$\circ\,$ $pp\rightarrow Vjj$ (a.k.a. $e^+e^-\rightarrow V \rightarrow 4j$) is central to EW precision measurements

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

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

$\;\small\rhd\,$ 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 $\;\small\rhd\,$ 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
$\star\,$ Goal: show how to reconstruct amplitudes in a manifestly spin- and little-group covariant form

$\circ\,$ $pp\rightarrow HHH$ previously unknown analytically, even at leading order (one loop).

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

$\circ$ $t\bar{t}H$, $HHH$ computed analytically (Form optimized) with unitarity and standard Feynman diagrams
$\phantom{\circ}$ techniques, and then cross checked with Recola2 and/or Open-Loops2

$\circ$ 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 and 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 (let cancellations happen numerically!)

$\circ$ Consider polynomials $f, g, h$ 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 mathematical observations and open questions:
$\quad\star$ $R_3$ is not an Integral Domain, i.e. it breaks $ab=0 \Rightarrow a = 0 \text{ or } b = 0$ (zero divisors)
$\quad\star$ $R_4$ 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 always for the denomiantors.
$\phantom{\circ}$ A denominator does not belong to $R_{Vjj}$ if one manifests $s_{56}=\langle 56\rangle [65]$,
$\phantom{\circ}$ which we show can always be partial fractioned (the physical pole is $\sqrt{s_{56}}$).

$\circ$ This approach is very similar to the massless case (just remove variables),
$\phantom{\circ}$ but does not generalise to cases with multiple massive legs, e.g. with two of them:
$\phantom{\circ}$ $p_{V_1} \cdot p_{V_2}$ cannot be eliminated through momentum conservation in favour of massless ones.

$\circ$ For $pp \rightarrow HHH$ we use the massive spinor-helicity (or spin-spinor) formalism,
$\phantom{\circ}$ albeit in a very simplified form since scalars have no states.

$$ \displaystyle \kern10mm R_{HHH} = \frac{\mathbb{F}\big[|1⟩_{\alpha}, [1|_{\dot\alpha}, |2⟩_{\alpha}, [2|_{\dot\alpha}, \boldsymbol{3}_{\alpha,\dot\alpha}, \boldsymbol{4}_{\alpha,\dot\alpha}, \boldsymbol{5}_{\alpha,\dot\alpha} \big]}{\big\langle |1\rangle[1|+|2\rangle[2| + \boldsymbol{3}_{\alpha,\dot\alpha} + \boldsymbol{4}_{\alpha,\dot\alpha} + \boldsymbol{5}_{\alpha,\dot\alpha}, \;\, \boldsymbol{3}_{\alpha,\dot\alpha} \boldsymbol{3}^{\dot\alpha,\alpha} - \boldsymbol{4}_{\alpha,\dot\alpha} \boldsymbol{4}^{\dot\alpha,\alpha}, \;\, \boldsymbol{4}_{\alpha,\dot\alpha} \boldsymbol{4}^{\dot\alpha,\alpha}- \boldsymbol{5}_{\alpha,\dot\alpha} \boldsymbol{5}^{\dot\alpha,\alpha} \big\rangle} $$

$\phantom{\circ}$ where $\boldsymbol{3}_{\alpha,\dot\alpha} \boldsymbol{3}^{\dot\alpha,\alpha} = \boldsymbol{4}_{\alpha,\dot\alpha} \boldsymbol{4}^{\dot\alpha,\alpha} = \boldsymbol{5}_{\alpha,\dot\alpha} \boldsymbol{5}^{\dot\alpha,\alpha} = 2 M_h^2$; $\boldsymbol{3}_{\alpha,\dot\alpha},\boldsymbol{4}_{\alpha,\dot\alpha},\boldsymbol{5}_{\alpha,\dot\alpha}$ are full-rank (unfactorizable).

$\circ$ While $pp \rightarrow ttH$ exposes the full complexity, including massive states

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

$\phantom{\circ}$ 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.

$\circ$ It is sometimes useful to map to a set of all massless momenta / spinors (e.g. recycle code),

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

$\phantom{\circ}$ 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

LCDs or Kinematic Poles

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

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

$\circ\,$ For $t\bar{t}H$, they 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).

