity of the Euler characteristic, we conclude that

\begin{equation*} \chi (X_{T^{\prime}}) = \chi (F) \cdot \chi (X_T) = - \chi (X_T) = \pm 1. \end{equation*}

This means that the 2-tree $T^{\prime}$ exhibits minimal kinematics, and the lemma is proved.

From the induction step in the proof above, we also see how to write down the three-term linear equations that define $X_T$ . We display these linear equations for the 2-trees with $n=6$ .

Example 2.2 ( $n=6$ ). Consider the two 2-trees in (10). Each of them defines a very affine threefold of Euler characteristic $-1$ . Explicitly, these two threefolds are given as follows:

\begin{align*}X_{T_1} & = V(\,p_{13} - p_{23}- 1,\, p_{14} - p_{34}- p_{13},\, p_{15} - p_{45}-p_{14}\,) \subset \,\, ({\mathbb{C}}^*)^6, \\ X_{T_2} & = V(\,p_{13}-p_{23}-1, \, p_{24}-p_{34}-p_{23} , \, p_{25}-p_{35}-p_{23}\,) \subset \,\, ({\mathbb{C}}^*)^6. \end{align*}

We now come to the converse direction, which asserts that 2-trees are the only maximal graphs $T$ satisfying $\chi (X_T) = \pm 1$ . This will follow from known results on graphs and matroids.

3. Horn matrices

A remarkable theorem due to June Huh [Reference HuhHuh14] characterizes very affine varieties $X \subset ({\mathbb{C}}^*)^m$ that have ML degree $1$ . The unique critical point $\hat {p}$ of the log-likelihood function on $X$ is given by the Horn uniformization, due to Kapranov [Reference KapranovKap91]. Huh’s result was adapted to the setting of algebraic statistics by Duarte et al. in [Reference Duarte and MariglianoDMS21]. For an exposition see also [Reference Huh and SturmfelsHS14, § 3]. The Horn uniformization can be written in concise notation as follows:

(11) \begin{equation} \hat {p} = \lambda \star (H s)^H. \end{equation}

Here, $H$ is an integer matrix with $m$ columns and $\lambda$ is a vector in ${\mathbb{Z}}^m$ . The pair $(H,\lambda )$ is an invariant of the variety $X$ , referred to as the Horn pair in [Reference Duarte and MariglianoDMS21]. The coefficients $s_{ij}$ in the log-likelihood function (aka Mandelstam invariants) form the column vector $s$ of length $m$ , so $Hs$ is a vector of linear forms in the coordinates of $s$ . The notation $(Hs)^H$ means that we regard each column of $H$ as an exponent vector, and we form $m$ Laurent monomials in the linear forms $H s$ . Finally, $\star$ denotes the Hadamard product of two vectors of length $m$ .

The formula for $\hat {p}$ given in (11) is elegant, but one needs to get used to it. We encourage our readers to work through Example 3.2, where $(Hs)^H$ is shown for two Horn matrices $H$ .

We now present the main result of this section, namely the construction of the Horn matrix $H = H_T$ for the very affine variety $X_T$ associated with any 2-tree $T$ on $[n-1]$ . The matrix $H_T$ has $3n-9$ rows and $m=2n-6$ columns, one for each edge of $T$ . It is constructed inductively as follows. If $n=4$ with $T = \{13,23\}$ , which represents the triangle graph, then

\begin{equation*}H_T = \begin{array}{c c c} & \begin{array} {ccc} & 13 \kern10pt & 23\\ \end{array} \\ & \left [ \begin{array}{ccc} & 1 \kern10pt & 0 \\ & 0 \kern10pt & 1\\ &-1 \kern10pt& -1\\ \end{array} \right ]\\ \mbox{} \end{array}. \end{equation*}

Now, for $k \geq 5$ , let $T^{\prime}$ be any 2-tree with $k$ vertices, where vertex $k$ is connected to $i$ and $j$ , and $T = T^{\prime} \,\backslash \{ik,jk\}$ as in the proof of Lemma 2.1. Then the Horn matrix for $T^{\prime}$ equals

\begin{align*}H_{T^{\prime}}\,\,\, & = \begin{array}{c c} & \begin{array}{@{} c c c c@{}} \kern-10pt 2k-\textrm {columns} & \kern18pt ik & \kern18pt jk \\ \end{array}\\ \begin{array}{c} 3k{-12} \,\textrm {rows} \\ \\ \\ \\ \end{array}\hspace {-1em} & \left [ \begin{array}{@{} c@{\quad} c@{\quad} c@{\quad} c@{}} &\kern20pt H_T & \kern28pt \textbf{ h}_{ij} & \textbf{ h}_{ij} \,\,\,\\ &\kern20pt \textbf { 0} & \kern28pt 1 & 0 \,\,\,\\ &\kern20pt \textbf { 0} & \kern28pt 0 & 1 \,\,\,\\ &\kern20pt \textbf { 0} & \kern28pt -1 & \!-1 \,\,\, \end{array} \right ]\\ \mbox{} \end{array} \end{align*}

where $\textbf { 0}$ is the zero row vector and $\textbf { h}_{ij}$ is the column of $H_T$ that is indexed by the edge $ij$ .

Before proving this theorem, we illustrate the construction of $H_T$ and the statement.

Example 3.2 ( $n=6$ ). We consider the two 2-trees that are shown in (10). In each case, the Horn matrix has nine rows and six columns. We find that the two Horn matrices are

For $H = H_{T_i}$ , the column vector $Hs$ has nine entries, each a linear form in six $s$ -variables. Each column of $H$ specifies an alternating product of these linear forms, and these are the entries of $(Hs)^H$ . By adjusting signs when needed, we obtain the six coordinates of $\hat {p}$ .

