Analytical Amplitudes from Numerical Exploration

arXiv:1904.04067 (JHEP) G. De Laurentis, D. Maître

arXiv:1910.11355 (JHEP) G. De Laurentis

arXiv:2002.04018 (JHEP) L. Budge, J. M. Campbell, G. De Laurentis, R. K. Ellis, S. Seth

Giuseppe De Laurentis

PhD supervisor: Daniel Maître

Summer Meeting - Freiburg\

A. Reconstruction of analytical spinor-helicity amplitudes

1. Motivation 2. Singular limits 3. Partial fractions 4. Ansatze fitting

B. Some applications

1. Six-gluon amplitude at one-loop 2. Two-loop five-parton finite remainders 3. Higgs + 4-parton (w/ finite top-mass) 4. Tree amplitudes from CHY formalism

A 1.1 Motivation

Cross sections at hadron colliders

$$d\hat{σ}_{n}=\frac{1}{2\hat{s}}dΠ_{n-2};(2π)^4δ^4\big(∑_{i=1}^n p_i\big);|\overline{\mathcal{A}(p_i,μ_F, μ_R)}|^2$$

Better predictions require both more loops and higher multiplicity.

IR singularities have to be cancelled between real and virtual corrections.

More loops $\rightarrow$ analytical complexity; more legs $\rightarrow$ algebraic complexity

Chromo-dynamics and kinematics

Color ordering at tree level and one loop F.A.Berends, W.Giele, Z.Bern, D.A.Kosower, …

$\mathcal{A}^{tree}_{n}({p_i, λ_i, a_i}) = ; g^{n-2} ∑_{σ\in S_n/Z_n} \text{Tr}(T^{a_σ(1)}\dots T^{a_σ(n)}) A^{tree}_n(σ(1^{λ_1}),\dots ,σ(n^{λ_n}))$

$A^{tree},, d_i,, c_i,, b_i,$ and $R$ are rational functions of kinematic invariants only.

Intermediate and final expressions

Brute force calculations are a mess:

Results are often much easier S.J.Parke, T.R.Taylor, F.A.Berends, W.T.Giele

$A^{tree}(1^{+}_{g}2^{+}_{g}3^{+}_{g}4^{-}_{g}5^{-}_{g}) = \frac{i,⟨45⟩^{4}}{⟨12⟩⟨23⟩⟨34⟩⟨45⟩⟨51⟩}$

Numerical and Analytical: pros and cons

Numerical calculations efficiently bypass algebraic complexity BlackHat [C.F.Berger et al.], CutTools [G.Ossola et al.], MadGraph [J.Alwall et al.], Rocket [W.T.Giele, G.Zanderighi], Samurai [P.Mastrolia et al.], NGluon [S.Badger et al.], OpenLoops [F.Cascioli et al.], …

Lowest-laying representations of the Lorentz group

$\boldsymbol{(j_{-},j_{+})}$	dim.	name	quantum field	kinematic variable
(0,0)	1	scalar	$h$	m
(0,1/2)	2	right-handed Weyl spinor	$\chi_{R,\alpha}$	$\lambda_\alpha$
(1/2,0)	2	left-handed Weyl spinor	$\chi_L^{,\dot\alpha}$	$\bar{\lambda}^{\dot\alpha}$
(1/2,1/2)	4	rank-two spinor/four vector	$A^\mu/A^{\dot\alpha\alpha}$	$P^\mu/P^{\dot\alpha\alpha}$
(1/2,0)$\oplus$(0,1/2)	4	bispinor (Dirac spinor)	$\Psi$	$u, v$

Spinor Covariants

Weyl spinors are sufficient for massless particles:

$\text{det}(P^{\dot\alpha\alpha})=m^2 \rightarrow 0 \quad \Longrightarrow \quad P^{\dot\alpha\alpha} = \bar\lambda^{\dot\alpha}\lambda^\alpha$.

