$\circ\,$ $pp\rightarrow Vjj$ (or similarly $e^+e^-\rightarrow V \rightarrow 4j$) is important for several EW precision measurements

$\circ\,$ Status for Drell-Yan plus jets (Vjj)

$\circ\,$ Status for $pp\rightarrow t\bar tH$

$\;\star\,$ one-loop: $q\bar q\rightarrow t\bar tH$ previously not known analytically;
$\kern15mm$ $gg\rightarrow t\bar t H$ known to $\mathcal{O}(\epsilon^2)$ in terms of form factors
Buccioni, Kreer, Liu, Tancredi '23 $\;\star\,$ two-loop: $q\bar q\rightarrow t\bar tH$ with quark-loop ($n_f$ part), known numerically (pySecDec)
Agarwal, Heinrich, Jones, Kerner, Klein, Lang, Magerya, Olsson '24 $\kern15mm$ $pp\rightarrow t\bar tH$ master integrals in LCA Febres Cordero, Figueiredo, Kraus, Page, Reina '23 $\circ\,$ Goal: show how to reconstruct amplitudes in a manifestly spin- and little-group covariant form

$\circ$ 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

$$ \underbrace{\mathcal{R}^{(2)}}_{\text{finite remainder}} = \mathcal{A}^{(2)}_R \underbrace{- \quad I^{(1)}\mathcal{A}^{(1)}_R \quad - \quad I^{(2)}\mathcal{A}^{(0)}_R}_{\text{divergent + convention-dependent finite part}} + \mathcal{O}(\epsilon) $$

$\phantom{\circ}$ $\mathcal{A}^{(1)}_R$ to order $\epsilon^2$ is still needed to build $\mathcal{R}^{(2)}$, but there is no real physical 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 $\mathbb{F}$

$\circ$ Original computation [1] was performed with Caravel

$\circ$ This computation started from the ancillaries files of [1] Abreu, Febres Cordero, Ita, Klinkert, Page, Sotnikov

$\rightarrow$ Assemble 5 helicity amplitudes into 3 categories: $\mathcal{R}_{\bar qQ\bar QqV}^{\text{NMHV}} ,\, \mathcal{R}_{\bar qggqV}^{\text{MHV}} ,\, \mathcal{R}_{\bar qggqV}^{\text{NMHV}}$

$\circ$ $ttH$ computed analytically (Form optimized) with unitarity, standard Feynman diagrams techniques,
$\phantom{\circ}$ and cross checked with Open-Loops

$\circ$ Amplitude should be gauge and Lorentz invariant, and spin and little-group covariant

${\color{red} ✗}$ gauge dependence, e.g. through reference vectors

${\color{red} ✗}$ tensor decompositions $\epsilon_\mu T^\mu$, polarizations are needed for simplifications

${\color{greeN} ✓}$ $\epsilon_\mu \rightarrow \epsilon_{\alpha\dot\alpha}$, $P^\mu \rightarrow \lambda_\alpha \tilde\lambda_{\dot\alpha}$; all $\alpha, \dot\alpha$ indices contracted; all $\lambda, \tilde\lambda$ random (subject to mom cons)

$\circ$ The singularity structure should be manifest in $\mathbb{C}$ (exprs will then be better behaved in $\mathbb{R}$ too)

${\color{red} ✗}$ Rational reparametrisations of the kinematics change the denominator structure

${\color{red} ✗}$ Forcing unphysical splits misses cancellations (e.g. even nor odd separation)

${\color{greeN} ✓}$ Chiral cancellations are required to obtain the true Least Common Denominator

${\color{greeN} ✓}$ Work off the real slice: $P^\mu \in \mathbb{C}^4$, $\lambda_\alpha \neq \tilde\lambda_{\dot\alpha}^\dagger$. In practice, $P^{\mu=y}\in i\mathbb{Q}\Rightarrow \lambda_{\alpha} \in \mathbb{F}_p \text{ or } \mathbb{Q}_p$

$\circ$ Focus only on final physical expressions

${\color{red} ✗}$ Unphysical intermediate steps may be unnecessarily complicated

${\color{red} ✗}$ Analytic manipulations at this complexity are unfeasible, even on "physical" results

${\color{greeN} ✓}$ Bypass all intermediate steps with numerical evaluations (cancellations happen numerically)

$\circ$ We must work with variables subject to constrains; the language is that of algebraic geometry.

