Two-Loop Five-Point Amplitudes
in the Spinor Helicity Formalism

Giuseppe De Laurentis
University of Edinburgh

arXiv:2311.10086, arXiv:2311.18752
(GDL, H. Ita, M. Klinkert, V. Sotnikov)
arXiv:24xx.xxxxx
(GDL, H. Ita, B. Page, V. Sotnikov)

ICHEP 2024




Find these slides at gdelaurentis.github.io/slides/ichep_jul2024

Introduction

Cross Sections at the Large Hadron Collider

$\circ\,$ Observations at the LHC are beautifully predicted by the Standard Model through,
$$ \require{color} \require{amsmath} σ_{2 \rightarrow n - 2} = \sum_{a,b} \int dx_a dx_b f_{a/h_1}(x_a, \mu_F) \, f_{b/h_2}(x_b, \mu_F) \;\hat{\sigma}_{ab\rightarrow n-2}(x_a, x_b, \mu_F, \mu_R) \, , \\ \hat{σ}_{n}=\frac{1}{2\hat{s}}\int d\Pi_{n-2}\;(2π)^4δ^4\big(\sum_{i=1}^n p_i\big)\;|\overline{\mathcal{A}(p_i,h_i,a_i,μ_F, μ_R)}|^2 \, . $$
$\phantom{\circ}\,$ At least to the extent with which we can compute $\mathcal{A} = \mathcal{A}^{(0)} + \alpha_{(s)}\mathcal{A}^{(1)} + \alpha^2_{(s)}\mathcal{A}^{(2)} + \dots$

Precision Physics Requires NNLO Corrections

$\circ\,$ NNLO corrections can still be large! Especially in the presence of loop-induced channels,
$\sigma^{\text{NNLO}}_{pp\rightarrow \gamma\gamma\gamma}$
Chawdhry, Czakon, Mitov, Poncelet ('19)
$d\sigma^{\text{NNLO}}_{pp\rightarrow W(\rightarrow \ell\nu)\gamma}/dy_\gamma$

Campbell, GDL, Ellis, Seth ('21)
$\circ\,$ High-multiplicity two-loop amplitudes required also because:

$\qquad\star$ At high energy, some radiation is more likely than no radiation (captured by resummation);
$\qquad\star$ As real-virtual-virtual contributions to emerging N$^{3}$LO computations;
$\qquad\star$ Many interesting kinematic regions are only accessible with extra radiation (e.g. $p_T$ distributions).

Status of Two-Loop Five-Point Amplitudes

Process Analytical Amplitudes Numerical Codes Cross Sections
$pp \rightarrow \gamma\gamma\gamma$ [3$\kern-2.2mm\phantom{x}^\star$, 4$\kern-2.2mm\phantom{x}^\star$, 5] [3$\kern-2.2mm\phantom{x}^\star$, 5] [1$\kern-2.2mm\phantom{x}^\star$, 2$\kern-2.2mm\phantom{x}^\star$]
$pp \rightarrow \gamma\gamma j$ [6$\kern-2.2mm\phantom{x}^\dagger$, 7$\kern-2.2mm\phantom{x}^\dagger$, 9] [6$\kern-2.2mm\phantom{x}^\dagger$] [8$\kern-2.2mm\phantom{x}^\dagger$]
$pp \rightarrow \gamma jj$ [10] [10]
$pp \rightarrow jjj$ [11$^\dagger$, 12, 13, 14] [11$^\dagger$,14] [15$\kern-2.2mm\phantom{x}^\dagger$, 16$\kern-2.2mm\phantom{x}^\dagger$]
$pp \rightarrow Wb\bar b$ [17$\kern-2.2mm\phantom{x}^\dagger$, 18$\kern-2.2mm\phantom{x}^{\dagger\ast}$, 19a$\kern-2.2mm\phantom{x}^\dagger$, 23$\kern-2.2mm\phantom{x}^\dagger$] [23$\kern-2.2mm\phantom{x}^\dagger$] [19a$\kern-2.2mm\phantom{x}^\dagger$, 19b$\kern-2.2mm\phantom{x}^\dagger$]
$pp \rightarrow Hb\bar b$ [20$^{\dagger\ast}$]
$pp \rightarrow Wj\gamma$ [21$^\star$]
$pp \rightarrow Wjj$ [17$\kern-2.2mm\phantom{x}^\dagger$, 23$\kern-2.2mm\phantom{x}^\dagger$] [23$\kern-2.2mm\phantom{x}^\dagger$]
$pp \rightarrow t\bar tH$ [22]
Legend: bold = full color; $\star$ = planar $\neq$ leading color; $\dagger$ = planar = leading color; $\ast$ = ($y_b \neq 0$, $m_b = 0; \text{or } W-\text{onshell}$)