$$ \lambda_\alpha=\frac{1}{\sqrt{p^0+p^3}}\begin{pmatrix}p^0+p^3 \\ p^1+ip^2\end{pmatrix} , , ;;; \lambda^\alpha=\epsilon^{\alpha\beta} \lambda_\beta =\frac{1}{\sqrt{p^0+p^3}}\begin{pmatrix}p^1+ip^2 \\ -p^0+p^3\end{pmatrix} $$

$\bar\lambda_{\dot\alpha}=\frac{1}{\sqrt{p^0+p^3}}\begin{pmatrix}p^0+p^3 \\ p^1-ip^2\end{pmatrix} , , ;;; \bar\lambda^{\dot\alpha}=\epsilon^{\dot\alpha\dot\beta}\bar\lambda_{\dot\beta}=\frac{1}{\sqrt{p^0+p^3}}\begin{pmatrix}p^1-ip^2 \\ -p^0+p^3\end{pmatrix}$

$$ \bar\lambda_{\dot\alpha} = (\lambda_\alpha)^* \quad if \quad p^i \in \mathbb{R}; \quad \quad \bar\lambda_{\dot\alpha} \neq (\lambda_\alpha)^* \quad if \quad p^i \in \mathbb{C} $$

Spinor Helicity Invariants

$$ s_{ij} = ⟨ij⟩[ji] $$

$$ ⟨i;|;(j+k);|;l] = (λ_i)^α (\not P_j + \not P_k )_{α\dotα} \barλ_l^\dotα $$

$$ ⟨i;|;(j+k);|;(l+m);|;n⟩ = (λ_i)^α (\not P_j + \not P_k )_{α \dot α} (\bar{\not P_l} + \bar{\not P_m} )^{\dot α α} (λ_n)_α $$

$$ tr_5(ijkl) = tr(\gamma^5 \not P_i \not P_j \not P_k \not P_l) = [i,|,j,|,k,|,l,|,i⟩ - ⟨i,|,j,|,k,|,l,|,i] $$

Why use complex momenta?

We want to be able to distinguish between different poles. For example, we want to tell apart:

$$ \frac{1}{⟨ij⟩} \quad and \quad \frac{1}{[ji]} $$

but with real momenta we would have:

$$ ⟨ij⟩ \sim [ji]^* \sim \sqrt{s_{ij}} $$

Singular limits $\Rightarrow$ poles of the amplitude

Let $\{r\}$ be a set of Lorentz invariants

$r_i \in \{ ⟨12⟩, ⟨13⟩, \dots, ⟨1|2+3|4], \dots, s_{123}, \dots \}$,

and let $, \mathbb{f} ,$ be the function we want to reconstruct.

We want to build phase space points where a single invariant from $\{r\}$ vanishes:

$r_i \rightarrow ε \ll 1, \quad r_{j \neq i} \sim \mathcal{O}(1), \quad \mathbb{f} \rightarrow ε^α ; \Rightarrow ; log(\mathbb{f}) \rightarrow α\cdot log(ε)$

$\Rightarrow$ The slope of the log-log plot gives us the type of singularity, if it exists.

Tree-level example

$\mathbb{f} = A^{tree}(1^{+}_{g}2^{+}_{g}3^{+}_{g}4^{-}_{g}5^{-}_{g}6^{-}_{g})$

Note: the invariant on the x-axis gets smaller from left to right.

The least common denominator

Studying all the limits yields the least common denominator for $\mathbb{f}$:

$\mathbb{f} = \frac{\mathcal{N_{LCD}}}{\mathcal{D_{LCD}}} = \frac{\mathcal{N_{LCD}}}{⟨12⟩⟨16⟩[16]⟨23⟩⟨34⟩[34][45][56]s_{234}s_{345}}$.

The complexity of the numerator depends on two parameters:

mass dimension; $\quad \quad \quad$ 2. little group scalings.

In this case $\mathcal{N_{LCD}}$ has mass dimension 10, and phase weights [-1, 0, -1, 1, 0, 1].

The ansatz has 1326 independent terms.

How big is the ansatz?

Easiest to count at constant null phase weights; the size of the ansatz is a function of:1. its mass dimension ($d$) $\quad\quad$ 2. multiplicity of phase space ($m$).

