$\circ\,$ $pp\rightarrow Vjj$ (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: $\mathcal{R}_{\bar qQ\bar QqV}^{\text{NMHV}} ,\, \mathcal{R}_{\bar qggqV}^{\text{MHV}} ,\, \mathcal{R}_{\bar qggqV}^{\text{NMHV}}$

$\circ$ $t\bar{t}H$ 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.

$\circ$ For example, consider polynomials in two variables $x, y$. They live in a polynomial ring:

$$ \displaystyle f(x,y), g(x, y), h(x, y) \in \mathbb{Q}[x, y] \, . $$

$\circ$ Now, localize them, e.g. on the unit circle $(x^2+y^2-1)$

$$ \displaystyle f(x,y) \approx g(x, y) + h(x, y) (x^2+y^2-1) \, , $$

$\phantom{\circ}$ we should consider $f(x,y)$ and $g(x, y)$ as equivalent, for any $h(x,y)$.

$\circ$ The structure is that of a polynomial quotient ring

$$ \displaystyle \mathbb{Q}[x, y] \big/ \big\langle x^2+y^2-1 \big\rangle \\[2mm] $$

$\phantom{\circ}$ its elements are equivalence classes of polynomials.

$\circ$ $\big\langle x^2+y^2-1 \big\rangle \subset \mathbb{Q}[x, y]$ is an example of an ideal, the infinite set of polynomials
$\phantom{\circ}$ $h(x, y) (x^2+y^2-1)$ that vanishes on the unit circle.

$\circ$ For $n$-point massless scattering, the quotient ring is

$$ \displaystyle \kern10mm R_{n} = \mathbb{F}\Big[|1⟩_{\alpha}, [1|_{\dot\alpha}, \dots, |n⟩_{\alpha}, [n|_{\dot\alpha} \Big] \Big/ \Big\langle {\textstyle \sum_{i=1}^n} |i\rangle[ i | \Big\rangle $$

$\circ$ The "unit circle" is now the codimension $4$ "momentum conservation" variety within a $4n$
$\phantom{\circ}$ dimensional space. On this variety we have equivalence relations such as

$$ \displaystyle \langle 1|2+3|1]=\langle 1|-1-4-5|1]=-\langle 1|4+5|1] \quad \text{in} \quad R_5 $$

$\circ$ The rational functions $r_i$ belong to the field of fractions of $R_n$,

$$ \displaystyle r_i(|i\rangle,[i|) = \frac{\mathcal{N}(|i\rangle,[i|)}{\mathcal{D}(|i\rangle,[i|)} \, , \quad r_i(|i\rangle,[i|) \in \text{Frac}(R_n) $$

$\circ$ Interesting mathematica observations and open questions:
$\quad\star$ $R_3$ is not an Integral Domain, i.e. it breaks $ab=0 \Rightarrow a = 0 \text{ or } b = 0$
$\quad\star$ $R_4$ is not an Unique Factorization Domain (which is why MHV = anti-MHV)
$\quad\star$ Conjecture: $R_{n\geq 5}$ is UFD. For instance, this would imply the denominators $\mathcal{D}$ are unique
$\phantom{\circ}$ Note: all polynomial rings are UFD, so clearly $R_4$ is not equivalent to one, e.g. $\mathbb{F}[s,t]$

$\circ$ For $pp \rightarrow V(\rightarrow \bar\ell\ell)jj$ the space is simpler than that of say $pp \rightarrow jjjj$, we don't want to use $R_6$.
$\phantom{\circ}$ Take the decay current to be $[5|\gamma^\mu|6\rangle$, and remove $p_{V\alpha\dot\alpha}=(5+6)_{\alpha\dot\alpha}$ by mom. cons.

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

$\phantom{\circ}$ This always holds for the numerator polynomials (and almost the denomiantors).

$\circ$ For $pp \rightarrow ttH$ we use the massive spinor-helicity (or spin-spinor) formalism

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

$\phantom{\circ}$ 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.

