# Rational maps and Connected formulas

Some time ago, I began to talk about Amplitudes and the scattering equations. Today a new paper by Cachazo, Guevara, Heydeman, Mizera, Schwarz and Wen appeared, and I thought this was an opportunity to understand a bit more this topic. Section 2 of this paper contains a very enlightening review of the rational maps, scattering equations and CHY formulas in arbitrary dimension.

We consider scattering of $n$ massless particles in arbitrary space-time dimensions, having momentum $p_i^{\mu}$ ($i=1 , \dots , n$) satisfying $p_i^2=0$ and $\sum p_i^{\mu} = 0$. From these, we construct the *scattering map* $$p^{\mu}(z) = \sum\limits_{i=1}^n p_i^{\mu} \prod\limits_{i \neq j} (z - \sigma_i)$$ where the $\sigma_i$ are fixed by the requirement that $p^2(z)=0$ for all $z$. One can see that $p^{\mu}(z)$ is a degree $n-2$ polynomial in $z$, so $p^2(z)$ has degree $2n-4$. It would seem that this imposes $2n-3$ conditions, but note that $p^2(\sigma_i)=0$ is automatic, so really there are $n-3$ constraints. These constraints are for all $i$: $$E_i := \sum\limits_{j \neq i} \frac{p_i \cdot p_j}{\sigma_i - \sigma_j} = 0 \, , $$ which are called the *scattering equations* (only $n-3$ of them are independent). The $-3$ here accounts for the $SL(2,\mathbb{C})$ symmetry of the scattering equations: we can fix three of the $\sigma_i$ as we want, and then the other $n-3$ are fixed up to permutation, leaving $(n-3)!$ solutions.

The scattering map contains the information about the $p_i^{\mu}$, which can be recovered using $$p_i^{\mu} = \frac{1}{2 \pi i} \oint_{\sigma_i} \frac{p^{\mu}(z)}{\prod\limits_{j=1}^n (z - \sigma_j)} \, . $$ This is a triviality, but the interesting part comes now. The Cachazo-He-Yuan formula states that the tree-level $n$-particle scattering amplitude of massless theories is given by $$\mathcal{A}_n = \int \mathrm{d} \mu_n \, \mathcal{I}_L \mathcal{I}_R$$ where $\mathcal{I}_L$ and $\mathcal{I}_R$ are factors that depend on the theory under consideration, and $$\mathrm{d} \mu_n = \delta (\sum p_i^{\mu}) \frac{\prod \delta(p_i^2) \prod ' \delta (E_i) \prod \mathrm{d} \sigma_i}{\mathrm{Vol} SL(2, \mathbb{C})} \, . $$ In other words, the measure is an integration over the $\sigma_i$ taking into account all the kinematic constraints given above (the prime indicates that only $n-3$ constraints should be considered out of the $n$ $E_i$). We stress that this is valid in any space-time dimension.

### Four dimensions

Now let's look at the four-dimensional case. The crucial observation here is that if $p^{\mu}$ is such that $p^2=0$, then one can use spinorial indices $p^{\mu} \rightarrow p^{\alpha \dot{\alpha}} = \sigma^{\alpha \dot{\alpha}}_\mu p^{\mu}$, and decompose the momentum using two spinors, $$p^{\alpha \dot{\alpha}} = \lambda^{\alpha} \tilde{\lambda}^{\dot{\alpha}} \, . $$ This is valid for all the $p_i^{\mu}$, and also for $$p^{\mu} (z) \rightarrow p^{\alpha \dot{\alpha}}(z) = \rho^{\alpha}(z) \tilde{\rho}^{\dot{\alpha}}(z) \, . $$ In this last equality, the degree $n-2$ can be shared in different ways between $\rho$ and $\tilde{\rho}$ : let's say the degrees are respectively $d$ and $n-2-d$. In that case we say we are in the $d$th sector. We can then parametrize the scattering map using the $d+1$ coefficients $\rho^{\alpha}_k$ of $\rho^{\alpha}$ and the $n-d-1$ coefficients $\tilde{\rho}^{\dot{\alpha}}_k$ of $\tilde{\rho}^{\dot{\alpha}}$. We then introduce a measure $$ \mathrm{d}\mu_{n,d}^{4D} \sim \prod \mathrm{d} \sigma_i \prod \mathrm{d}^2 \rho_k \prod \mathrm{d}^2 \tilde{\rho}_k \, , $$ where the $\sim$ means that I ommit a bunch of delta function and other scalar factors. The CHY measure is then recovered by summing over the sectors. For instance, amplitudes in $\mathcal{N}=4$ SYM decompose as $$\mathcal{A}^{\mathcal{N}=4}_n = \sum\limits_{d=1}^{n-3} \mathcal{A}^{\mathcal{N}=4}_{n,d} \, . $$ I do not explain here how the partial amplitudes $ \mathcal{A}^{\mathcal{N}=4}_{n,d}$ are computed, but note that the $d$th sector has $n-2-2d$ units of helicity violation.

### Six dimensions

Let's now turn briefly to six dimensions. The little group was $U(1)$ in four dimensions, and is now $\mathrm{Spin}(4) \sim SU(2) \times SU(2)$. This means that a similar decomposition of $p^{\mu}$ will take place, but with more redundancy. Introducing an angle bracket $\langle \rangle$ taking care of little-group indices contractions, we can write $$p^{AB}(z) = \langle \rho^A(z) \rho^B(z) \rangle$$ with $A,B = 1,2,3,4$. In the case where $n$ is even and the two polynomials above have the same degree, we can obtain an analog of the sector with no helicity violation measure, $\mathrm{d} \mu^{6D}_{n \, \textrm{even}}$. The case of odd $n$ is quite different, in part because this sector does not exist. The details are the object of the paper cited in the introduction.