[1] Chawdhry, Czakon, Mitov, Poncelet '19

[3] Abreu, Page, Pascual, Sotnikov '20

[5] Abreu, GDL, Ita, Klinkert, Page, Sotnikov '23

[7] Chawdhry, Czakon, Mitov, Poncelet '21

[9] Agarwal, Buccioni, von Manteuffel, Tancredi '21

[11] Abreu, Febres Cordero, Ita, Page, Sotnikov

[13] GDL, Ita, Klinkert, Sotnikov '23

[15] Czakon, Mitov, Poncelet '21'

[17] Abreu, Febres Cordero, Ita, Klinkert, Page, Sotnikov '21

[19a, 19b] Hartanto, Poncelet, Popescu, Zoia '22

[21] Badger, Hartanto, Kryś, Zoia '22

[23] GDL, Ita, Page, Sotnikov - To appear '24

[2] Kallweit, Sotnikov, Wiesemann '20

[4] Chawdhry, Czakon, Mitov, Poncelet '20

[6] Agarwal, Buccioni, von Manteuffel, Tancredi '21

[8] Chawdhry, Czakon, Mitov, Poncelet '21

[10] Badger, Czakon, Hartanto, Moodie, Peraro, Poncelet, Zoia '23

[12] Agarwal, Buccioni, Devoto, Gambuti, von Manteuffel, Tancredi '23

[14] GDL, Ita, Sotnikov '23

[16] Chen, Gehrmann, Glover, Huss, Marcoli '22

[18] Badger, Hartanto, Zoia '21

[20] Badger, Hartanto, Kryś, Zoia '21

[22] Catani, Devoto, Grazzini, Kallweit, Mazzitelli, Savoini '22

Numerical Computation

Numerical Generalized Unitarity @ 2 Loops
Ita (‘15) Abreu, Febres Cordero, Ita, Page, Zeng (‘17)

$\circ$ The integrand Ansatz is matched to products of trees on cuts
$$ \require{color} \displaystyle \sum_{\text{states}} \, \prod_{\text{trees}} A^{\text{tree}}(\lambda, \tilde\lambda, \ell)\big|_{\text{cut}_{\Gamma}} = \sum_{\substack{\Gamma' \ge \Gamma, \\ i \in M_\Gamma' \cup S_\Gamma'}} \kern-2mm {\color{red}{c_{\,\Gamma',i}(\lambda, \tilde\lambda)}} \, \frac{m_{\Gamma',i}(\lambda\tilde\lambda, \ell)}{\displaystyle \prod_{j\in P_{\Gamma'} / P_{\Gamma}} \rho_{j}(\lambda\tilde\lambda, \ell)}\Bigg|_{\text{cut}_\Gamma} $$
$\star$ Numerical Berends-Giele recursion for LHS, solve for coeffs. in RHS.
$\star$ IBP reduction = decomposition on RHS, $\; m_{\Gamma,i} \in M_\Gamma \cup S_\Gamma $
$\circ$ The SLC (non-planar) cut-hierarchy is significantly larger than the LC (planar) one, e.g.

Partial Amplitudes & Finite Remainders

$\circ$ Amplitude (integrands) can be written as (for a suitable choice of master integrals)

