Reconstruction of analytical amplitudes across loops, legs and theories

arXiv:1904.04067 (JHEP) arXiv:1910.11355

with Daniel Maître

IPPP Internal Seminar

A. Reconstruction of analytical spinor-helicity amplitudes

1. Motivation 2. Singular limits 3. Partial fractions and ansatze

4. Rational coefficients in QCD

B. Tree amplitudes from the CHY formalism

1. Motivation 2. Scattering equations 3. CHY-Integrands 4. Amplitudes

A 1.1 Motivation

Cross sections at hadron colliders

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

Powers of coupling in pure gluon scattering.

Chromo-dynamics and kinematics

Color ordering at tree level and one loop [1, 2, …]:

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

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

Intermediate and final expressions

Brute force calculations are a mess:

Results are often much easier [6, 7]:

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

Numerical and Analytical: pros and cons

Numerical calculations efficiently bypass algebraic complexity, see: BlackHat [8], CutTools [9], MadGraph [10], Rocket [11], Samurai [12], NGluon [13], OpenLoops [14]…

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)	$\Psi$	$u, v$

Spinors

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

$$ s_{ij} = ⟨ij⟩[ji] $$

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

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

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

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

We need 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 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.

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.

Partial fraction decomposition #3

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

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:

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: $tr_5(2345)1_\mu - tr_5(1345)2_\mu + tr_5(1245)3_\mu - tr_5(1235)4_\mu + tr_5(1234)5_\mu = 0$

Six-gluon one-loop rational coefficients

Three mass triangle

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

An expression was previously obtained in [16].

We obtain an explicitly rational expression, like for $\mathcal{N}=1$ SUSY in [17].

Pole structure

Do we need square roots of momentum invariants?

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

A six-gluon one-loop rational part

An analytical expression for this component was previously obtained in [18] from Feynman diagrams.

The pole structure

is a bit of a mess:

The rational part

Five-partons two-loop remainders (preliminary results)

The same strategy can be applied to other rational coefficients, e.g. two-loop finite remainders given in [19].

Example: uubggg pmpmp Nf1 #3

and

are equivalent.

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.

4. Amplitudes

Numerical amplitudes

We now have all the building blocks to compute amplitudes.

With the open source packages this is straightforward:

>>> 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

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

Analytical reconstruction

Using the same strategy previously discussed we can reconstruct analytical expressions. For example:

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.

Amplitudes in the CHY formalism can be computed via:

$\circ$ the scattering equations, numerically solved via elimination theory;

$\circ$ CHY-Integrads, which make manifest relations between theories;

Analytical expressions are then accessible using the above strategy.

Outlook

Application to the following quantities is well within reach:

$\circ$ six-parton one-loop amplitudes;

$\circ$ five-parton two-loop amplitudes;

for which we have already reconstructed several components.

Additionally it might be of interest to:

$\circ$ implement analytical expressions on GPGPU for fast evaluations;

$\circ$ reconstruct the seven-gluon one-loop coefficients;

$\circ$ assemble the NNLO three-jets cross-section;

$\circ$ apply the reconstruction technique to double copy structures.

Thank you for your attention!

Reconstruction of analytical amplitudes across loops, legs and theories </font face>

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