$|s_{ij}| = \frac{m(m-3)}{2}$ $\quad\quad$ $|tr_5| = {m-1 \choose 4}$

$\left(\mkern -9mu \begin{pmatrix}, |s_{ij}| , \\ , d/2 , \end{pmatrix} \mkern -9mu \right) \leq$ ansatz size $\leq \left(\mkern -9mu \begin{pmatrix} , |s_{ij}| , \\ , d/2 , \end{pmatrix} \mkern -9mu \right) + |tr_5| \left(\mkern -9mu \begin{pmatrix} , |s_{ij}| , \\ , (d-4)/2 , \end{pmatrix} \mkern -9mu \right)$

The upper bound is saturated for $(\forall m \wedge d \leq 4)$ and for $(\forall d \wedge m \leq 5)$. Otherwise it is an overcounting due to Schouten identity for 4-momenta: $tr_5(2345)1_\mu - tr_5(1345)2_\mu + tr_5(1245)3_\mu - tr_5(1235)4_\mu + tr_5(1234)5_\mu = 0$

Partial fractioning

Except for the easiest cases, we should really think about $\mathbb{f}$ as:

$\mathbb{f} = \sum_i \frac{\mathcal{N}_i}{\mathcal{D}_i} = \sum_i \frac{\mathcal{N}_i}{\mathcal{R}_i\mathcal{S}_i}$,

where $\mathcal{R}_i$ are products of subsets of $\mathcal{D_{LCD}}$ (i.e. real poles), and $\mathcal{S}_i$ are products of factors not in $\mathcal{D_{LCD}}$ (i.e. spurious poles).

Doubly singular limits

The required information can be accessed from doubly singular limits.

We now want phase space points where two invariants vanish:

$r_i \rightarrow ε \ll 1, \quad r_j \rightarrow ε \ll 1, \quad \mathbb{f} \rightarrow ε^α ; \Rightarrow ; log(\mathbb{f}) \rightarrow α\cdot log(ε)$

In general we cannot guarantee uniqueness anymore, even with complex momenta. $\exists ;r_{k \neq i, j} \sim \epsilon$.

Information from doubly singular limits

The first number is the slope of the log-log plot in the limit, the second number is the degeneracy of the phase space in the limit.

Which poles share the same denominator?

The following is single line of the table in the previous slide:

Diagramatic representation of relation between poles

Partial fraction decomposition #1

(Spoiler: $ \quad \small \mathcal{N_1} = 1i⟨4|2+3|1]^3, \quad \mathcal{N_2} = -1i⟨6|1+2|3]^3$)

Where does $⟨2|1+6|5]$ come from?

$\{⟨12⟩, [56]\}_{\epsilon} , \cap , \{⟨16⟩, s_{234}\}_{\epsilon} , \cap , \{⟨23⟩, [45]\}_{\epsilon} , \cap , \{[34], s_{234}\}_{\epsilon} , \dots$

Partial fraction decomposition #2

(Spoiler: $ \quad \small \mathcal{N_1} = 1i[23]^3⟨56⟩^3, \quad \mathcal{N_2} = -1i[12]^3⟨45⟩^3, \quad \mathcal{N_3} = 1is_{123}^3 $)

$⟨1|2+3|4] = \{⟨12⟩, [34]\}_{\epsilon} , \cap , \{⟨23⟩, s_{234}\}_{\epsilon} , \cap , \{⟨16⟩, [45]\}_{\epsilon} , \cap , \dots$ $⟨3|1+2|6] = \{⟨23⟩, [16]\}_{\epsilon} , \cap , \{⟨12⟩, s_{345}\}_{\epsilon} , \cap , \{⟨34⟩, [56]\}_{\epsilon} , \cap , \dots$

Both of these partial fractions correspond to some BCFW shift, but we are not limited to these representations.

($\small \mathcal{N_1} = 1i[12]⟨45⟩⟨4|2+3|1]^2, , \mathcal{N_2} = 1i[23]⟨56⟩⟨6|1+2|3]^2, , \mathcal{N_3} = 1i⟨4|2+3|1]⟨6|1+2|3]s_{123}$)

We now have no spurious poles, but $s_{234}$ and $s_{345}$ appear in the same denominator, although the doubly singular limit suggests they shouldn’t.