$$ \displaystyle A(\lambda, \tilde\lambda, \ell) = \sum_{\substack{\Gamma,\\ i \in M_\Gamma \cup S_\Gamma}} \, c_{\,\Gamma,i}(\lambda, \tilde\lambda, \epsilon) \, \frac{m_{\Gamma,i}(\lambda\tilde\lambda, \ell)}{\textstyle \prod_{j} \rho_{\,\Gamma,j}(\lambda\tilde\lambda, \ell)} \;\; \xrightarrow[]{\int d^D\ell} \;\; \sum_{\substack{\Gamma,\\ i \in M_\Gamma}} \frac{ \sum_{k=0}^{\text{finite}} \, {\color{red}c^{(k)}_{\,\Gamma, i}}(\lambda, \tilde\lambda) \, \epsilon^k}{\prod_j (\epsilon - a_{ij})} \, {\color{orange}I_{\Gamma, i}}(\lambda\tilde\lambda, \epsilon) $$
$\circ$ $\Gamma$: topologies $\quad\circ$ $M_\Gamma$: master integrands $\quad\circ$ $S_\Gamma$: surface terms
$\circ$ All physical information is contained in the finite remainders, at two loops
Weinzierl ('11)
$$ \underbrace{\mathcal{R}^{(2)}}_{\text{finite remainder}} = \mathcal{A}^{(2)}_R \underbrace{- \quad I^{(1)}\mathcal{A}^{(1)}_R \quad - \quad I^{(2)}\mathcal{A}^{(0)}_R}_{\text{divergent + convention-dependent finite part}} + \mathcal{O}(\epsilon) $$
Catani ('98) Becher, Neubert ('09) Gardi, Magnea ('09)
$\phantom{\circ}$ $\mathcal{A}^{(1)}_R$ to order $\epsilon^2$ is still needed to build $\mathcal{R}^{(2)}$, but there is no real reason to reconstruct it.
$\circ$ Finite remainder as a weighted sum of pentagon functions Chicherin, Sotnikov ('20) Chicherin, Sotnikov, Zoia ('21)
$$ \textstyle \mathcal{R}(\lambda, \tilde\lambda) = \sum_i \color{red}{r_{i}(\lambda,\tilde\lambda)} \, \color{orange}{h_i(\lambda\tilde\lambda)} $$
$\circ$ Goal: reconstruct $\color{red}{r_{i}(\lambda,\tilde\lambda)}$ from numerical samples in a field $\mathbb{F}$
$\mathbb{F}_p$: von Manteuffel, Schabinger ('14); $\phantom{\mathbb{F}_p}$ Peraro ('16)
$\mathbb{C}$: GDL, Maitre ('19); $\mathbb{Q}_p$: GDL, Page ('22)





Analytic & Geometric Structure





based on:
GDL, Page (JHEP 12 (2022) 140)

Least Common Denominator

(i.e. geometry at codimension one)

$\circ$ Polynomials belong to the the covariant quotient ring of spinors,

$$\displaystyle \kern10mm R_n = \mathbb{F}\big[|1⟩, [1|, \dots, |n⟩, [n|\big] \big/ \big\langle \sum_i |i⟩[i| \big\rangle$$
$\circ\,$ The rational function $r_i$ belong to the field of fractions of $R_n$,


$$ \displaystyle r_i(|i\rangle,[i|) = \frac{\mathcal{N}(|i\rangle,[i|)}{\prod_j \mathcal{D}_j^{q_{ij}}(|i\rangle,[i|)} $$
$\circ\,$ The $\mathcal{D}_j$ are related to the letters of the symbol alphabet

Abreu, Dormans, Febres Cordero, Ita, Page ('18)
$$ \displaystyle \{\mathcal{D}_{\{1,\dots,35\}}\} = \bigcup_{\sigma \; \in \; \text{Aut}(R_5)} \sigma \circ \big\{ \langle 12 \rangle, \langle 1|2+3|1] \big\} $$
$\kern0mm\color{green}\text{Identical to 1-loop!}$
The codimension one variety
$\langle x^3 + y^3 - z^2 \rangle$ in $\mathbb{R}[x,y,z]$
$$ \displaystyle \kern5mm \{D_j\} = \kern-3mm \bigcup_{\sigma \; \in \; \text{Aut}(R_6)} \sigma \circ \big\{ \langle 12 \rangle, \langle 1|2+3|1], \langle 1|2+3|4], s_{123}, \Delta_{12|34|56}, ⟨3|2|5+6|4|3]-⟨2|1|5+6|4|2] \big\} $$
$\kern0mm\color{green}\text{New letter! Can we get rid of it?}$
Poles & Zeros $\;\Leftrightarrow\;$ Irreducible Varieties $\;\Leftrightarrow\;$ Prime Ideals
Physics $\kern18mm$ Geometry $\kern18mm$ Algebra
Multivariate Partial Fraction Decompositions

(i.e. geometry at codimension greater than one)