For the second 2-tree $T_2$ , the six coordinates of the critical point $\hat {p}$ are

\begin{align*}&\begin{array}{lllll@{\quad}lll}\hat {p}_{13} & = & \dfrac{s_{13}}{s_{13} + s_{23} + s_{24} + s_{34} + s_{25} + s_{35}} & & & \hat {p}_{23} & = & - \dfrac {s_{23} + s_{24} + s_{25} + s_{34} + s_{35}}{s_{13} + s_{23} + s_{24} + s_{34} + s_{25} + s_{35}} \\[10pt]\end{array}\\&\def\lumina\hspace*{5.5pc}\begin{array}{lll}\hat {p}_{24} & = & - \dfrac{(s_{23} + s_{24} + s_{25} + s_{34} + s_{35}) s_{24}}{(s_{13} + s_{23} + s_{24} + s_{34} + s_{25} + s_{35})(s_{24}+s_{34})} \\ \hat {p}_{34} & = & \dfrac {(s_{23} + s_{24} + s_{25} + s_{34} + s_{35}) s_{34}}{(s_{13} + s_{23} + s_{24} + s_{34} + s_{25} + s_{35})(s_{24}+s_{34})} \\[10pt]\hat {p}_{25} & = & - \dfrac{(s_{23} + s_{24} + s_{25} + s_{34} + s_{35}) s_{25}}{(s_{13} + s_{23} + s_{24} + s_{34} + s_{25} + s_{35})(s_{25}+s_{35})} \\ \hat {p}_{35} & = & \dfrac {(s_{23} + s_{24} + s_{25} + s_{34} + s_{35}) s_{35}}{(s_{13} + s_{23} + s_{24} + s_{34} + s_{25} + s_{35})(s_{25}+s_{35})}.\end{array}\end{align*}

For the first 2-tree $T_1$ , the six coordinates of the critical point $\hat {p}$ are

\begin{align*}&\begin{array}{llcll@{\quad}llc}\hat {p}_{13} & = & \dfrac {s_{13} \,{+}\, s_{14} \,{+}\, s_{34}\,{+}\, s_{15} \,{+}\, s_{45}}{s_{13} \,{+}\, s_{23} \,{+}\, s_{14} \,{+}\, s_{34} \,{+}\, s_{15} \,{+}\, s_{45}} & & & \hat {p}_{23} & = & - \dfrac {s_{23}}{s_{13} \,{+}\, s_{23} \,{+}\, s_{14} \,{+}\, s_{34} \,{+}\, s_{15} \,{+}\, s_{45}} \\[10pt]\end{array}\\&\def\lumina\hspace*{5.5pc}\begin{array}{lll}\hat {p}_{14} & = & \dfrac {(s_{13} \,{+}\, s_{14} \,{+}\, s_{34}\,{+}\, s_{15} \,{+}\, s_{45}) (s_{14} \,{+}\, s_{15} \,{+}\, s_{45})}{(s_{13} \,{+}\, s_{23} \,{+}\, s_{14} \,{+}\, s_{34} \,{+}\, s_{15} \,{+}\, s_{45})(s_{14} \,{+}\, s_{15} \,{+}\, s_{34} \,{+}\, s_{45})}\\[10pt]\hat {p}_{34} & = & -\dfrac { (s_{13} \,{+}\, s_{14} \,{+}\, s_{34}\,{+}\, s_{15} \,{+}\, s_{45}) s_{34}} {(s_{13} \,{+}\, s_{23} \,{+}\, s_{14} \,{+}\, s_{34} \,{+}\, s_{15} \,{+}\, s_{45}) (s_{14} \,{+}\, s_{34} \,{+}\, s_{15} \,{+}\, s_{45}) }\\[10pt]\hat {p}_{15} & = & \phantom {-} \dfrac {( s_{13} \,{+}\,s_{14} \,{+}\, s_{34} \,{+}\, s_{15} \,{+}\, s_{45}) (s_{14} \,{+}\, s_{15} \,{+}\, s_{45}) s_{15}}{(s_{13} \,{+}\, s_{23}\,{+}\, s_{14} \,{+}\, s_{34} \,{+}\, s_{15} \,{+}\, s_{45}) (s_{14} \,{+}\, s_{34} \,{+}\, s_{15} \,{+}\, s_{45})(s_{15} \,{+}\, s_{45})}\\[10pt]\hat {p}_{45} & = & -\dfrac {( s_{13} \,{+}\,s_{14} \,{+}\, s_{34} \,{+}\, s_{15} \,{+}\, s_{45}) (s_{14} \,{+}\, s_{15} \,{+}\, s_{45}) s_{45}}{(s_{13} \,{+}\, s_{23}\,{+}\, s_{14} \,{+}\, s_{34} \,{+}\, s_{15} \,{+}\, s_{45}) (s_{14} \,{+}\, s_{34} \,{+}\, s_{15} \,{+}\, s_{45})(s_{15} \,{+}\, s_{45})}.\end{array}\end{align*}

We note that these $\hat {p}_{ij}$ satisfy the trinomial equations given for $X_{T_1}$ , respectively $X_{T_2}$ , in Example 2.2.

Proof of Theorem 3.1 . We start by reviewing the Horn uniformization (11) in the version proved by Huh [Reference HuhHuh14]. Let $X \subset ({\mathbb{C}}^*)^m$ be any very affine variety. The following are equivalent:

1. (i) the variety $X$ has ML degree $1$ ;

2. (ii) there exists $\lambda =(\lambda _1, \ldots ,\lambda _m) \in ({\mathbb{C}}^*)^m$ and a matrix $H = (h_{ij})$ in $ {\mathbb{Z}}^{\ell \times m}$ with zero column sums and left kernel $A$ , such that the monomial map

   \begin{equation*} ({\mathbb{C}}^*)^\ell \,\to \, ({\mathbb{C}}^*)^m, \,\, \,{\bf q}=(q_1,\ldots ,q_\ell ) \,\mapsto \,\biggl (\lambda _1 \underset {i=1}{\overset {\ell }{\prod }}q_i^{h_{i1}},\, \ldots ,\,\lambda _m \underset {i=1}{\overset {\ell }{\prod }}q_i^{h_{im}} \biggr ),\end{equation*}
   maps the $A$ -discriminantal variety $\Delta _{A}$ dominantly onto $X$ .

Constructing a Horn uniformization proves that the variety $X$ has ML degree $1$ . The corresponding Horn map (11) is precisely the unique critical point of the log-likelihood function. The following argument thus also serves as an alternative proof of Theorem1.4.