This means that in the limit of $\{s_{234}, s_{345}\} \rightarrow \epsilon$ the individual terms will behave like $\epsilon^{-2}$ but the sum as $\epsilon^{-1}$.

Different representations can be exploited to ensure numerical stability.

Spinor ansatz

Let’s consider the first representation:

$\mathbb{f}=$ $\frac{\mathcal{N_1}}{[16]⟨23⟩⟨34⟩[56]⟨2|1+6|5]s_{234}}+$ $\frac{\mathcal{N_2}}{⟨12⟩⟨16⟩[34][45]⟨2|1+6|5]s_{345}}$

$[⟨24⟩⟨24⟩⟨24⟩[12][12][12],$ $⟨24⟩⟨24⟩⟨34⟩[12][12][13],$ $⟨24⟩⟨24⟩⟨45⟩[12][12][15],$ $⟨24⟩⟨34⟩⟨34⟩[12][13][13],$ $⟨24⟩⟨34⟩⟨45⟩[12][13][15],$ $⟨24⟩⟨45⟩⟨45⟩[12][15][15],$ $⟨34⟩⟨34⟩⟨34⟩[13][13][13],$ $⟨34⟩⟨34⟩⟨45⟩[13][13][15],$ $⟨34⟩⟨45⟩⟨45⟩[13][15][15],$ $⟨45⟩⟨45⟩⟨45⟩[15][15][15]]$,

much smaller than the one for $\mathcal{N_{LCD}}$ which had more than 1000 terms!

Gaussian elimination

$\mathbb{f}=$ $\frac{\mathcal{N_1}}{[16]⟨23⟩⟨34⟩[56]⟨2|1+6|5]s_{234}}+$ $\frac{\mathcal{N_2}}{⟨12⟩⟨16⟩[34][45]⟨2|1+6|5]s_{345}}$

Isolate the first term by generating phase space points in the limit of, say, $s_{234} \rightarrow \epsilon$.

Then we can reconstruct the numerical coefficient of the ansatz entries by Gaussian elimination.

$$ M_{ij}c_j = \mathbb{f}(P_i) \quad \text{i.e.} \quad \begin{pmatrix} \leftarrow ansatz(P_1) \rightarrow \\ \leftarrow ansatz(P_2) \rightarrow \\ \dots \end{pmatrix} \cdot \begin{pmatrix} c_1 \\ c_2 \\ \dots \end{pmatrix} = \begin{pmatrix} \mathbb{f}(P_1) \\ \mathbb{f}(P_2) \\ \dots \end{pmatrix} $$

We get:

Brief litterature review

First complete numerical calculation R.K.Ellis, W.T.Giele, G.Zanderighi

Pheno applications to 4-jet production Z.Bern, G.Diana, L.J.Dixon, F.FebresCordero, S.Hoeche, D.A.Kosower, H.Ita, D.Maitre, K.Ozeren S.Badger, B.Biedermann, P.Uwer, V.Yundin

NMHV analytical cut constructible part R.Britto, B.Feng, P.Mastrolia

NMHV analytical rational part Z.Xiao, G.Yang, C.Zhu

More litterature info David C. Dunbar

Our results

First complete analytical results presented with a single notation.

All expressions are manifestely gauge invariant and rational (no square roots).

Best speed up is a factor of ~75 in the split NMHV configuration (compared to numerical computations in BlackHat).

Worst one is a factor of ~2 in the alternated NMHV configuration.

Bottleneck: numerators of subleading poles.

Three mass triangle

$\mathcal{D_{LCD}} = ⟨12⟩[12]⟨34⟩[34]⟨56⟩[56]⟨1|3+4|2]^4⟨3|1+2|4]^4⟨5|1+2|6]^4Δ_{135}^3$

We obtain an explicitly rational expression, like for $\mathcal{N}=1$ SUSY by N.E.J.Bjerrum-Bohr, D.C.Dunbar, W.B.Perkins.