$\langle xy^2 + y^3 - z^2 \rangle$
$\cap$
$\langle x^3 + y^3 - z^2 \rangle$
$=$
$\begin{gather}\langle 2y^3-z^2, x-y \rangle \cap \langle y^3-z^2, x \rangle \\ \cap \langle z^2, x+y \rangle\end{gather}$
$\circ$ When is a partial fraction decomposition possible? (an example)

$$ \frac{\mathcal{N}}{(\prod_j \mathcal{D}_j^{q_j})\times\langle 4|1+3|4]\langle 5|1+4|5]} \stackrel{?}{=} \frac{\mathcal{N}_1}{(\prod_j \mathcal{D}_j^{q_j})\times\langle 4|1+3|4]} + \frac{\mathcal{N}_2}{(\prod_j \mathcal{D}_j^{q_j})\times\langle 5|1+4|5]} $$
$\circ$ Compute primary decompositions and check if $\mathcal{N}$ vanishes on all branches (Hilbert's Nullstellensatz)

$$ J = K \cap \bar K \cap L \cap \bar L \cap M \quad \text{or} \quad V(J) = V(K) \cup V(\bar K) \cup V(L) \cup V(\bar L) \cap V(M) $$
$$ J = \big\langle \langle 4|1+3|4], \langle 5|1+4|5] \big\rangle \qquad K = \big\langle \langle14\rangle,\langle15\rangle,\langle45\rangle,[23] \big\rangle \quad L = \big\langle \langle ij\rangle \; \forall \; i,j\in\{1,\dots 5\} \big\rangle \\[2mm] M = \big\langle \langle 4|1+3|4], \langle 5|1+4|5], |1+4|5\rangle\langle14\rangle + |5|4\rangle\langle15\rangle, \langle\rangle \leftrightarrow [] \big\rangle $$
For a fleshed out example with open-source code see GDL (ACAT '22)







Analytic Reconstruction





also based on:
GDL, Ita, Page, Sotnikov (to appear)

Decorrelating Kinematic Residues

$\circ\,$ Change basis from a subset of the pentagon coefficients $r_{i \in \mathcal{B}}$ to $\mathbb{Q}$-linear combinations $\tilde r$,

$$ R = r_j h_j = r_{i\in \mathcal{B}} M_{ij} h_j = \tilde{r}_{i} \, O_{ii'}M_{i'j} \, h_j \, , \qquad O_{ii'}, M_{ij}\in \mathbb{Q} $$
$\phantom{\circ}\,$ Notation: [mass dimension], {Little-group weights}
$\circ\,$ By Gaussian elimination, partition the space:

$$ \text{span}(r_{i \in \mathcal{B}}) = \underbrace{\text{column}(\text{Res}(r_{i \in \mathcal{B}}, \mathcal{D}_k^m))}_{\text{functions with the singularity}} \;\;\; \oplus \, \underbrace{\text{null}(\text{Res}(r_{i \in \mathcal{B}}, \mathcal{D}_k^m))}_{\text{functions without the singularity}} $$
$\circ\,$ Search for linear combinations that remove as many singularities as possible

$$ \kern12mm \displaystyle O_{i'i} = \bigcap_{k, m} \, \text{nulls}(\text{Res}(r_{i \in \mathcal{B}}, \mathcal{D}_k^m)) $$

Spinor-Helicity Results

$\circ$ The 5-gluon MHV rational functions fit in 3 pages of the appendix,
$$ \tilde{r}^{\text{MHV}}_{\text{first 5 of 115}} = \left\{ \frac{⟨45⟩^2}{⟨12⟩⟨13⟩⟨23⟩}, \frac{⟨45⟩^3}{⟨12⟩^2⟨34⟩⟨35⟩}, \frac{⟨45⟩^3}{⟨12⟩⟨15⟩⟨23⟩⟨34⟩}, \frac{[14][12][35]}{⟨23⟩[45]^3}, \frac{⟨45⟩^2⟨24⟩}{⟨12⟩^2⟨23⟩⟨34⟩}, \dots \right\} \text{+ symmetries}$$
$\circ$ The $pp\rightarrow Wjj$ functions are now 1.9 MB (from 1.3 GB),
$\kern4mm$
$\phantom{\circ}$ Since PentagonsFunction++ can take permutations of the 1-mass basis we only need one $M_{ij}$ per partial
$\phantom{\circ}$ (another 2 MB overall). We now have fast and stable floating-point evaluations in double precision!