Fix a 2-tree $T$ on $[n-1]$ , with associated variety $X= X_T$ in $ ({\mathbb{C}}^*)^m$ , where $m=2n-6$ . Let $H = H_T$ be the $(3n-9) \times (2n-6)$ matrix constructed previously. Its left kernel is given by

\begin{equation*} A = \left\{\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1 & 1 & 1 & 0 & 0 & 0 & \cdots & 0 & 0 & 0 \\[4pt] 0 & 0 & 0 & 1 & 1 & 1 & \cdots & 0 & 0 & 0 \\[4pt]\vdots & \vdots & \vdots & & & & \ddots & & \\[4pt] 0 & 0 & 0 & 0 & 0 & 0 & \cdots & 1 & 1 & 1 \\ \end{array}\right\}. \end{equation*}

This matrix has $n-3$ rows. Its toric variety is the $(n-4)$ -dimensional linear space

\begin{equation*} X_A = \bigl \{ \,(\,t_1,t_1,t_1,\,t_2,t_2,t_2,\,\ldots ,\,t_{n-3},t_{n-3},t_{n-3}\,) \,:\, t_1,t_2,\ldots ,t_{n-3} \in ({\mathbb{C}}^*)^{3n-9}\,\bigr \} \quad \textrm {in} \,\, \,{\mathbb{P}}^{3n-10}. \end{equation*}

The $A$ -discriminantal variety is the variety projectively dual to $X_A$ . Here, it is the linear space

\begin{equation*} \Delta _A = \bigl \{\, q = (q_1,q_2,\ldots , q_{3n-9}) \ : \ q_{3i+1}+q_{3i+2}+q_{3i+3}=0 \,\,\,\textrm {for} \,\,\, i=0,1,\ldots ,n-4 \,\bigr \}. \end{equation*}

Disregarding $\lambda$ for now, the monomial map in item (ii) above takes $q \in \Delta _A$ to the vector

\begin{equation*} p = \bigl (\,p_{13}, p_{23},\,\ldots \,,\,p_{ik} \,,\,\, p_{jk}, \,\ldots \,\, \bigr ) = \biggl (\, \frac {q_1}{q_3}, \,\frac {q_2}{q_3}, \, \ldots \,,\, \,p_{ij} \cdot \frac {q_{3k-8}}{q_{3k-6}}\,,\,\, p_{ij} \cdot \frac {q_{3k-7}}{q_{3k-6}} ,\, \ldots \biggr ). \end{equation*}

Using the trinomial equations that define $\Delta _A$ , we can write this as follows

\begin{equation*} p = \biggl ( \frac {q_1}{-(q_1+q_2)}, \frac {q_2}{-(q_1+q_2)},\,\ldots \,,\,\, p_{ij} \cdot \frac {q_{3k-8}}{-(q_{3k-8}+q_{3k-7})}\,,\,\,p_{ij} \cdot \frac {q_{3k-7}}{-(q_{3k-8}+q_{3k-7})}, \, \ldots \, \biggr ). \end{equation*}

The vector $p$ satisfies the equation $\,p_{13} + p_{23} + 1=0\,$ and it satisfies all subsequent equations $p_{ik} + p_{jk} + p_{ij} = 0$ that arise from the construction of the 2-tree $T$ .

In a final step, we need to adjust the signs of the coordinates in order for $p$ to satisfy the equations $p_{ik} - p_{jk} - p_{ij} = 0$ that cut out $X_T$ ; see e.g. Example 2.2. This is done by replacing $\hat {p}_{lm} = \lambda _{lm} p_{lm}$ for appropriate signs $\lambda _{lm} \in \{-1,+1\}$ , or, equivalently, by replacing $p$ with the Hadamard product $\hat {p} = \lambda \star p$ for an appropriate sign vector $\lambda \in \{-1,+1\}^{2n-6}$ .

The signs can be constructed recursively as follows. We set $\lambda _{13}=-1, \lambda _{23}=1$ and, with the same notation as earlier, we define $\lambda _{ik}=-\lambda _{ij}$ and $\lambda _{jk}=\lambda _{ij}.$ Indeed, if $(\hat {p}_{ij},\hat {p}_{ik},\hat {p}_{jk})$ satisfy $p_{ik} + p_{jk} + p_{ij} = 0$ , then $(\lambda _{ij}\hat {p}_{ij},\lambda _{ik}\hat {p}_{ik},\lambda _{jk} \hat {p}_{jk})$ satisfy $p_{ik} - p_{jk} - p_{ij} = 0$ . This now gives the desired birational map from $\Delta _A$ onto $X_T$ . That map furnishes the rational formula for the unique critical point $\hat {p}$ . To obtain the version $\,\hat {p} = \lambda \star (Hs)^H$ seen in (11), we note that the map $s \mapsto Hs$ parametrizes the linear space $\Delta _A$ . This completes the proof.

We conclude this section by making the coordinates of $\hat {p}= \lambda \star (Hs)^H$ more explicit. Given any 2-tree $T$ and any edge $ij$ of $T$ , we write $[s_{ij}]$ for the sum of all Mandelstam invariants $s_{lm}$ where $lm$ is any descendant of the edge $ij$ in $T$ . Here descendant refers to the transitive closure of the parent-child relation in the iterative construction of $T$ : the new edges $ik$ and $jk$ are children of the old edge $ij$ . In this case we call $\{i,j,k\}$ a triangle of the 2-tree $T$ . This triangle is an ancestral triangle of an edge $lm$ of $T$ if $lm$ is a descendant of the edge $ik$ . The entries of the vector $H s$ are the linear forms $[s_{ij}]$ and their negated sums $-[s_{ik}] - [s_{jk}]$ .

Corollary 3.3. The evaluation of the Plücker coordinate $p_{lm}$ at the critical point of $L_T$ equals

(12) \begin{equation} \hat {p}_{lm} = \underset {}{\prod } \lambda _{ik}\frac {[s_{ik}]}{[s_{ik}]+[s_{jk}]}, \end{equation}

where the product runs over all ancestral triangles $\{i,j,k\}$ of the edge $lm$ , and $\lambda _{ik} \in \{-1,+1\}$ is the sign chosen in the proof of Theorem 3.1 .

This is a corollary of Theorem3.1. The proof is given by inspecting the Horn matrix $H_T$ . It is instructive to rewrite the rational functions $\hat {p}_{ij}$ in Example 3.2 using the notation in (12).

4. Amplitudes

In this section we take a step towards particle physics. We define an amplitude $m_T$ for any 2-tree $T$ . This is a rational function in the Mandelstam invariants $s_{ij}$ . When $T$ is planar, $m_T$ is a degeneration of the biadjoint scalar amplitude $m_n$ . We saw this in Example 1.5. Cachazo and Early in [Reference Cachazo and EarlyCE21] introduced minimal kinematics as a means to study such degenerations.

We shall express the amplitude $m_T$ in terms of the Horn matrix $H=H_T$ . Given any 2-tree $T$ , the entries of the column vector $s$ are the $2n-6$ Mandelstam invariants $s_{ij}$ . The entries of the column vector $H s$ are $3n-9$ linear forms in $s$ . Our main result is the following theorem.

Theorem 4.1. Fix a 2-tree $T$ on $[n-1]$ . The amplitude $m_T$ associated with $T$ equals

(13) \begin{equation} m_T = \prod _{\{i\lt j\lt k\}} \frac {[s_{ik}] + [s_{jk}]}{ [s_{ik}] \cdot [s_{jk}] }. \end{equation}

The product is over all triangles in $T$ . This rational function of degree $3-n$ is the product of $n-3$ of the linear forms in $Hs$ divided by the product of the other $2n-6$ linear forms in $Hs$ .