Pole structure: are there radicals?

How to avoid square roots

All branch cuts should have been taken care of by unitarity cuts.

We should be able to explain this without using square roots.

$\Delta_{135} = (K_1 \cdot K_2)^2 - K_1^2 K_2^2$

$(\Omega_{351})^2 \equiv (2s_{12}s_{56}-(s_{12}+s_{56}-s_{45})s_{123})^2 = 4s_{123}^2\Delta_{135}-4s_{12}s_{56}\langle 4|1+2|3]\langle 3|1+2|4]$

$(\Pi_{351})^2 \equiv (s_{123}-s_{124})^2 = 4\Delta_{135}-4\langle 4|1+2|3]\langle 3|1+2|4]$

The three-mass triangle rational coefficient

(Showing ⟨1|3+4|2], ⟨3|1+2|4] and ⟨5|1+2|6] quadruple and triple poles)

A six-gluon one-loop rational part

The pole structure

is a bit of a mess:

The rational part

(Showing ⟨1|2+3|1], ⟨1|2+3|6] and symmetric poles)

Preview of refitted results into spinor helicity of [S. Abreu, J. Dormans, F. Febres Cordero, H. Ita, B. Page, V. Sotnikov]

Expression complexity

Using the Mandelstam expressions as numerical input, we reconstruct them in spinor helicity.

Example: uubggg pmpmp Nf1 #3

and

are equivalent.

Example: ggggg mpmpp Nf1 # 9

<img src="ggggg_mpmpp_nf1_9.png”; style="max-width:512px;float:center;border:none;“> and $-1\frac{[12]³[15][23]⟨25⟩³[35]³}{[13]⁴[25]⟨5|1+2|5]³}+\frac{97}{12}\frac{[12]⁴⟨25⟩[35]⁴}{[13]⁴[25]³⟨5|1+2|5]}$ $+\frac{13}{3}\frac{[12]⁴⟨15⟩[15][35]⁴}{[13]⁴[25]⁴⟨5|1+2|5]}+\frac{1}{4}\frac{[12]⁴⟨15⟩[15]⟨25⟩[35]⁴}{[13]⁴[25]³⟨5|1+2|5]²}$ $-\frac{3}{2}\frac{[12]²⟨25⟩²[25][35]²}{[13]²[25]⟨5|1+2|5]²}+\frac{7}{4}\frac{[12]³⟨25⟩²[35]³}{[13]³[25]⟨5|1+2|5]²}$ $-\frac{43}{3}\frac{[12]³⟨25⟩[35]³}{[13]³[25]²⟨5|1+2|5]}$ $-\frac{25}{3}\frac{[12]³⟨15⟩[15][35]³}{[13]³[25]³⟨5|1+2|5]}$ $-\frac{3}{2}\frac{[12]⟨25⟩[25][35]}{[13][25]⟨5|1+2|5]}$ $+4\frac{[12]²⟨25⟩[35]²}{[13]²[25]⟨5|1+2|5]}$ $-\frac{15}{2}\frac{[12]²[35]²}{[13]²[25]²}$ $+\frac{7}{2}\frac{[12][35]}{[13][25]}$ $-\frac{2}{3}$

are equivalent.

Biggest linear systems to solve

Recall that size of the linear system is directly related to the mass dimension of the numerators by this formula

By performing the partial fraction decomposition before fitting the numerators the size of the biggest systems to solve is significantly reduced.

Numerical stability

Errors made by double-precision evaluations over $1k$ $collider$ phase space points.

Example of cut diagram

Only singularity involving $m_{top}$ (from pentagon contributions)

$16 |S_{1×2×3×4}| = −s_{12} , s_{23} , s_{34} , \langle 1 |2 + 3|4] , \langle 4|2 + 3|1] + m^2_{top} , tr_5(1234)^2$

We can generate point near this singularity in a similar fashion.

Structure of the coefficients

The massive external leg (the Higgs) is easily accomodated by considering it as a pair of massless particles (think decay products). In the end all dependance on $P_{Higgs}$ is removed by using momentum conservation.