$\phantom{\circ}$ The covariant rings are

$$ \displaystyle \kern10mm R_{Vjj} = \mathbb{F}\big[|1⟩_{\alpha}, [1|_{\dot\alpha}, |2⟩_{\alpha}, [2|_{\dot\alpha}, |3⟩_{\alpha}, [3|_{\dot\alpha}, |4⟩_{\alpha}, [4|_{\dot\alpha}, [5|_{\dot\alpha}, |6⟩_{\alpha} \big] \Big/ \big\langle {\textstyle \sum_{i=1}^4} [5|i]\langle i |6\rangle \big\rangle $$

$\phantom{\circ}$ where we took the the $V$ current to be $[5|\gamma^\mu|6\rangle$ and removed $(5+6)_{\alpha\dot\alpha}$ by mom. cons.; and

$$ \displaystyle \kern10mm R_{ttH} = \frac{\mathbb{F}\big[|1⟩_{\alpha}, [1|_{\dot\alpha}, |2⟩_{\alpha}, [2|_{\dot\alpha}, |\boldsymbol{3}^I⟩_{\alpha}, [\boldsymbol{3}^I|_{\dot\alpha}, |\boldsymbol{4}_J⟩_{\alpha}, [\boldsymbol{4}_J|_{\dot\alpha}, \boldsymbol{5}_{\alpha\dot\alpha} \big]}{\big\langle \sum_{i,I,J} |i\rangle[i|, \langle \boldsymbol{3}|\boldsymbol{3}⟩ +[\boldsymbol{3}|\boldsymbol{3}], \langle \boldsymbol{3}|\boldsymbol{3}⟩-\langle \boldsymbol{4}|\boldsymbol{4}⟩, \langle \boldsymbol{4}|\boldsymbol{4}⟩ +[\boldsymbol{4}|\boldsymbol{4}]\big\rangle} $$

$\phantom{\circ}$ where $\langle \boldsymbol{3}^I|\boldsymbol{3}^J⟩=m\epsilon^{JI} \text{ and } [\boldsymbol{3}^I|\boldsymbol{3}^J]=\bar{m}\epsilon^{IJ}$; we are setting $m=\bar{m}$ and the tops on-shell.
$\phantom{\circ}$ Note: we need only reconstruct a single choice, say $I=J=1$, the other follow by covariance.

$\circ$ Helicity amplitudes are Lorentz invariant; minimal ansätze are build in the invariant sub-rings

$$ \displaystyle \mathcal{R}_{Vjj} = \frac{\mathbb{F}\big[ \langle ij\rangle : \, 1\leq i< j\leq 6, i,j \neq 5, \; [ij] : 1\leq i< j\leq 5 \big]}{\big\langle {\textstyle \sum_{i=1}^4} [5|i]\langle i |6\rangle, 34 \text{ Schouten identities} \big\rangle} $$

$$ \displaystyle \mathcal{R}_{ttH} = \mathbb{F}\big[ \underbrace{\langle 12\rangle, \langle \boldsymbol{3}1\rangle ... ⟨2|\boldsymbol{3}|2] ... ⟨2|\boldsymbol{3}|\boldsymbol{4}|2⟩}_{37\; \text{invariants}} \big]\Big/ \big\langle \underbrace{⟨2|\boldsymbol{3}|2]⟨2|\boldsymbol{4}|1]-⟨2|\boldsymbol{3}|1]⟨2|\boldsymbol{4}|2]-[1|2]⟨2|\boldsymbol{3}|\boldsymbol{4}|2⟩, ...}_{\text{more than} \; 90 \; \text{generators}} \big\rangle $$

Least Common Denominator

(i.e. geometry at codimension one)

Poles & Zeros $\;\Leftrightarrow\;$ Irreducible Varieties $\;\Leftrightarrow\;$ Prime Ideals
Physics $\kern18mm$ Geometry $\kern18mm$ Algebra

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

$$ R = r_j h_j = r_{i\in \mathcal{B}} M_{ij} h_j = \tilde{r}_{i} \, O_{ii'}M_{i'j} \, h_j \, , \qquad O_{ii'}, M_{i'j}\in \mathbb{Q} $$