In order for this theorem to make sense, we first need a definition of the amplitude $m_T$ . We shall work in the framework of beyond-planar MHV amplitudes developed by Arkani-Hamed et al. in [Reference Arkani-Hamed, Bourjaily, Cachazo, Postnikov and TrnkaABCPT15]. Our point of departure is the observation that every 2-tree on $[n-1]$ defines an on-shell diagram. Here we view $T$ as a list of $n-2$ triples $ijk$ , starting with $123$ and ending with $12n$ . The last triple $12n$ is special because it uses the vertex $n$ . Moreover, it does not appear in the formula (13). For a concrete example, we identify the 2-trees $T_1$ and $T_2$ in (10) with the following two on-shell diagrams, one planar and one non-planar:

(14) \begin{equation} T_1 : \{123, 134, 145, 126 \} \qquad \textrm {and} \qquad T_2 : \{123,234,235, 126\}. \end{equation}

With each triple $ijk$ in $T$ we associate the row vector $p_{jk} e_i - p_{ik} e_j + p_{ij} e_k$ , and we define $M_T$ to be the $(n-2) \times n$ matrix whose rows are these vectors for all triples in $T$ . For instance,

\begin{equation*} M_{T_2} = \!\!\!\! \begin{array}{c@{\quad} c@{\quad} c@{\quad} c} & \begin{matrix} \hskip-6pt 1 \phantom{0000} & 2\phantom{0000} & 3\phantom{0000} & 4\phantom{0000} & 5\phantom{000} & \phantom{0}6 \\ \end{matrix} \\ & \left [ \begin{array}{@{\hskip3pt}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} p_{23} & -p_{13} & \phantom {-} p_{12} & 0 & 0 & 0 \\[4pt] 0 & \phantom {-} p_{34} & - p_{24} & \,p_{23}\, & 0 & 0 \,\\[4pt] 0 & \phantom {-}p_{35} & -p_{25} & 0 & \,p_{23} \,& 0 \\[4pt] p_{26} & - p_{16} & 0 & 0 & 0 & \,p_{12} \end{array} \right ]\\ \mbox{} \end{array}. \end{equation*}

Note that the kernel of $M_T$ coincides with the row span of the matrix $X$ in (1). This implies that there exists a polynomial $\Delta (M_T)$ of degree $n-3$ in the Plücker coordinates such that the maximal minor of $M_T$ obtained by deleting columns $i$ and $j$ is equal to $\,\pm \,p_{ij} \cdot \Delta (M_T)$ .

Lemma 4.2. For any 2-tree $T$ , the gcd of the maximal minors of the matrix $M_T$ equals

(15) \begin{equation} \Delta (M_T) = \prod _{ij} p_{ij}^{v_T(ij)-1}, \end{equation}

where the product is over all edges of $T$ , and $v_T(ij)$ is the number of triangles containing $ij$ .

Following [Reference Arkani-Hamed, Bourjaily, Cachazo, Postnikov and TrnkaABCPT15, (2.15)], we define the integrand associated with the 2-tree $T$ to be

(16) \begin{equation} \mathcal{I}_T = \frac { \Delta (M_T)^2 } { \prod _{ijk \in T} p_{ij} p_{ik} p_{jk} }. \end{equation}

This is a rational function of degree $-n$ in the Plücker coordinates. Lemma 4.2 now implies the following corollary.

Corollary 4.3. Given any 2-tree $T$ , the associated integrand equals

(17) \begin{equation} \mathcal{I}_T = \prod _{ij} p_{ij}^{v_T(ij)-2}. \end{equation}

The integrands for the 2-trees $T_1$ and $T_2$ from our running example in (10) and (14) are

\begin{equation*} \mathcal{I}_{T_1} = \frac {1}{ p_{16} p_{26} p_{23} p_{34} p_{45} p_{15}} \qquad \textrm {and} \qquad \mathcal{I}_{T_2} = \frac {p_{23}}{ p_{16} p_{26} p_{13} p_{24} p_{34} p_{25} p_{35}}. \end{equation*}

Note that $\mathcal{I}_{T_1}$ is a Parke–Taylor factor for $n=6$ . We are now using the unconventional labeling $1,n,2,3,4,\ldots ,n-1$ for the vertices of the $n$ -gon. Any triangulation of this $n$ -gon is a 2-tree $T$ . Such 2-trees are called planar. To be precise, the triangulation consists of the three edges of the triangle $\{1,n,2\}$ together with the $2n-6$ edges given by the 2-tree $T$ .

Corollary 4.4. For any planar 2-tree $T$ , our integrand equals the Parke–Taylor factor.

(18) \begin{equation} \quad \mathcal{I}_T = \frac {1}{p_{1n} \, p_{n2} \,p_{23} \,p_{34} \, \cdots \, p_{n-2,n-1} \,p_{n-1,1}} = PT(1,n,2,3,\ldots , n-1) . \end{equation}

We now define the amplitude associated with a 2-tree $T$ to be the following expression:

(19) \begin{equation} m_T = - \frac {\mathcal{(I}_T)^2}{\textrm {Hess}(L_T)}( \hat {p} ). \end{equation}

Here $\textrm {Hess}(L_T)$ is the determinant of the Hessian of the log-likelihood function in (7). The numerator and the denominator are evaluated at the critical point $\hat {p}$ which is given by Theorem3.1.

Example 4.5 ( $n=6$ ). Let $T = T_2$ be the non-planar 2-tree in our running example. Then

\begin{align*}- \textrm{Hess}(L_T)({\hat {p}}) &= \frac {(s_{13} + s_{23} + s_{24} + s_{25} + s_{34} + s_{35})^7 \,(s_{24} + s_{34})^3\,(s_{25} + s_{35})^3} {s_{13} \,(s_{23} + s_{24} + s_{25} + s_{34} + s_{35})^5 \, s_{24} \,s_{34} \, s_{25} \,s_{35}}\\[5pt] & = \frac { ([s_{13}]+[s_{23}])^7}{[s_{13}]^1 \, [s_{23}]^5} \cdot \frac {([s_{24}]+[s_{34}])^3}{[s_{24}]^1 \, [s_{34}]^1} \cdot \frac {([s_{25}]+[s_{35}])^3}{[s_{25}]^1 \, [s_{35}]^1}. \end{align*}

The numerator $\mathcal{(I}_T)^2(\hat {p})$ is the same expression but with each exponent increased by one. Therefore $m_T$ is equal to the ratio given by the Horn matrix $H_T$ , as promised in Theorem 4.1.