Outlook

Taming the Complexity Growth
$\circ$ For every leg or mass, the number of letters in the spinor alphabet grows, as well their mass dimension;

$\circ$ The LCD Ansatz size also grows quickly with
multiplicity (m) and mass dimension (d):

GDL, Maître ('20)
$$ \displaystyle \kern2mm \text{Ansatz size} \geq \small \left(\mkern -9mu \begin{pmatrix}\, m(m-3)/2 \, \\ \, d/2 \, \end{pmatrix} \mkern -9mu \right) $$









$\circ\,$ We can retain control by iterating surface by surface
Campbell, GDL, Ellis, ('22)$\;$ GDL, Page ('22);$\;$ GDL, Maître ('19);$\;$
$$ \begin{alignedat}{2} & r^{(139 \text{ of } 139)}_{\bar{u}^+g^+g^-d^-(V\rightarrow \ell^+ \ell^-)} = & \qquad\qquad & {\small \text{Variety (scheme?) to isolate term(s)}} \\[2mm] & +\frac{7/4(s_{24}-s_{13})⟨6|1+4|5]s_{123}(s_{124}-s_{134})}{⟨1|2+3|4]⟨2|1+4|3]^2 Δ_{14|23|56}} & \qquad\qquad & \Big\langle ⟨2|1+4|3]^2, Δ_{14|23|56} \Big\rangle \\[1mm] & -\frac{49/64⟨3|1+4|2]⟨6|1+4|5]s_{123}(s_{123}-s_{234})(s_{124}-s_{134})}{⟨1|2+3|4]⟨2|1+4|3]Δ^2_{14|23|56}} + \dots & \qquad\qquad & \Big\langle Δ_{14|23|56} \Big\rangle \end{alignedat} $$
$\circ\,$ Partial fraction decomposition and numerator insertions from e.g.:
$$ \sqrt{\big\langle ⟨2|1+4|3], Δ_{14|23|56} \big\rangle} = \big\langle s_{124}-s_{134}, ⟨2|1+4|3] \big\rangle \, , \\[1mm] \big\langle ⟨1|2+3|4], ⟨2|1+4|3] \big\rangle = \big\langle ⟨1|2+3|4], ⟨2|1+4|3], (s_{13}-s_{24})\big\rangle \cap \big\langle ⟨12⟩, [34] \big\rangle $$
Thank you
for your attention!

These slides are powered by:
markdown, html, revealjs, hugo, mathjax, github

Backup Slides

Full $N_C $ motivation
3 is not that big! And certainly not close to $\infty$
Slc contributions to $pp\rightarrow jjj$ should be similar to blue curve.
Expect $\mathcal{O}(10\%)$ effect on duble-virtual hard function,
this is scheme dependant.
Effect on $\sigma^{\text{NNLO}}$ depends on size of $\mathcal{H}^{(2)}$.
Pheno. Goal
Stable and fast evaluations for cross sections
C++ Code available at
gitlab.com/five-point-amplitudes/FivePointAmplitudes-cpp
Analytics available at
zenodo.org/records/10142295 & zenodo.org/records/10231547
with Mathematica, Python and C++ scripts.

Constraints from Poles
Bootstrapping trees (?)

$\circ$ The degree of divergence / vanishing on various surfaces imposes strong constraints, e.g.
$ A^{\text{tree}}_{q^+g^+g^+\bar q^-g^-g^-} = \frac{\mathcal{N(\text{m.d.} = 6\,,\; \text{p.w.} = [-1, 0, 0, 1, 0, 0])}}{\langle 12\rangle\langle 23\rangle\langle 34\rangle [45][56][61]s_{345}}$
$\circ$ Pretend this is un unknown integral coefficient, $\mathcal{N}$ has 143 free parameters.
$\circ$ List the various prime ideal, such as




$ \big\langle \langle 12\rangle, \langle 23\rangle, \langle 13\rangle \big\rangle, \; \big\langle |1\rangle \big\rangle, \; \big\langle \langle 12\rangle, |1+2|3]\big\rangle, \dots$
$\phantom{\circ}$ and impose that $\mathcal{N}$ vanishes to the correct order. We determine it up to an overall constant.
GDL, Page ('22)
$\circ$ Likewise, the ansatz for $A^{\text{tree}}_{g^+g^+g^+ g^-g^-g^-}$ shrinks $1326 \rightarrow 1$, etc..




Effectively, we can compute trees, just from their poles orders.
Note: compared to BCFW there is no information about residues.