The coefficients are Taylor expasions in $m_{top}$:

$C^{(0)} + m^2_{top} C^{(2)}$.

with $C^{(0)}$ and $C^{(2)}$ resabling the six-gluon coefficients.

4. Tree Amplitudes from CHY formalism

Brief overview

Massless $n-$point tree amplitudes are given by

$\textit{A}_n , = , z_1^4 \cdot i \sum_{j = 1}^{(n-3)!} \frac{I_{\scriptscriptstyle CHY}(z^{(j)}(k); k; ϵ)}{\det(ϕ_{rst}^{ijk})(z^{(j)}(k); k)}$

The $z’s$ are related to the momenta through the scattering equations.

The only process dependant part is $I_{CHY}$, the rest depends only on multiplicity.

For instance:

(See backup slides for specific definitions of functions)

Numerical amplitudes

Developed two python packages:

>>> pip install seampy
>>> pip install lips

seampy from Scatterin Equations and AMplitudes with PYthon
lips from Lorentz Invarinat Phase Space

Arbitrary-precision floating-point amplitudes with a couple of lines of code

>>> from seampy import theories, NumericalAmplitude
>>> from lips import Particles

>>> theories
[YM, EG, BS, BI, NLSM, Galileon, CG, DF2]

>>> oDF2Amp = NumericalAmplitude(theory="DF2", helconf="+++++")
>>> oParticles = Particles(5)

>>> oDF2Amp(oParticles)
mpc(real='#nbr', imag='#nbr')

Analytical reconstruction

First complete analytical expressions for $(DF)^2$ theory at five point and more.
For example:

$$ \begin{gathered} A_{(\text{DF})^2}(1^+,\,2^+,\,3^+,\,4^+,\,5^+) =\\
\frac{i[12]⟨13⟩⟨25⟩[35]^2}{⟨12⟩^2⟨34⟩⟨45⟩}+\frac{i[14][24][35]}{⟨12⟩⟨35⟩}+\\
(12345 → 23451)\,+\,(12345 → 34512)\,+\\
(12345 → 45123)\,+\,(12345 → 51234)\,+\\
\frac{2i[15][23]⟨4|1+2|4]}{⟨12⟩⟨34⟩⟨45⟩}+\\
\frac{2i[12][45]⟨3|1+5|3]}{⟨15⟩⟨23⟩⟨34⟩}+\\
\frac{2i[12][15][34]}{⟨23⟩⟨45⟩}\phantom{+} \end{gathered} $$

$(DF)^2$ double copies into conformal gravity.

(CG is the zero-mass limit of a mass deformed theory which reproduces EG at infinite mass)

>>> oDF2Amp(oParticles) mpc(real='#nbr', imag='#nbr')

Analytical reconstruction

First complete analytical expressions for $(DF)^2$ theory at five point and more. For example:

$(DF)^2$ double copies into conformal gravity.

(CG is the zero-mass limit of a mass deformed theory which reproduces EG at infinite mass)

Summary

Spinor-helicity rational coefficients can be reconstructed using:

$\circ$ singular limits for to the least common denominator;

$\circ$ doubly singular limits for partial fraction decompositions;

$\circ$ generic ansatze in singular limits for numerators;

Applications involve tree-level, one-loop and two-loop expressions alike.

It is possible to accomodate masses (both internal/external).

Thank you for your attention!

B 1.1 Motivation

Hidden relations …

Relations between amplitudes in different theories are often hidden in a standard Lagrangian quantum field theory formulation.

$\mathcal{L_{Biadjoint Scalar}} = 2∂^µΦ^{aa′}∂_µΦ^{aa′}+\frac{y}{3}f^{abc}\tilde f^{a′b′c′}Φ^{aa′}Φ^{bb′}Φ^{cc′}$

E.g. the above are related by the KLT relations [20], more generally known as double copy relations.