Proof of Theorem 4.1 . Fix a 2-tree $T$ on $[n-1]$ . For any edge $ij$ of $T$ , we write $a_{ij}$ for the number of descendants of that edge. For any triangle $\{i , j \lt k\}$ of $T$ , we set $b_k = a_{ik} + a_{jk}+1$ .

A key combinatorial lemma about 2-trees is that $a_{ij}+1$ equals the sum of the integers $2 \bigl (2-v_T(lm) \bigr )$ where $lm$ runs over the set $\textrm {dec}_T(ij)$ of descendants of the edge $ij$ . To be precise,

(20) \begin{equation} \qquad a_{ij} + 1 = \sum _{lm \in \textrm {dec}_T(ij)} \!\!\!\!\! 2 (2- v_T(lm)) \,\,\quad \hbox{for all edges $ij \not=12$.} \end{equation}

The initial edge $12$ is excluded. Recall that $v_T(lm)$ is the number the triangles containing the edge $lm$ . We prove (20) by induction on the construction of $T$ , after checking it for $n \leq 5$ . Indeed, suppose a new vertex $n$ enters the 2-tree, with edges $rn$ and $sn$ . Then $v_T(rs)$ increases by $1$ and $v_T(rn) = v_T(sn)=1$ . Otherwise $v_T$ is unchanged. For all ancestors $ij$ of $rs$ , the right hand side and the left hand side of (20) increase by $2$ . For $ij \in \{rn,sn\}$ , both sides are $2$ . For all other edges $ij$ , the two sides remain unchanged. Hence (20) is proved.

We now evaluate $(\mathcal{I}_T)^2$ at $\hat {p}$ by plugging (12) into the square of (17). This gives

\begin{equation*} (\mathcal{I}_T)^2(\hat {p}) = \biggl (\, \prod _{lm} (\hat {p}_{lm})^{v_T(lm) - 2}\, \biggr )^{\! 2} = \prod _{lm} \prod _{ijk} \biggl (\frac {[s_{ik}] + [s_{jk}]}{[s_{ik}]} \biggr )^{\! 2(2 - v_T(lm))}, \end{equation*}

where the inner product is over ancestral triangles $ijk$ of $lm$ . Switching the products yields

\begin{align*} (\mathcal{I}_T)^2(\hat {p}) & = \prod _{ijk}\left [ \prod _{lm \in \textrm {dec}_T(ik)} \Bigl (\frac {[s_{ik}] + [s_{jk}]}{[s_{ik}]} \Bigr )^{\! 2(2 - v_T(lm))} \cdot \prod _{lm \in \textrm {dec}_T(jk)} \Bigl (\frac {[s_{ik}] + [s_{jk}]}{[s_{jk}]} \Bigr )^{\! 2(2 - v_T(lm))} \right ] \\[5pt] & = \prod _{ijk}\left [ \Bigl (\frac {[s_{ik}] + [s_{jk}]}{[s_{ik}]}\Bigr )^{a_{ik}+1} \cdot \Bigl (\frac {[s_{ik}] + [s_{jk}]}{[s_{jk}]}\Bigr )^{a_{jk}+1} \right ]\!. \end{align*}

where the product is over all triangles $ijk$ of $T$ . In conclusion, we have derived the formula

(21) \begin{equation} (\mathcal{I}_T)^2(\hat {p}) = \prod _{ijk} \frac { (\,[s_{ik}] + [s_{jk}]\,)^{b_k+1}}{ [s_{ik}]^{a_{ik}+1}\, [s_{jk}]^{a_{jk}+1}}, \end{equation}

In order to prove Theorem4.1, we must show that the Hessian at the critical point equals

(22) \begin{equation} \textrm {Hess}(L_T)(\hat {p}) = - \prod _{ijk } \frac { (\,[s_{ik}] + [s_{jk}]\,)^{b_k}}{ [s_{ik}]^{a_{ik}}\, [s_{jk}]^{a_{jk}}}. \end{equation}

The proof is organized by an induction on $k$ , where the Mandelstam invariants $s_{ij}$ are transformed as we deduce the desired formula for $k$ from the corresponding formula for $k-1$ .

As a warm-up, it is instructive to examine the case $k=4$ , where the Hessian is a $1 \times 1$ matrix. The entry of that matrix is the second derivative of (4) evaluated at (5). We find

\begin{equation*} \textrm {Hess}(L_T)(\hat {p}) = \frac {\partial^2 L}{\partial x_1^{\,2}} (\hat {p}) = -\frac {s_{13}}{\hat x_1^2} - \frac {s_{23}}{(\hat x_1-1)^2} = - \frac {(s_{13} + s_{23})^3}{s_{13} s_{23}}. \end{equation*}

This equation matches (22), and it serves as the blueprint for the identity in (24) below.

Our first step towards (22) is to get rid of the minus sign. To this end, we write $\mathcal{H}$ for the Hessian matrix of the negated scattering potential $L_T$ , evaluated at $\hat {p}$ . Its entries are

\begin{equation*} \mathcal{H}_{ii} = \underset {\ell=1}{\overset {n}{\sum }} \frac {s_{i \ell }}{\hat {p}_{i \ell }^{\,2}} \qquad \textrm {and} \qquad \mathcal{H}_{ij} = -\frac {s_{ij}}{\hat {p}_{ij}^{\,2}} \quad \textrm {for} \,\,\,i \not = j. \end{equation*}

We shall prove that $\textrm {det}(\mathcal{H})$ equals the product on the right-hand side of (22). This will be done by downward induction. Let $k$ be the last vertex, connected to earlier vertices $i,j$ . We display the rows and columns of the Hessian that are indexed by the triangle $\{i,j,k\}$ :

\begin{equation*} \mathcal{H} = \left\{\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \ldots & \ldots & \ldots & \ldots & \ldots & 0 \\ \ldots & \ldots +\frac {s_{ij}}{\hat {p}_{ij}^2}+\frac {s_{ik}}{\hat {p}_{ik}^2} & \ldots & -\frac {s_{ij}}{\hat {p}_{ij}^2} & \ldots & -\frac {s_{ik}}{\hat {p}_{ik}^2} \\ \ldots & \ldots & \ldots & \ldots & \ldots & 0 \\ \ldots &-\frac {s_{ij}}{\hat {p}_{ij}^2} & \ldots & \ldots +\frac {s_{ij}}{\hat {p}_{ij}^2}+\frac {s_{jk}}{\hat {p}_{jk}^2} & \ldots & -\frac {s_{jk}}{\hat {p}_{jk}^2} \\ \ldots & \ldots & \ldots & \ldots & \ldots & 0 \\ 0 & -\frac {s_{ik}}{\hat {p}_{ik}^2} & 0 & -\frac {s_{jk}}{\hat {p}_{jk}^2} & 0 & \frac {s_{ik}}{\hat {p}_{ik}^2}+\frac {s_{jk}}{\hat {p}_{jk}^2} \end{array}\right\}.\end{equation*}