$\circ$ $|\boldsymbol{3}^I⟩_{\alpha}$ is basically two copies of a massless spinor, we can think of this through the map

$$ \displaystyle 1 \rightarrow 1, 2 \rightarrow 2, \boldsymbol{3} \rightarrow 3+4, \boldsymbol{4} \rightarrow 5+6, \boldsymbol{5} \rightarrow 7+8 $$

$\phantom{\circ}$ but if we want neat expressions we must be careful not to overparametrise the space!

$\circ$ To not make this too abstract, we are after expressions like these, but for the MI coefficients.

$\circ$ For $Vjj$ there are 5 amplitudes (showing 3)

$$ {A}_g^{(0)}(1^{+}_\bar{q}, 2^{+}_g, 3^{+}_g, 4^{-}_q, 5^{+}_\bar{\ell}, 6^{-}_\ell) = \frac{⟨46⟩^2}{⟨12⟩⟨23⟩⟨34⟩⟨65⟩} \, , \\[6mm] {A}_g^{(0)}(1^{+}_\bar{q}, 2^{+}_g, 3^{-}_g, 4^{-}_q, 5^{+}_\bar{\ell}, 6^{-}_\ell) = \frac{⟨13⟩⟨3|1+2|5]^2}{⟨12⟩⟨23⟩[65]⟨1|2+3|4]s_{123}} \; + \; (123456\rightarrow \overline{432165}) \, , \\[6mm] {A}_q^{(0)}(1^{+}_\bar{q}, 2^{+}_{q'}, 3^{+}_{\bar{q}'}, 4^{-}_q, 5^{+}_\bar{\ell}, 6^{-}_\ell) = -\frac{[12]⟨46⟩⟨3|1+2|5]}{⟨23⟩[23]⟨56⟩[56]s_{123}}+(123456\rightarrow 156423)\phantom{+} $$

$\circ$ For $q\bar{q}\rightarrow t\bar{t}H$ there is only a single amplitude

$$ {A}_{ttH}^{(0)}(1^{+}_q, 2^{-}_\bar{q}, 3_t, 4_\bar{t}, 5_H)^I_J = \frac{⟨2|𝟑|1]⟨𝟑^I𝟒_J⟩-[𝟑^I1][1𝟒_J]⟨12⟩}{s_{12}(s_{12𝟑}-m_t²)} + (12345\rightarrow\overline{21345},12435,\overline{21435}) $$

$\phantom{\circ}$ where for clarity I have not suppressed the spin indices. Symmetries are made manifest.

$\phantom{\circ}$ Note: The amplitude is spin covariant, just like it is little group covariant!
$\phantom{\circ} \kern7.2mm$ We need only obtain a single choice, say $I=J=1$, the other follows.

$\circ$ We can always factorize a polynomial into products of irreducible factors, to some powers

$$ \displaystyle r_i(|i\rangle,[i|) = \frac{\mathcal{N}(|i\rangle,[i|)}{\prod_j \mathcal{D}_j^{q_{ij}}(|i\rangle,[i|)} % \, , \quad r_i(|i\rangle,[i|) \in \text{Frac}(R_n) $$

$\phantom{\circ}$ For the numerators this is generally not particularly useful (when in least common denominator form)
$\phantom{\circ}$ The denominator factors $\mathcal{D}_j$ are conjectured to be (mostly) related to the letters of the symbol alphabet

$\circ$ Convert your alphabet from independent Mandelstam invariants to redudant spinors brackets

$\circ$ Factorization and extra chiral cancellations are key for simplification in gauge amplitudes

Least Common Denominator

(i.e. geometry at codimension one)

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

$\boldsymbol{Vjj}$ and $\boldsymbol{t\bar{t}H}$ LCDs

$\circ\,$ The irreducible denominator factors $\mathcal{D}_j \text{ for } Vjj$ (modding out by permutation orbits) read

$$ \displaystyle \mathcal{D}_{Vjj} \subset \kern-3mm \bigcup_{\sigma \; \in \; \text{Aut}(R_6)} \sigma \circ \big\{ \langle 12 \rangle, \langle 1|2+3|1], \langle 1|2+3|4], s_{123}, \Delta_{12|34|56}, ⟨3|2|5+6|4|3]-⟨2|1|5+6|4|2] \big\} $$

$\phantom{\circ}\,$ where only the last one is new at two loops.