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

$\circ\,$ By Gaussian elimination, partition the space:

$$ \text{span}(r_{i \in \mathcal{B}}) = \underbrace{\text{column}(\text{Res}(r_{i \in \mathcal{B}}, \mathcal{D}_k^m))}_{\text{functions with the singularity}} \;\;\; \oplus \, \underbrace{\text{null}(\text{Res}(r_{i \in \mathcal{B}}, \mathcal{D}_k^m))}_{\text{functions without the singularity}} $$

Reconstruction from Conjectured Properties

(for planar five-point one-mass amplitudes - all properties checked a posteriori)

$\circ\,$ Denominator pairs $\{\mathcal{D}_i, \mathcal{D}_j\}$ can be cleanly separated:

$$ \frac{\mathcal{N}}{\mathcal{D}_i^{q_i}\mathcal{D}_j^{q_j}\mathcal{D}_{\text{rest}}} \rightarrow \frac{\mathcal{N}_i}{\mathcal{D}_i^{q_i}\mathcal{D}_{\text{rest}}} + \frac{\mathcal{N}_j}{\mathcal{D}_j^{q_j}\mathcal{D}_{\text{rest}}} $$

$\phantom{\circ}\,$ Examples of $\{\mathcal{D}_i, \mathcal{D}_j\}$ are:

$\qquad\star\,$ Any pairs of $s_{ijk}$ or $\Delta_{ij|kl|mn}$ or $\langle i|j|p_V|k|i]-\langle j|l|p_V|k|j]$
$\qquad\star\,$ Any conjugate pair $\{\langle i|j+k|l], \langle l|j+k|i]\}$ or cyclic $\{\langle i|j\rangle, [i|j]\}$
$\qquad\star\,$ Pairs of the form $\{\Delta_{ij|kl|mn}, \langle c|a+b|d] \text{ or } \langle ab \rangle \text{ or } [ab] \}$ unless $\{ab\}$ are $\{ij\}$ or $\{kl\}$ or $\{mn\}$

$\circ\,$ Other denominator pairs $\{\mathcal{D}_i, \mathcal{D}_j\}$ can be separated to order $\kappa$

$$ \frac{\mathcal{N}}{\mathcal{D}_i^{q_i}\mathcal{D}_j^{q_j}\mathcal{D}_{\text{rest}}} \rightarrow \sum_{\kappa - q_j\leq m \leq q_i}\frac{\mathcal{N}_i}{\mathcal{D}_i^{m}\mathcal{D}_j^{\kappa - m}\mathcal{D}_{\text{rest}}} $$

$\qquad\star\,$ E.g. $\Delta_{ij|kl|mn}^4, \langle i|k+l|j]^5$ are separable to order 5.

${\color{greeN} ✓}$ Reconstruction only required 50k $\mathbb{F}_p$ samples $\;{\color{greeN} ✓}$Already simpler than original ones ($\sim$20MB)
$\;{\color{red} ✗}$ Results are unstable and sub-optimal, e.g. numbers like this appeared

Iterated Pole Subtraction

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

$\circ$ Multivariate partial fraction decompositions follow from varieties where pairs of denominator factors vanish

$$ \frac{\mathcal{N}}{\mathcal{D}_1\mathcal{D}_2} \stackrel{?}{=} \frac{\mathcal{N}_2}{\mathcal{D}_1} + \frac{\mathcal{N}_1}{\mathcal{D}_2} \; \Longleftrightarrow \; \mathcal{N} \stackrel{?}{\in} \big\langle \mathcal{D}_1, \mathcal{D}_2 \big\rangle \, \text{ i.e. } \; \mathcal{N} \stackrel{?}{=} \mathcal{N}_1 \mathcal{D}_1 + \mathcal{N}_2 \mathcal{D}_2 $$

$\circ\,$ Retain control by iteratively fitting residues on varieties (using $p$-adic numbers $\mathbb{Q}_p$, get $\mathbb{F}_p$ vals for nums)

$$ \begin{alignedat}{2} & r^{(139 \text{ of } 139)}_{\bar{u}^+g^+g^-d^-(V\rightarrow \ell^+ \ell^-)} = & \qquad\qquad & {\small \text{Variety (scheme?) to isolate term(s)}} \\[2mm] & +\frac{7/4{\color{blue}(s_{24}-s_{13})}⟨6|1+4|5]s_{123}{\color{green}(s_{124}-s_{134})}}{⟨1|2+3|4]⟨2|1+4|3]^2 Δ_{14|23|56}} + ... & \qquad\qquad & \Big\langle ⟨2|1+4|3]^2, Δ_{14|23|56} \Big\rangle \\[1mm] % & -\frac{49/64⟨3|1+4|2]⟨6|1+4|5]s_{123}(s_{123}-s_{234})(s_{124}-s_{134})}{⟨1|2+3|4]⟨2|1+4|3]Δ^2_{14|23|56}} + \dots & \qquad\qquad & \Big\langle Δ_{14|23|56} \Big\rangle \end{alignedat} $$