Since $k$ is a terminal node in the 2-tree $T$ , its two edges satisfy a variant of (5), namely

(23) \begin{equation} \hat {p}_{ik} = \frac {s_{ik}}{s_{ik} \!+\! s_{jk}} \cdot \hat {p}_{ij} \quad \textrm {and} \quad \hat {p}_{jk} = \frac {s_{jk}}{s_{ik} \!+\! s_{jk}} \cdot \hat {p}_{ij} . \end{equation}

Using these identities, the lower right entry of the matrix $\mathcal{H}$ can be written as follows:

(24) \begin{equation} \mathcal{H}_{kk} = \frac {s_{ik}}{\hat {p}_{ik}^2}+\frac {s_{jk}}{\hat {p}_{jk}^2} = \frac {s_{ik}}{\frac {s_{ik}^2 \hat {p}_{ij}^2}{(s_{ik}+s_{jk})^2}}+\frac {s_{jk}}{\frac {s_{jk}^2 \hat {p}_{ij}^2}{(s_{ik}+s_{jk})^2}} = \frac {(s_{ik}+s_{jk})^3}{s_{ik} \,s_{jk}} \cdot \frac {1}{ \hat {p}_{ij}^2} . \end{equation}

By factoring out $\mathcal{H}_{kk}$ from the last row, the lower right entry becomes $1$ . We add multiples of the last row to rows $i$ and $j$ , so as to cancel their last entries. The resulting upper left block with one fewer row and one fewer column is the Hessian matrix of the scattering potential for the 2-tree which is obtained from $T$ by removing vertex $k$ and its two incident edges $ik$ and $jk$ . However, in the new matrix $s_{ij}$ is now replaced by $s_{ij} + s_{ik} + s_{jk}$ . Note that this sum equals $[s_{ij}]$ if $v_T(ij)=2$ . Proceeding inductively, more and more terms get added, and eventually each Mandelstam invariant $s_{lm}$ is replaced by the corresponding sum $[s_{lm}]$ .

In the end, we find that the determinant of $\mathcal{H}$ is equal to the product of the quantities

(25) \begin{equation} \frac {[s_{ik}]}{\hat {p}_{ik}^2}+\frac {[s_{jk}]}{\hat {p}_{jk}^2} = \frac {([s_{ik}]+[s_{jk}])^3}{[s_{ik}]\, [s_{jk}]} \cdot \frac {1}{ \hat {p}_{ij}^2} , \end{equation}

where $ijk$ ranges over all triangles of the 2-tree $T$ . A combinatorial argument like that presented above shows that this product is equal to (22). This completes the proof.

Remark 4.6. Our proof rests on the fact that the Hessian is the product of the expressions (25). Another way to get this is to directly triangularize the scattering equations $\nabla L_T=0$ . Using Corollary 3.3 and the trinomials defining $X_T$ , we see that $\nabla L_T=0$ is equivalent to

\begin{equation*} \frac {[s_{ik}]}{p_{ik}} \, + \, \frac {[s_{jk}]}{p_{jk}} = 0 \quad \hbox{for all triangles $ijk$ of $T$.} \end{equation*}

This is a triangular system of $n-3$ equations in the unknowns $x_1,x_2,\ldots ,x_{n-3}$ . The Jacobian of this system is upper triangular, and its determinant is the product of the expressions (25).

5. On-shell diagrams and hypertrees

On-shell diagrams were developed by Arkani-Hamed, Bourjaily, Cachazo, Postnikov and Trnka [Reference Arkani-Hamed, Bourjaily, Cachazo, Postnikov and TrnkaABCPT16]. They encode rational parts, or leading singularities [Reference CachazoCacha08], occurring in scattering amplitudes for $\mathcal{N}=4$ Super Yang–Mills theory [Reference Arkani-Hamed, Bourjaily, Cachazo, Postnikov and TrnkaABCPT16, § 4.7]. These are defined using the spinor-helicity formalism on a product of Grassmannians, $\textrm {Gr}(2,n) \times \textrm {Gr}(2,n)$ . We focus on the special case of MHV on-shell diagrams, where leading singularities are rational functions in the coordinates $p_{ij}$ ( $=\langle ij\rangle$ , in physics notation [Reference Arkani-Hamed, Bourjaily, Cachazo, Postnikov and TrnkaABCPT15, § 2.3]) of a single Grassmannian $\textrm {Gr}(2,n)$ . The resulting discontinuities of MHV amplitudes are, for us, CHY integrands.

In [Reference Arkani-Hamed, Bourjaily, Cachazo, Postnikov and TrnkaABCPT15, § 2.2], an identification was proposed between on-shell diagrams and certain collections of $n-2$ triples in $[n]=\{1,\ldots , n\}$ . Among these on-shell diagrams are the hypertrees of Castravet–Tevelev [Reference Castravet and TevelevCT13]. This identification is again noted in [Reference TevelevTev, Lemma 9.5].

In algebraic geometry, hypertrees represent effective divisors on the moduli space $\mathcal{M}_{0,n}$ . In physics, one can also consider on-shell diagrams for higher Grassmannians $\textrm {Gr}(k,n)$ ; cf. [Reference Franco, Galloni, Penante and WenFGPW15]. The analogs to hypertrees are now $(n-k)$ -element collections of $(k+1)$ -sets in $[n]$ . These define effective divisors on the configuration spaces $X(k,n) = \textrm {Gr}(k,n)^o/({\mathbb{C}}^*)^n$ . It would be interesting to examine these through the lens of [Reference Castravet and TevelevCT13, Reference TevelevTev]. In this paper, we stay with $k=2$ .

We define a hypertree to be a collection $T$ of $n-2$ triples $\Gamma _1,\ldots , \Gamma _{n-2}$ in $[n]$ such that:

1. (a) each $i\in [n]$ appears in at least two triples; and

2. (b) $\big \vert \bigcup _{i\in S}\Gamma _i\big \vert \,\ge \, \vert S\vert+2\,$ for all non-empty subsets $S \subseteq [n-2]$ .