Partial Fraction Decompositions

$\circ$ For integral coefficients, we can't rely on the Ansatz to shrinks to an overall constant.
$\circ$ Partial fraction decompositions (PFDs) are a popular method to tame algebraic complexity.
$\circ$ In my opinion, a PFD algorithm needs



$1.$ to say if two poles $W_a$ and $W_b$ are separable into different fractions;
$2.$ ideally, to answer $(1.)$ without having access to an analytic expression.
$\circ$ Hilbert's nullstellensatz: if $\mathcal{N}$ vanishes on all branches of $\langle W_a, W_b \rangle$, then the PFD is possible$\kern-3mm\phantom{x}^\dagger$.
$\circ$ Generalizing to powers $>\kern-1mm 1$ can be done via symbolic powers and the Zariski-Nagata Theorem.
GDL, Page ('22)
$\circ$ Similarly, generalizing to non-radical ideals requires ring extensions.
Campbell, GDL, Ellis ('22)
Issue: evaluations on singular surfaces are expensive, but are not always needed!
Opportunity: we get more than partial fraction decompositions.







$\kern-4mm\phantom{x}^\dagger$ $\langle W_a, W_b\rangle$ needs to be radical.

Beyond Partial Fractions

$\circ$ $\color{red}\text{Case 0}$: the ideal does $\color{green}\text{not involve denominator factors}$.
E.g. a 6-point function $c_i$ has a pole at $⟨1|2+3|4]$ but not at $⟨4|2+3|1]$,
yet it is regular on the irreducible surface $V(\big\langle ⟨1|2+3|4], ⟨4|2+3|1] \big\rangle)$. Then


$\displaystyle c_i \sim \frac{⟨4|2+3|1]}{⟨1|2+3|4]} + \mathcal{O}(⟨1|2+3|4]^0) \; \text{ instead of } \; c_i \sim \frac{1}{⟨1|2+3|4]} + \mathcal{O}(⟨1|2+3|4]^0)$
$\circ$ $\color{red}\text{Case 1}$: the $\color{green}\text{degree of vanishing is non-uniform}$ across branches, for example:

$\displaystyle \frac{s_{14}-s_{23}}{⟨1|3+4|2]⟨3|1+2|4]}$
has a double pole on the first branch, and a simple pole on the second branch of

$\big\langle⟨1|3+4|2], ⟨3|1+2|4]\big\rangle_{R_6} = \big\langle ⟨13⟩, [24] \big\rangle_{R_6} \cap \big\langle ⟨1|3+4|2], ⟨3|1+2|4], (s_{14}-s_{23})\big\rangle_{R_6}$
$\circ$ $\color{red}\text{Case 2}$: ideal is $\color{green}\text{non-radical}$ (example on last slide)


$\displaystyle \small \kern0mm \sqrt{\big\langle {\color{black}⟨3|1+4|2]}, {\color{black}Δ_{23|14|56}} \big\rangle_{R_6}} = \big\langle {\color{black}⟨3|1+4|2]}, {\color{black}s_{124}-s_{134}} \big\rangle_{R_6} $

The Numerator Ansatz

$\circ\,$ The numerator Ansatz takes the form
GDL, Maître ('19)
$\displaystyle \text{Num. poly}(\lambda, \tilde\lambda) = \sum_{\vec \alpha, \vec \beta} c_{(\vec\alpha,\vec\beta)} \prod_{j=1}^n\prod_{i=1}^{j-1} \langle ij\rangle^{\alpha_{ij}} [ij]^{\beta_{ij}}$
$\phantom{\circ}$ subject to constraints on $\vec\alpha,\vec\beta$ due to: 1) mass dimension; 2) little group; 3) linear independence.

$\circ\,$ Construct the Ansatz via the algorithm from Section 2.2 of GDL, Page ('22)
Linear independence = irreducibility by the Gröbner basis of a specific ideal.
$\circ\,$ Efficient implementation using open-source software only

Gröbner bases $\rightarrow$ constrain $\vec\alpha,\vec\beta$
Decker, Greuel, Pfister, Schönemann

Integer programming $\rightarrow$ enumerate sols. $\vec\alpha,\vec\beta$
Perron and Furnon (Google optimization team)





$\circ\,$ Linear systems solved w/ CUDA over $\mathbb{F}_{2^{31}-1}$ ($t_{\text{solving}} \ll t_{\text{sampling}}$) w/ linac (coming soon-ish)