$\circ\,$ For $HHH$, they are

$$ \small \begin{gathered} \mathcal{D}_{HHH} = \big\{ ⟨1|2⟩, [1|2], ⟨2|𝟓|1], ⟨2|𝟒|1], ⟨2|𝟑|1], ⟨1|𝟑|2], [1|𝟑|𝟓|1], ⟨1|𝟑|𝟓|1⟩, ⟨1|𝟓|𝟒|2⟩, [2|𝟒|𝟓|1], Δ_{12|𝟑|𝟒|𝟓}, \\ ⟨2|𝟑|𝟒|𝟓|1], ⟨1|𝟓|𝟒|𝟑|2], ⟨1|2⟩[1|2]⟨1|𝟓|𝟒|𝟑|2]⟨2|𝟑|𝟒|𝟓|1]+m_t^2\text{tr}_5(1|2|𝟑|𝟒)^2, \\ ⟨1|𝟑|2]⟨2|𝟒|𝟓|1⟩[1|𝟑|2⟩[2|𝟒|𝟓|1]+m_t^2\text{tr}_5(1|2|𝟑|𝟒)^2 \big\} \end{gathered} $$

$\circ\,$ 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

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

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

$\phantom{\circ}$ while $R_{HHH}$ has 20 invariants, subject to 122 constraints.

$\circ\,$ The authors of arXiv:2509.14350 seem to tackle a very similar problem for

$\quad\small\rhd\,$ “[...] finding possible terms in an action, or many other applications.”
$\quad\small\rhd\,$ They say “[...] the awareness in the physics community of the possible structures of the rings of invariants thus arising is rather low, to our knowledge. In particular, the possibility of having relations among invariants has received very little attention in physics.”

$\phantom{\circ}\,$ The key concept is that the ring we consider are not freely generated.

$\circ\,$ With Ben in arXiv:2203.04269 we showed that these rings are “Cohen–Macaulay”

$\quad\small\rhd\,$ Follows from quotienting a polynomial ring by a maximal-codimension ideal
$\quad\small\rhd\,$ Implies e.g. that symbolic powers of max-codim ideals match normal powers
$\kern20.9mm$ that all max-codim ideals are equi-dimensional

$\circ\,$ The authors of arXiv:2509.14350 state that invariant rings are “Gorenstein”, which implies CM

$\quad\small\rhd\,$ “all rings of the type we are discussing are Gorenstein”
$\quad\small\rhd\,$ “Gorenstein is for rings what Calabi–Yau is for manifolds; the spaces of invariants are in fact (non-compact) Calabi-Yau varieties”

What futher practical information can we learn from the mathematics literature?

$\circ\,$ The numerator Ansatz takes the form (massless case)

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

$\phantom{\circ}$ 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)

Preview of Linac

work in collaboration with Jack Franklin, to appear

cuda_row_reduce(A, field_characteristic=primes[0], verbose=False)  # A is a 2D numpy.ndarray

$\circ\,$ Performance on a laptop GPU of approx. 60 CPU cores
$\circ\,$ Performance on a workstation GPU of approx. 600 CPU cores
$\circ\,$ Tested on systems up to 100k equations and unknowns (takes 45 minutes).

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

$Vjj$ 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 can 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

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

Iterated Pole Subtraction (another example)

$\circ$ Example from triple-Higgs

$$ \hat d^{++}_{12\times 3 \times 4}= \frac{\mathcal{N} \leftarrow 2794 \text{ free parameters }}{⟨12⟩²⟨1|𝟓|𝟒|𝟑|2]⟨2|𝟑|𝟒|𝟓|1]Δ_{12|𝟑|𝟒|𝟓}} $$

$\circ$ We can prove $⟨1|𝟓|𝟒|𝟑|2], ⟨2|𝟑|𝟒|𝟓|1]$ can be separated, their primary decomposition reads

$$ \big\langle ⟨1|𝟓|𝟒|𝟑|2], ⟨2|𝟑|𝟒|𝟓|1] \big\rangle = \big\langle ⟨1|𝟓|𝟒|𝟑|2], ⟨2|𝟑|𝟒|𝟓|1], \text{tr}_5 \big\rangle \cap \big\langle ⟨1|𝟓|𝟒|𝟑|2], ⟨2|𝟑|𝟒|𝟓|1], s_{2𝟑}, s_{1𝟓} \big\rangle $$

$\phantom{\circ}$ Generate two phase space points, one for each branch, and verify the numerator vanishes.

$\circ$ Similarly, with four evaluations we can prove $⟨1|𝟓|𝟒|𝟑|2], Δ_{12|𝟑|𝟒|𝟓}$ can be separated,

$$ \big\langle ⟨1|𝟓|𝟒|𝟑|2] , \, Δ_{12|𝟑|𝟒|𝟓} \big\rangle= \big\langle M_H, \; 𝟓_{\alpha\dot\alpha}𝟒^{\dot\alpha\beta} \big\rangle \cap \big\langle M_H, \; 𝟒^{\dot\alpha\alpha}𝟑_{\alpha\dot\beta} \big\rangle \cap \big\langle \langle 1 | 𝟑 | 2], \; \langle 1 | 𝟒 | 2], \; \langle 1 | 𝟑 | 𝟒 | 1 \rangle, [2 | 𝟑 | 𝟒 | 2] \big\rangle \cap \big\langle ??? \big\rangle $$

$\phantom{\circ}$ Although we don't have a complete set of generators for the last branch, we can still sample it.