Hypertrees have the same number of triples as 2-trees, but 2-trees are not hypertrees because some $i$ appears in only one triple. However, if axiom (a) for hypertrees is dropped but (b) is kept, one obtains all nonzero leading singularities of MHV amplitudes. In particular, 2-trees satisfy (b), and one might view them as hypertrees in a weak sense. A hypertree is called irreducible if the inequality in (b) is strict for $2 \leq |S| \leq n-3$ ; see [Reference Castravet and TevelevCT13, Definition 1.2].

Example 5.1 ( $n=6$ ). The 2-trees in (14) are not hypertrees. The following is a hypertree:

(26) \begin{equation} T = \{\,123,345,156 ,246\,\}, \end{equation}

but it is not a 2-tree. The hypertree $T$ corresponds to the octahedral on-shell diagram in [Reference Cachazo, Early, Guevara and MizeraCEGM19, Figure 1.1]. See also [Reference Castravet and TevelevCT13, Figure 2] and [Reference He, Yan, Zhang and ZhangHYZZ18, § 5.2]. This hypertree is irreducible, in the sense defined above, and it will serve as our running example throughout this section.

Many of the concepts for 2-trees from the previous sections make sense for hypertrees $T$ . We define the $(n-2) \times n$ matrix $M_T$ as in § 4. The rows of $M_T$ are the vectors $p_{jk} e_i - p_{ik} e_j + p_{ij} e_k$ for $\{i,j,k\} \in T$ . These span the kernel of the $2 \times n$ matrix $X$ in (1). The gcd of the maximal minors of $M_T$ is a polynomial $\Delta (M_T)$ of degree $n-3$ in the Plücker coordinates $p_{ij}$ . The equation $\Delta (M_T)=0$ defines a divisor in $\mathcal{M}_{0,n}$ , namely the hypertree divisor. If $T$ is a 2-tree then the hypertree divisor is a union of Schubert divisors $\{p_{ij} = 0\}$ , as seen in Lemma 4.2. For the geometric application in [Reference Castravet and TevelevCT13], this case is uninteresting. Instead, Castravet and Tevelev focus on hypertree divisors that are irreducible; see [Reference Castravet and TevelevCT13, Theorem 1.5].

Example 5.2 (Irreducible hypertree). The hypertree $T$ in (26) is irreducible. Its matrix is

\begin{equation*} M_T = \begin{array}{c c c c} & \begin{matrix} \hskip-6pt 1 \phantom{0000} & 2\phantom{0000} & 3\phantom{0000} & 4\phantom{0000} & 5\phantom{000} & \phantom{0}6 \\ \end{matrix} \\ & \left [ \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} \,p_{23} & -p_{13} & \, p_{12} \,& 0 & 0 & 0\, \\[4pt] 0 & 0 & \,p_{45} \,& -p_{35} & \phantom {-} p_{34} & 0 \\[4pt] \, p_{56} & 0 & 0 & 0 & -p_{16} & \,p_{15} \,\\[4pt] 0 & \phantom {-} p_{46} & 0 & -p_{26} & 0 &\, p_{24} \, \end{array} \right ]\\ \mbox{} \end{array}. \end{equation*}

The hypertree divisor for $T$ is an irreducible surface in the threefold $\mathcal{M}_{0,6}$ . It is defined by

\begin{equation*} \Delta (M_T) = p_{12} p_{35} p_{46} \,-\, p_{13} p_{26} p_{45}. \end{equation*}

This polynomial is irreducible in the coordinate ring of $\textrm {Gr}(2,6)$ . See also [Reference Castravet and TevelevCT13, Figure 2]. The corresponding on-shell diagram appears in [Reference Arkani-Hamed, Bourjaily, Cachazo, Postnikov and TrnkaABCPT15, (2.18)] where different labelings are used.

For every hypertree $T$ , we define the CHY integrand $\,\mathcal{I}_T\,$ by the formula in (16). This is a rational function of degree $-n$ in the Plücker coordinates $p_{ij}$ . The scattering potential for $T$ is the log-likelihood function $L_T$ in (7) where the sum is over all pairs $(i,j)$ that are contained in some triple of $T$ . Thus we set $s_{ij} = 0$ in (2) for all non-edges $ij$ of $T$ . The hypertree $T$ in (26) has three non-edges, namely $14$ , $25$ and $36$ . Furthermore, we use the same formulas as in (19) to define the hypertree amplitude for $T$ . To be precise, we set

(27) \begin{equation} m_T = \sum _{\hat {p}} \frac {\mathcal{(I}_T)^2}{\textrm {Hess}(L_T)}( \hat {p} ), \end{equation}

where the sum ranges over all critical points $\hat {p}$ of the scattering potential $L_T$ . This is a rational function of degree $3-n$ in the Mandelstam invariants $s_{ij}$ where $(i,j)$ appears in $T$ .

Example 5.4 (Hypertree amplitude). We compute the amplitude $m_T$ for the hypertree $T$ in (26). The generic $n=6$ scattering potential $L$ has six critical points $\hat {p}$ , but the restricted scattering potential $L_T$ has only two. The expression (27) also makes sense for $L$ . We have

(28) \begin{equation} m_T = \sum _{\hat {p}} \frac {1}{{\rm Hess}(L)}\left (\frac { (p_{12} p_{35} p_{46} \,-\, p_{13} p_{26} p_{45})^2}{( p_{12}p_{13}p_{23})(p_{34}p_{35}p_{45})(p_{15}p_{16}p_{56})(p_{24}p_{26}p_{46})}\right )^{\! 2}(\hat {p}). \end{equation}

This evaluates to a rational function in the Mandelstam invariants. For generic $s_{ij}$ , we find

\begin{align*} m_T & = \dfrac {1}{s_{16} s_{24} s_{35}}+\dfrac {1}{s_{16} s_{23} s_{45}}+\dfrac {1}{s_{13} s_{26} s_{45}}+\dfrac {1}{s_{15} s_{23} s_{46}}+\dfrac {1}{s_{12} s_{35} s_{46}}+\dfrac {1}{s_{15} s_{26} s_{34}} + \dfrac {1}{s_{12} s_{34} s_{56}}\\[6pt] &\quad +\dfrac {1}{s_{12} s_{35} s_{124}} + \dfrac {1}{s_{24} s_{35} s_{124}} +\dfrac {1}{s_{12} s_{56} s_{124}} +\dfrac {1}{s_{24} s_{56} s_{124}} +\dfrac {1}{s_{13} s_{24} s_{56}} + \dfrac {1}{s_{15} s_{34} s_{125}} +\dfrac {1}{s_{12} s_{46} s_{125}} \\[6pt] &\quad +\dfrac {1}{s_{15} s_{46} s_{125}} +\dfrac {1}{s_{13} s_{26} s_{134}} +\dfrac {1}{s_{26} s_{34} s_{134}}+\dfrac {1}{s_{12} s_{34} s_{125}} + \dfrac {1}{s_{34} s_{56} s_{134}}+\dfrac {1}{s_{15} s_{23} s_{145}}+\dfrac {1}{s_{15} s_{26} s_{145}}\\[6pt] &\quad +\dfrac {1}{s_{23} s_{45} s_{145}} +\dfrac {1}{s_{26} s_{45} s_{145}}+\dfrac {1}{s_{13} s_{56} s_{134}} + \dfrac {1}{s_{16} s_{35} s_{235}}+\dfrac {1}{s_{23} s_{46} s_{235}}+\dfrac {1}{s_{35} s_{46} s_{235}}+\dfrac {1}{s_{13} s_{24} s_{245}}\\&\quad +\dfrac {1}{s_{16} s_{24} s_{245}} +\dfrac {1}{s_{16} s_{23} s_{235}} + \dfrac {1}{s_{16} s_{45} s_{245}}+\dfrac {1}{s_{13} s_{45} s_{245}}. \end{align*}

