$\phantom{\circ}\,$ Observations at the LHC are beautifully predicted by the Standard Model

$$ \require{color} \require{amsmath} σ_{2 \rightarrow n - 2} = \sum_{a,b} \int dx_a dx_b f_{a/h_1}(x_a, \mu_F) \, f_{b/h_2}(x_b, \mu_F) \;\hat{\sigma}_{ab\rightarrow n-2}(x_a, x_b, \mu_F, \mu_R) \, , \\ \hat{σ}_{n}=\frac{1}{2\hat{s}}\int d\Pi_{n-2}\;(2π)^4δ^4\big(\sum_{i=1}^n p_i\big)\;|\overline{\mathcal{A}(p_i,h_i,a_i,μ_F, μ_R)}|^2 \, . $$

$\phantom{\circ}\,$ at least to the extent with which we can compute $\mathcal{A} = \mathcal{A}^{(0)} + \alpha_{(s)}\mathcal{A}^{(1)} + \alpha^2_{(s)}\mathcal{A}^{(2)} + \dots$

$\circ\,$ Theoretical uncertainties already larger than experimental ones, especially at higher points

$\sigma^{\text{tot.}}_{pp \, \rightarrow \, Z \, + \, n\,j}:$

$\circ\,$ NNLO is essential for agreement with experiment, e.g.

$\frac{d\sigma_{pp \, \rightarrow \, Z \, + \, \geq 1 \, b \text{ jet}}}{d |\eta|^{b-\text{jet}_1}}:$

$\circ\,$ Besides this, only two other cross-section studies at NNLO, only for the process $q\bar q'\rightarrow Wb\bar b$

$\circ\,$ Phenomenology can be hindered by complexity of results. It's hard to do Monte Carlo integration
$\phantom{\circ}\,$ and verify IR cancellations when you have to evaluate >1GB of files in higher precision.

$\circ$ 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 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

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

$\circ$ Amplitude should be gauge and Lorentz invariant, 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

$\circ$ 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

${\color{red} ✗}$ If a function is neither even nor odd, forcing the split misses cancellations

${\color{greeN} ✓}$ Chiral cancellations yield 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

${\color{greeN} ✓}$ Bypass all intermediate steps with numerical evaluations

Least Common Denominator

(i.e. geometry at codimension one)

$ \displaystyle \kern5mm \{D_j\} \subset \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\} $

$\kern0mm\color{green}\text{New letter!}$

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 requires $\mathbb{F}_p$ 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

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

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

$\circ\,$ Partial fraction decomposition and numerator insertions from e.g. (see appendix of paper):

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

For a fleshed out example with open-source code see GDL (ACAT '22)

$\circ$ The $pp\rightarrow Vjj$ coefficient functions are now 1.9 MB (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.

$\circ$ Amplitudes at antares-results, with human readable expr., and CI tests for full amplitude in real kinematics

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)