$\circ\,$ The $\mathcal{D}_j \text{ for } t\bar{t}H$ read

$$ \displaystyle \kern-10mm \mathcal{D}_{ttH} = \big\{ \langle 12 \rangle, [12], s_{123}, \dots, (s_{123}-m^2), \langle 1|\boldsymbol{3}|1], \dots, \\[2mm] \kern30mm \langle 1|\boldsymbol{3}|\boldsymbol{4}| 2 \rangle, \dots, \langle 1|\boldsymbol{3}|1+2|\boldsymbol{4}| 2], \dots, \Delta_{12|34|5}, \dots \Delta_{12|3|4|5} \big\} $$

$\phantom{\circ}\,$ note that there is no dependence on the top states (this looks like 3 massive scalars).

$\circ\,$ Challenge: in LCD form the numerators are intractably complicated.
$\phantom{\circ}\,$ For $Vjj$ the most complicated $\bar{q}^+g^-g^+q^-$ function had a mass dimension ($\approx$ poly. degree) of 114,
$\phantom{\circ}\,$ and little group weights $\{3, -12, 12, -3, -1, 1\}$. The ansatz size is approx. 25M.
$\phantom{\circ}\,$ Note how different from zero the little group weights are, chiral invariants are important!

$\circ\,$ 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 (abusing notation for residue):

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

$\circ\,$ With $p$-adic numbers this would be straight forward, set $\mathcal{D}_j\propto p$ and evaluate the function

$$ r_{i\in \mathcal{B}} = \sum_{m = 1}^{\text{max}_i(q_{ik})} \frac{e^k_{im}}{p^m} + \mathcal{O}(p^0) \text{ is a number in } \mathbb{Q}_p $$

See Particles._singular_variety or Ideal.point_on_variety to generate the configuration

$\circ\,$ We can't do this with only finite fields. Instead, build Laurent expansions around $t_{\mathcal{D}_k}$ (use more slices)

$$ r_{i \in \mathcal{B}} = \sum_{m = 1}^{\text{max}_i(q_{ik})} \frac{e^k_{im}}{(t-t_{\mathcal{D}_k})^m} + \mathcal{O}((t-t_{\mathcal{D}_k})^0) $$

$\phantom{\circ}\,$ strictly formal over $\mathbb{F}_p$, but convergent over $\mathbb{Q}_p$ for $(t-t_{\mathcal{D}_k}) \propto p$

$\circ\,$ Issue what if the letter does not have a factor linear in $t$?

$$ r_{i \in \mathcal{B}} = \sum_{m = 1}^{\text{max}_i(q_{ik})} \frac{c^k_{im} t + d^k_{im}}{(t^2+a_kt+b_k)^m} + \mathcal{O}((t^2+a_kt+b_k)^0) $$

$\circ\,$ From these coefficients, build null spaces used in the search for simple functions

$$ \text{null}(\text{Res}(r_{i \in \mathcal{B}}, \mathcal{D}_k^m))_{ij} \text{ from } \text{ rref } (d^k_{m})_{i,\text{slice}_j} $$

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

$\circ$ General construction for Lorentz-Invariant sub-rings through elimination theory

$\quad\star$ Build a ring with both covariant and invariant variables

$$ \mathbb{F}\big[ |i\rangle, [i|, \langle i j\rangle , [ij] \big] $$

$\quad\star$ Define relations among variables (on top of existing constraints)

$$ \big\{ \langle ij \rangle - \epsilon^{\beta\alpha} \lambda_{i\alpha} \lambda_{j, \beta}, [ij] - \tilde\lambda_{i\dot\alpha} \epsilon^{\dot\alpha\dot\beta} \tilde\lambda_{j, \dot\beta} \big\} $$