$\circ$ Fit $⟨1|𝟓|𝟒|𝟑|2]$ residue by sampling in limit $⟨1|𝟓|𝟒|𝟑|2] \rightarrow 0$

$$ \hat d^{++}_{12\times 3 \times 4} = \frac{\mathcal{N} \leftarrow 112 \text{ free parameters }}{⟨12⟩²⟨1|𝟓|𝟒|𝟑|2]} + \mathcal{O}(⟨1|𝟓|𝟒|𝟑|2]^0) $$

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

$\phantom{\circ}\,$ Ongoing project with a masters' student in Edinburgh to improve our ability to compute them.

$\circ\,$ 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++.

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

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

$\quad\small\rhd$ Pheno ready results for the hard functions are available at FivePointAmplitudes.

$\circ$ For $t\bar{t}H$ and $HHH$, efficient Fortran implementation of the analytic expressions in MCFM

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

Backup slides.

Effective Pentagons (another non UFD example)

$\circ$ As mentioned, pentagons can be reduced to a combination of boxes,

$$ \def\mt{m} \def\mh{M_H} \def\spa#1.#2{\left\langle#1\,#2\right\rangle} \def\spb#1.#2{\left[#1\,#2\right]} \begin{eqnarray} &&E_0(p_1,p_2,p_3,p_4;\mt)= c^{(1)} D_0(p_2,p_3,p_4;\mt) +c^{(2)} D_0(p_{12},p_3,p_4;\mt) \\ &+&c^{(3)} D_0(p_1,p_{23},p_4;\mt) +c^{(4)} D_0(p_1,p_2,p_{34};\mt) +c^{(5)} D_0(p_1,p_2,p_3;\mt)\, . \end{eqnarray} $$

$\circ$ We find it useful to write the box coefficients in terms of effective pentagons $\hat e$ and boxes $\hat d$

$$ d^{h_1h_2}_{p_a\times p_b \times p_c } = \sum_{i=\{i_1,i_2\}} c^{(i)} \hat e_{p_x \times p_y \times p_z \times p_w}+ \hat d^{h_1h_2}_{p_a\times p_b \times p_c } $$

$\phantom{\circ}$ where the sum involves the two pentagons that pinch to the given box.

$\circ$ The coefficients $\hat e$ and $\hat d$ are not uniquely defined, but $\hat e$ has the property of capturing
$\phantom{\circ}$ the residue of the poles that mix top-mass and kinematic dependence.
$\phantom{\circ}$ The non-uniqueness comes from, e.g.

$$ ⟨1|2⟩[1|2]⟨1|𝟓|𝟒|𝟑|2]⟨2|𝟑|𝟒|𝟓|1]+m_t^2\text{tr}_5(1|2|𝟑|𝟒)^2=0 $$

Example of Code Syntax for Codim-2 Limit

$\circ$ This is just a couple of pip install's aways

field = Field("padic", 2 ** 31 - 1, 5)
oPs8pt = Particles(8, field=field, seed=0)
oPs8pt._singular_variety(("s_34-s_56", "s_56-s_78", '⟨1|7+8|5+6|3+4|2]', '⟨2|3+4|5+6|7+8|1]'),
                         (field.digits, field.digits, 1, 1), seed=0,
                         generators=('s_34-s_56', 's_56-s_78', '⟨1|7+8|5+6|3+4|2]', 
                                     '⟨2|3+4|5+6|7+8|1]', 'tr5(1|2|3+4|5+6)'))
oPs8pt.m_t = field.random()
oPs8pt.m_h = "sqrt(s_34)"
oPs5pt = oPs.cluster([[1, ], [2, ], [3, 4], [5, 6], [7, 8]])
from antares_results.HHH.ggHHH.pp import coeffs as coeffs_pp
coeffs_pp[’d_12x3x4’](oPsC)

130808068*2147483647^-1 + 687356881 + 792807618*2147483647 + 696603492*2147483647^2 + O(2147483647^3)

The denominator goes like $p^2$, but the coefficient goes like $p^{-1} \Rightarrow$ the numerator vanishes linearly.

$\circ$ The output is a $p$-adic number, i.e. a Laurent series in powers of the prime.
$\phantom{\circ}$ With finite fields we cannot do this (with just one evaluation)! It would be dividing by zero.