.. are made explicit

By comparison in the CHY formalism [21, 22, 23] these relations are made explicit in the structure of the CHY-Integrands:

Loosely speaking we have [24]:

$A = \int I_{CHY} \quad$ and $\quad A_{EG} = A_{YM} \tilde A_{YM} / A_{BS}$

Independent formulation

$\circ$ it does not rely on the Lagrangian or Feynman diagrams;

$\circ$ it is more efficient than a brute-force computation;

$\circ$ it is less efficient than recursion relations;

$\circ$ but the latter in some cases are not available.

B 1.2 General Introduction

Scattering equations and Riemann sphere

The backbone of the CHY formalism are the scattering equations:

The $k$'s are the momenta, and the $z$'s live on the Riemann sphere.

Möbius invariance or redundancy

The scattering equations $f_a$ are invariant under Möbius transformations:

A Möbius transformation is obtained by performing the following operations on the Riemann sphere:

In general this allows us to fix three of the $z$'s:

Amplitudes from integrals

The scattering amplitudes $\textit{A}_n$ are given by:

With the following definitions:

which make $\textit{A}_n$ independent of the choice of $\{i, j, k\}$ and $\{r, s, t\}$.

Amplitudes from sums

In practice we want an algebraic equation of the amplitudes.

Change variables $z_a \rightarrow f_a$, $,$ pick up a Jacobian:

$$ϕ_{ab}, = \frac{∂ f_a}{∂ z_b}= \begin{cases} \frac{2 k_a \cdot k_b}{(z_a - z_b)^2} & a \neq b ; ,\\\
- \sum\limits_{j \in A \backslash {a}} \frac{2 k_a \cdot k_j}{(z_a - z_j)^2} & a = b ; . \end{cases}$$

Integrate over the delta functions, set $\{z_1, z_2, z_3\} = \{\infty, 1, 0\}$ get:

$\textit{A}_n , = , z_1^4 \cdot i \sum_{j = 1}^{(n-3)!} \frac{I_{\scriptscriptstyle CHY}(z^{(j)}(k); k; ϵ)}{\det(ϕ_{rst}^{ijk})(z^{(j)}(k); k)}$

B 2. The scattering equations

The original scattering equations

This part of the computation depends only on the scattering multiplicity! No information about the thoery is required.

Solving the scattering equations (SE) is not trivial.

If we simply take the common denominator we obtain a polynomial of much higher degree than necessary.

The polynomial scattering equations

Instead we are going to consider the polynomial SE $h_{m}$ from [25]

For example, for $n = 5$:

$h_2 = z_2 z_3 k_{\{1,2,3\}}^2+z_2 z_4 k_{\{1,2,4\}}^2+z_3 z_4 k_{\{1,3,4\}}^2$

Elimination theory

Needed for solving systems of polynomial equations. Just like Gaussian elimination is used for the linear case.

We can rewrite the two polynomial SE for $n = 5$ as:

Using the following simplified notation:

Five-point solutions

Now taking the determinant of the matrix form of the SE we get:

This is the required quadratic in $ z_4 / z_3 $ .

By reintroducing $z_2$ and setting it to 1 we can then solve for $z_4$ and $z_3$:

What happens at six point?

Now we have three polynomial scattering equations:

We want to eliminate 4 variables: $$ \{1, z_2\} \otimes \{1, z_3\} = \{1, z_2, z_3, z_2 z_3\} $$

But we only have 3 equations:

Completing the system of equations

The solution is to introduce new variables and new equations:

Now we have the same number of equations and variables, write:

Problem: the determinant is sextic

Abel–Ruffini theorem: no algebraic solution for order higher than 4.

In general: $\;$ order det $\;\sim\;$ # of SE solutions $\;\sim\;$ $(n-3)!$

Solution: use a numerical root-finding algorithm

Two open source packages: seampy, lips

>>> hms(6)
⎡                     s₁₂⋅z₂ + s₁₃⋅z₃ + s₁₄⋅z₄ + s₁₅⋅z₅                     ⎤
⎢s₁₂₃⋅z₂⋅z₃ + s₁₂₄⋅z₂⋅z₄ + s₁₂₅⋅z₂⋅z₅ + s₁₃₄⋅z₃⋅z₄ + s₁₃₅⋅z₃⋅z₅ + s₁₄₅⋅z₄⋅z₅⎥
⎣     s₁₂₃₄⋅z₂⋅z₃⋅z₄ + s₁₂₃₅⋅z₂⋅z₃⋅z₅ + s₁₂₄₅⋅z₂⋅z₄⋅z₅ + s₁₃₄₅⋅z₃⋅z₄⋅z₅     ⎦