$\quad\star$ Compute a lexicographical Groebner basis with invariants > covariants

$\circ$ We obtain the following invariant rings

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

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

$\circ\,$ 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)

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

Example

$\circ\,$ Start from the function

$$ \displaystyle f^{\text{ex}} = \frac{\mathcal{N}^{\text{ex}}}{⟨14⟩^2[14]^2 s_{56} ⟨1|2+4|3]^2⟨2|1+4|3]^4⟨2|1+3|4]^2Δ_{14|23|56}^4} $$

$\phantom{\circ}\,$ The numerator Ansatz has size 104$\,$128

$\circ\,$ Clean up the $Δ_{14|23|56}$ Gram residue

$$ \displaystyle f^{\text{ex}} = \frac{\mathcal{N}^{\text{ex}}_1}{⟨14⟩^2[14]^2s_{56}⟨2|1\!+\!4|3]^4Δ_{14|23|56}^4 \,} + \frac{\mathcal{N}^{\text{ex}}_2}{⟨14⟩^2[14]^2s_{56}⟨2|1+4|3]^4⟨1|2\!+\!4|3]^2⟨2|1\!+\!3|4]^2} $$

$\circ\,$ Split $s_{14}$ and impose symmetry

$$ \displaystyle f^{\text{ex}} = \frac{\mathcal{N}^{\text{ex}}_{3}}{⟨14⟩^2 s_{56} ⟨2|1+4|3]^4Δ_{14|23|56}^4} + \frac{\mathcal{N}^{\text{ex}}_{4}}{⟨14⟩^2 s_{56} ⟨1|2+4|3]^2⟨2|1+4|3]^4⟨2|1+3|4]^2} + (123456\rightarrow \overline{432165}) $$

$\circ\,$ Impose degree bound on poles at codimension two

$$ \displaystyle f^{\text{ex}} = \sum_{k=0}^3 \frac{\mathcal{N}^{\text{ex}}_{5,k}}{⟨14⟩^2 s_{56} ⟨2|1+4|3]^{1+k} Δ_{14|23|56}^{4-k}} + \frac{\mathcal{N}^{\text{ex}}_6}{⟨14⟩^2 s_{56}⟨1|2+4|3]^2⟨2|1+4|3]^4⟨2|1+3|4]^2} + (123456\rightarrow \overline{432165}) $$

The Ansatz now has size 13$\,$532, almost a factor of 10 simpler.

Multivariate Partial Fractions

$\circ$ We want a mathematically rigorous answer to the question

$$ \frac{\mathcal{N}}{\mathcal{D}_1\mathcal{D}_2} \stackrel{?}{=} \frac{\mathcal{N}_2}{\mathcal{D}_1} + \frac{\mathcal{N}_1}{\mathcal{D}_2} $$

$\phantom{\circ}$ without knowing $\mathcal{N}$ analytically. The complexity should not depend on $\mathcal{N}$ (besided numerical evaluations).
$\phantom{\circ}$ The complexity will depend on $\mathcal{D}_1, \mathcal{D}_2$

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

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

$$ \langle xy^2 + y^3 - z^2 \rangle + \langle x^3 + y^3 - z^2 \rangle = \langle xy^2 + y^3 - z^2, x^3 + y^3 - z^2 \rangle = \langle 2y^3-z^2, x-y \rangle \cap \langle y^3-z^2, x \rangle \cap \langle z^2, x+y \rangle $$

$\phantom{\circ}$ This is a primary decomposition. If $\mathcal{N}$ vanishes on all branches, than the partial fraction decomposition exists.