$\circ\,$ Partial fraction decomposition and numerator insertions from e.g.:

$$ \sqrt{\big\langle ⟨2|1+4|3], Δ_{14|23|56} \big\rangle} = \big\langle {\color{green}(s_{124}-s_{134})}, ⟨2|1+4|3] \big\rangle \, , \\[1mm] \big\langle ⟨1|2+3|4], ⟨2|1+4|3] \big\rangle = \big\langle ⟨1|2+3|4], ⟨2|1+4|3], {\color{blue}(s_{13}-s_{24})}\big\rangle \cap \big\langle ⟨12⟩, [34] \big\rangle $$

Core Tools - Fully Open Source

For fleshed out examples see e.g. GDL (ACAT '22) or Appendix B of 2504.19909

Install from github (git clone) or PyPI (pip install); use of Jupyter is recommended.

$\circ$ pyadic
$\quad\rightarrow$ Finite field $\mathbb{F}_p$ and $p$-adic $\mathbb{Q}_p$ number types, including field extensions
$\quad\rightarrow$ rational number reconstruction (Wang's EEA, LGRR, MQRR)
$\quad\rightarrow$ univariate and multivariante Newthon & univariate Thiele interpolation algorithms in $\mathbb{F}_p$

$\circ$ syngular (in the backhand Singular is used for many operations)
$\quad\rightarrow$ object-oriented algebraic geometry (Field, Ring, Quotient Ring, Ideal)
$\quad\rightarrow$ ring-agnostic monomials and polynomials (with support for unicode characters, e.g. spinor brackets)
$\quad\rightarrow$ multivariate solver (Ideal.point_on_variety), under- and over-constrained systems OK
$\quad\rightarrow$ a semi-numerical prime and primary ideal test (assumes equi-dimensionality of ideal)

$\circ$ lips (Lorentz invariant phase space)
$\quad\rightarrow$ phase space points over any field ($\mathbb{Q}, \mathbb{Q}[i], \mathbb{R}, \mathbb{C}, \mathbb{Q}_p, \mathbb{F}_p$), including internal and external masses
$\quad\rightarrow$ evaluate any Mandelstam or spinor expression (custom ast/regex parser)
$\quad\rightarrow$ generation of any special kinematic configuration (wrapper around Ideal.point_on_variety)

$\circ$ The $pp\rightarrow Vjj$ coefficient functions are now 1.9 MB (down from 1.4 GB), fast and stable.
$\phantom{\circ}$ Matrices $M_{ij}$ account for another 2 MB overall. Transcendental basis at PentagonFunctions++.

$\circ$ The complexity split is: quarks NMHV: 100 KB, gluons MHV: 200 KB, gluons NMHV: 1.6 MB.

$\circ$ The largest numbers are: quarks NMHV and gluons MHV: 3-digit, gluons NMHV: 12 digits.

$\circ$ Pheno ready results for the hard functions are available at FivePointAmplitudes.

A Numerical CAS for Computations in Q-Rings

(partially work in progress)

$\circ$ antares (automated numerical to analytical reconstruction software)
$\rightarrow$ Univariate slicing, LCD determination, basis change, multivariate partial fractioning strategies,
$\phantom{\rightarrow}$ constraining of numerators, Ansatz generation and fitting strategies
$\rightarrow$ Most operations do not require defining the variables (or redundancies), only being able to evaluate them.

$\circ$ antares-results (human readable exprs in docs) with CI tests for coefficients and/or full amplitudes

Thank you
for your attention!

$\circ\,$ The numerator Ansatz takes the form

$\displaystyle \text{Num. poly}(\lambda, \tilde\lambda) = \sum_{\vec \alpha, \vec \beta} c_{(\vec\alpha,\vec\beta)} \prod_{j=1}^n\prod_{i=1}^{j-1} \langle ij\rangle^{\alpha_{ij}} [ij]^{\beta_{ij}}$

$\phantom{\circ}$ 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

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