>>> oParticles = Particles(6)
>>> num_ss = {str(s): oParticles.compute(str(s)) for s in mandelstams(6)}

>>> sols = solve_scattering_equations(6, num_ss)
>>> len(sols)
6
>>> sols[0]
{'z3': mpc(real='#nbr', imag='#nbr'),
 'z4': mpc(real='#nbr', imag='#nbr'),
 'z5': mpc(real='#nbr', imag='#nbr')}

Recursion for the elimination theory matrix

$$ \mkern -24mu M_i= \left( \begin{array}{ccccccc} M_{i-1} & M_{i-1}^{z_{i-3}} & 0 & \dots & 0 & 0\\
0 & M_{i-1} & M_{i-1}^{z_{i-3}} & \dots & 0 & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots\\
0 & 0 & 0 & \dots & M_{i-1} & M_{i-1}^{z_{i-3}}\\
\end{array} \right), \quad M_4=H, \quad H = \left( \begin{array}{c} h_1\\
h_2\\
\vdots\\
h_{n-3}\\
\end{array} \right) $$

$M_i$ is of dimensions $(i−4)×(i−3)$ when written in terms of $M_{i−1}$. After a derivative is taken the relevant $z_i$ is assumed to be set to zero.

B 3. CHY-Integrands

Quick recap

Recall the expression for the amplitudes:

$\textit{A}_n , = , z_1^4 \cdot i \sum_{j = 1}^{(n-3)!} \frac{I_{\scriptscriptstyle CHY}(z^{(j)}(k); k; ϵ)}{\det(ϕ_{rst}^{ijk})(z^{(j)}(k); k)}$

$\circ\mkern -10mu \checkmark$ Theory independent part $z^{(j)}(k)$.

$\circ\mkern -10mu ✗$ Theory dependent part $I_{\scriptscriptstyle CHY}(z^{(j)}(k); k; ϵ)$.

Summary of considered theories

Definitions (scalars)

The Parke-Taylor-like factor $C_n$ is defined as:

Theories involving this factor are color ordered.

The $W_1$ factor from [29]:

Definitions (matrices)

$$ B_{ab} = \begin{cases} \frac{2 ϵ_a \cdot ϵ_b}{(z_a - z_b)} & a \neq b ; ,\\\
0 & a = b ; , \end{cases} \quad\quad C_{ab} = \begin{cases} \frac{2 ϵ_a \cdot k_b}{(z_a - z_b)} & a \neq b ; ,\\\
- \sum\limits_{j \in A \backslash \{a\}} \frac{2 ϵ_a \cdot k_j}{(z_a - z_j)} & a = b ; . \end{cases} $$

The Pfaffian

The determinant of anti-symmetric matrices can be written as the square of a polynomial. This polynomial is called the Pfaffian.

Note: non trivial to get the Pfaffian from the determinant, due to sign ambiguity.

The reduced Pfaffian

The matrices $A$ and $\Psi$ have two null vectors, therefore their determinant and Pfaffian are zero.

Instead use:

$\Psi_{ij}^{ij}$ denotes deletion of rows and columns $i$ and $j$.

An easy identity to spot

Born-Infeld, non linear sigma model and Galileon amplitudes involve the Pfaffian of the $n\times n$ matrix $A$.

The Pfaffian of an odd-size matrix is zero, meaning in these theories only even point functions are non-vanishing.

More specifically, for Born-Infeld amplitudes this is a consequence of helicity conservation.

Analytical Amplitudes from Numerical Exploration

Table of Contents

A 1.1 Motivation

4. Tree Amplitudes from CHY formalism

Summary

B 1.1 Motivation

B 1.2 General Introduction

B 2. The scattering equations

B 3. CHY-Integrands