Iterated Pole Subtraction

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

$\circ\,$ Iteratively reconstruct a residues at a time using $p$-adic numbers to get $\mathbb{F}_p$ samples for the residues

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

$\circ\,$ We get more than just partial fraction decomposition, we cna identify numerator insertions from e.g.:

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

$\circ\,$ Interesting and non-trivial bevhavior also at 5-point 3-mass

$$ \def\spa#1.#2{\left\langle#1\,#2\right\rangle} \def\spb#1.#2{\left[#1\,#2\right]} \def\spaa#1.#2.#3{\langle\mskip-1mu{#1} | #2 | {#3}\mskip-1mu\rangle} \def\spbb#1.#2.#3{[\mskip-1mu{#1} | #2 | {#3}\mskip-1mu]} \def\spab#1.#2.#3{\langle\mskip-1mu{#1} | #2 | {#3}\mskip-1mu]} \def\spba#1.#2.#3{[\mskip-1mu{#1} | #2 | {#3}\mskip-1mu\rangle} \def\spaba#1.#2.#3.#4{\langle\mskip-1mu{#1} | #2 | #3 | {#4}\mskip-1mu\rangle} \def\spbab#1.#2.#3.#4{[\mskip-1mu{#1} | #2 | #3 | {#4}\mskip-1mu]} \def\spabab#1.#2.#3.#4.#5{\langle\mskip-1mu{#1} | #2 | #3 | {#4}| {#5} \mskip-1mu]} \def\spbaba#1.#2.#3.#4.#5{[\mskip-1mu{#1} | #2 | #3 | {#4}| {#5}\mskip-1mu\rangle} \def\tr#1.#2{\text{tr}(#1|#2)} \def\qb{\bar{q}} \def\Qb{\bar{Q}} \def\cA{{\cal A}} \def\slsh{\rlap{$\;\!\!\not$}} \def\three{{\bf 3}} \def\four{{\bf 4}} \def\five{{\bf 5}} \begin{align}\label{eq:decomp_spaba1351_spbab2542} \big\langle \spaba1.\three.\five.1,\, \spbab2.\five.\four.2 \big\rangle = \; &\big\langle \, \spab1.\three.2,\, \spab1.\four.2,\, \spaba1.\three.\five.1,\, \spbab2.\five.\four.2 \, \big\rangle\; \cap \\ &\big\langle \, \spaba1.\three.\five.1,\, \spbab2.\five.\four.2, |\five|2]\langle1|\three| - |1+\three|2]\langle1|\five| \, \big\rangle \;, \nonumber \end{align} \\ \text{because: } |\five|2]\spaba1.\three.\five.1[2| + |1\rangle\spbab2.\five.\four.2\langle1|\five| = \spab1.\five.2 \Big( |\five|2]\langle1|\three| - |1+\three|2]\langle1|\five| \Big) \, , $$

$\phantom{\circ}\,$ or between the triangle and box Grams

$$ \begin{gather}\label{eq:decomp_delta12_34_5_and_delta_12_3_4_5} \big\langle \Delta_{12|34|5},\,\Delta_{12|3|4|5} \big\rangle = \big\langle s_{34},\, \tr1+2.{\three+\four}^2 \big\rangle \cap \big\langle \Delta_{12|34|5},\, \tr1+2.{\three-\four}^2 \big\rangle \, . \end{gather} $$

Challenges

$\circ\,$ Can we guess the constraints? If not, can we verify them with numerical evaluations?
$\phantom{\circ}\,$ $\mathbb{Q}_p$ evaluations can be costly (probably depending on implementation). Xia, Yang ('25) say they are not!

$\circ\,$ Ideal intersection can be highly non-trivial:

$$ \mathcal{N} \in \langle q_1, q_2 \rangle \cap \langle q_3, q_4 \rangle \stackrel{?}{=} \langle q_1q_3, q_1q_4, q_2q_3, q_2 q_4\rangle $$

$\phantom{\circ}\,$ Unfortunately not always. This is called a complete intersection when it holds.

$\circ\,$ Therefore, either:

$\quad\star\,$ we compute the intersection explicitly (can be prohibitively hard)

$\quad\star\,$ or we have to make a choice of which constrain we manifest

$\circ\,$ Computing primary decompositions with these many variables is hard, Singular can't do it on its own

$\circ\,$ Even constructing the ansatz requires a GB, which in some cases Singular doesn't easily give

$\circ\,$ And of course computing the reduction to MIs of the amplitude is not easy in the first place.

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