We next impose the constraints $s_{14} = s_{25} = s_{36}=0$ coming from the hypertree $T$ . Some poles now become spurious. By collecting distinct maximal nonzero residues and then canceling spurious poles, we obtain the Feynman diagram expansion for the hypertree amplitude:

\begin{align*} m_T & = \frac {1}{s_{16} s_{24} s_{35}}+\frac {1}{s_{16} s_{23} s_{45}}+\frac {1}{s_{13} s_{26} s_{45}}+\frac {1}{s_{15} s_{23} s_{46}}+\frac {1}{s_{12} s_{35} s_{46}}+\frac {1}{s_{13} s_{24} s_{56}}+\frac {1}{s_{12} s_{34} s_{56}}\\[6pt] &\quad +\frac {1}{s_{15} s_{26} s_{34}} + \frac { \phantom {/} s_{15}\,+\,s_{45 }\phantom {/} }{s_{15} s_{23} s_{26} s_{45}}+\frac {\phantom {/}s_{12}\,+\,s_{24}\phantom {/} }{s_{12} s_{24} s_{35} s_{56}} +\frac {\phantom {/}s_{12}\,+\,s_{15}\phantom {/} }{s_{12} s_{15} s_{34} s_{46}} + \frac {\phantom {/}s_{13}\,\,+s_{34}\phantom {/} }{s_{13} s_{26} s_{34} s_{56}}+\frac {\phantom {/}s_{13}\,+\,s_{16}\phantom {/} }{s_{13} s_{16} s_{24} s_{45}}\\&\quad +\frac { \phantom {/}s_{16}\,+\,s_{46}\phantom {/}}{s_{16} s_{23} s_{35} s_{46}}. \end{align*}

This is the rational function (27), where we sum over the two critical points of $L_T$ . This amplitude has $12$ poles, $24$ compatible pairs, and $14$ Feynman diagrams, in bijection with the faces of a rhombic dodecahedron shown in Figure 1. By comparison, the amplitude in (9) has $ 9$ poles and is a sum over $14$ Feynman diagrams, given combinatorially by the associahedron.

Figure 1.

Combinatorics of the octahedral hypertree amplitude $m_T$ .

It was shown in [Reference Arkani-Hamed, Bourjaily, Cachazo, Postnikov and TrnkaABCPT15, § 3.2] that, for any on-shell diagram $T$ , the CHY integrand $\mathcal{I}_T$ decomposes as a sum of Parke–Taylor factors $PT(\alpha _1,\ldots , \alpha _n)$ . Therefore, $m_T$ is a linear combination of biadjoint amplitudes $m(\alpha ,\beta )$ as $\alpha ,\beta$ range over pairs of cyclic orders on $\lbrack n\rbrack$ that are both cyclic shuffles of the triples in $T$ . See [Reference Cachazo, He and YuanCHYB14] for details of this construction. Such a decomposition with seven terms is shown in [Reference Arkani-Hamed, Bourjaily, Cachazo, Postnikov and TrnkaABCPT15, (3.12)] for the integrand $\mathcal{I}_T$ in (28).

This example raises several questions for future research. The first concerns the ML degree of an arbitrary hypertree $T$ . By this we mean the number of complex critical points of the function $L_T$ . The hypertree $T$ in (26) has $\textrm {MLdegree}(T)=2$ .

Returning to the title of this paper, we ought to be looking for minimal kinematics.

The following computation shows that we can indeed do this for our running example.

Example 5.8. In Example 5.4 we set the three-particle pole $s_{234}= s_{23} + s_{34} + s_{24}$ to zero, in addition to $s_{14} = s_{25} = s_{36}=0$ . This results in a dramatic simplification of the amplitude:

\begin{equation*} m_T = \frac { \left (s_{12}+s_{45}\right ) \left (s_{13} + s_{46} \right ) \left ( s_{26} + s_{35} \right ) }{ s_{12} s_{13} s_{26} s_{35} s_{45} s_{46} }. \end{equation*}

The ML degree is now $1$ . By Huh’s theorem [Reference HuhHuh14], the critical point is given by a Horn pair $(H,\lambda )$ . In short, the subspace $\{s_{14} = s_{25} = s_{36} = s_{234} = 0\}$ exhibits minimal kinematics.

Finally, all of our questions extend naturally from $\textrm {Gr}(2,n)$ to $\textrm {Gr}(k,n)$ . Using physics acronyms, we seek to extend our amplitudes $m_T$ from CHY theory [Reference Cachazo, He and YuanCHY14] to CEGM theory [Reference Cachazo, Early, Guevara and MizeraCEGM19]. For example, the CEGM potential on the $4$ -dimensional space $X(3,6) = \textrm {Gr}(3,6)^o/({\mathbb{C}}^*)^6$ is

\begin{equation*} L = \sum _{1\le i\lt j\lt k\le 6} \!\! \log (p_{ijk}) \cdot \mathfrak{s}_{ijk}. \end{equation*}

This log-likelihood function is known to have $26$ critical points; see e.g. [Reference Sturmfels and TelenST21, Proposition 5]. We now restrict to the kinematic subspace $\{\mathfrak{s}_{135}=\mathfrak{s}_{235}=\mathfrak{s}_{246}=\mathfrak{s}_{256}=\mathfrak{s}_{356}=\mathfrak{s}_{245}=0\}$ . Then the ML degree drops from $26$ to $1$ . In short, this subspace exhibits minimal kinematics.

Revisiting the case $k=2$ once more, here is a complexity question about graphs for $\mathcal{M}_{0,n}$ .