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Mon, 01 Jan 0001 00:00:00 +0000

Analytic Structure and Reconstruction in QCD: Two-Loop $\boldsymbol{pp \to Vjj}$ and One-Loop $\boldsymbol{q\bar{q}\rightarrow t\bar{t}H}$

Giuseppe De Laurentis

University of Edinburgh

Vjj: JHEP 06 (2025) 093

(GDL, H. Ita, B. Page, V. Sotnikov)

ttH: arXiv:2504.19909

(J. Campbell, GDL, K. Ellis)

Caravel Meeting 2025

UZH

Find these slides at gdelaurentis.github.io/slides/caravelmeeting2025

Numerical Computation

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 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 $\mathbb{F}$

$\mathbb{F}_p$: von Manteuffel, Schabinger ('14); $\phantom{\mathbb{F}_p}$ Peraro ('16)
$\mathbb{C}$: GDL, Maitre ('19); $\mathbb{Q}_p$: GDL, Page ('22)

Setting up the Calculation

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

$$ \require{color} \displaystyle \sum_{\text{states}} \, \prod_{\text{trees}} A^{\text{tree}}(\lambda, \tilde\lambda, \ell)\big|_{\text{cut}_{\Gamma}} = \sum_{\substack{\Gamma' \ge \Gamma, \\ i \in M_\Gamma' \cup S_\Gamma'}} \kern-2mm {\color{black}{c_{\,\Gamma',i}(\lambda, \tilde\lambda)}} \, \frac{m_{\Gamma',i}(\lambda\tilde\lambda, \ell)}{\displaystyle \prod_{j\in P_{\Gamma'} / P_{\Gamma}} \rho_{j}(\lambda\tilde\lambda, \ell)}\Bigg|_{\text{cut}_\Gamma} $$

C++ code

Abreu, Dormans,

Febres Cordero, Ita

Kraus, Page, Pascual,

Ruf, Sotnikov ('20)

$\rightarrow$ Numerical Berends-Giele recursion for LHS, solve for coeffs. in RHS.
$\rightarrow$ IBP reduction = decomposition on RHS, $\; m_{\Gamma,i} \in M_\Gamma \cup S_\Gamma$

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

1. Wrote a Python script to split the 1.4 GB ancillaries into >10k files
2. Compile into 18.2 GB of C++ binaries (for reference Caravel compiles into approx. 5 GB)
3. Obtain $\mathbb{F}_p$ evaluations of the form factors (each takes approx. 1 sec per point)
4. Recombine triplets of form factors into six-point helicity amplitudes (incl. decays)

$\rightarrow$ 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

Buccioni, Lang, Lindert, Maierhöfer, Pozzorini, Zhang, Zoller

Analytic & Geometric Structure

algebro-geometric formulation for physicists in:
GDL, Page (JHEP 12 (2022) 140)

see also Sturmfeld et al. “Spinor-Helicity Varieties”:
arXiv:2406.17331

Massless Scattering

$\circ$ 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 $ab=0 \Rightarrow a = 0 \text{ or } b = 0$
$\quad\star$ $R_4$ 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. $\mathbb{F}[s,t]$

Choosing the Appropriate Covariant Q-Ring

$\circ$ 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

Shadmi, Weiss Ochirov; Arkani-Hamed, Huang, Huang;

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

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

Conde, Marzolla Conde, Joung, Mkrtchyan;

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

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

Examples of Trees

$\circ$ 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.

Spinor Alphabets

$\circ$ 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

Abreu, Dormans, Febres Cordero, Ita, Page ('18)

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

From work in progress with S. Abreu, X. Liu, P.F. Monni

Mandelstam letters
$s_{12}$
$s_{123}$
$s_{12} - s_{123} - s_{345} + s_{45}$
$-s_{12} + s_{123}$
$s_{12}(s_{123} - s_{56}) - s_{123}(s_{123} + s_{34} - s_{56})$
$\displaystyle\frac{ s_{12}\left(s_{16}(s_{23} - s_{234})s_{34} + s_{23}^{2}(\cdots) + \cdots\right) + s_{123}(\cdots) + s_{23}(\cdots) }{ \sqrt{(-s_{12} + s_{123} - s_{23})^2\cdots} }$

$\Rightarrow$

Spinor letters
$\langle 1\,2\rangle[1\,2]$
$s_{123}$
$\langle 3\,|\,6\rangle[3\,|\,6]$
$\langle 3\,|\,1{+}2\,|\,3]$
$\langle 3\,|\,1{+}2\,|\,4]\langle 4\,|\,1{+}2\,|\,3]$

$\operatorname{tr}_5(2,3,4,5)$

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

Least Common Denominator

(i.e. geometry at codimension one)

$\circ\,$ We can now determine the least common denominators (LCDs),

$$ \displaystyle \mathcal{D} = \prod_j \mathcal{D}_j^{q_{ij}}(|i\rangle,[i|) \, . $$

$\phantom{\circ}\,$ Obtain the $q_{ij}$ from a univariate slice $\vec\lambda(t)$, i.e. a 1D curve.

$\circ$ The curve must intersect all varieties $V(\langle \mathcal{D}_j \rangle)$, e.g.

$\phantom{\circ}\,$ Solve for $a_i, b_i$ such that constraints are satisfied.

$\circ\,$ Publicly impelemented, see antares, lips, syngular

$\phantom{\circ}\,$ do_codimension_one_study(func, slice, denoms)
$\phantom{\circ}\,$ Particles.univariate_slice or Ring.univariate_slice

Space has dimension $4n-4$,

$\mathcal{D}_j = 0$ have dimension $4n-5$,

$\vec\lambda(t)$'s have dimension 1.

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!

Basis Change from Laurent Coefficients

$\circ\,$ 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):

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

$\circ\,$ Search for linear combinations that remove as many singularities as possible

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

Laurent Series or p(z)-adic expansion

$\circ\,$ 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 $t$?

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

see also Fontana, Peraro ('23)

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

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

Analytic Reconstruction

Invariant Quotient Rings

$\circ$ 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

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

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

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)

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

127187555379407704220939486282289348327703498501718808908391691454242601886997968263623652083189652150273

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

GDL, Maître ('19) GDL, Page ('22)

$\circ$ 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

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

$\cap$

$=$

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

$\phantom{\circ}$ 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)

GDL, Maître ('19) GDL, Page ('22) Chawdhry ('23) Xia, Yang ('25)

$\circ\,$ 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)

Conclusions
&
Outlook

Spinor-Helicity Amplitudes Results

$\circ$ 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!

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Mon, 01 Jan 0001 00:00:00 +0000

Analytic Structure and Reconstruction in QCD: Two-Loop $\boldsymbol{pp \to Vjj}$ and One-Loop $\boldsymbol{q\bar{q}\rightarrow t\bar{t}H}$

Giuseppe De Laurentis

University of Edinburgh

Vjj: arXiv:2503.10595

(GDL, H. Ita, B. Page, V. Sotnikov)

ttH: arXiv:2504.19909

(J. Campbell, GDL, K. Ellis)

CERN QCD Seminar

Geneva, CH

Find these slides at gdelaurentis.github.io/slides/cernjune2025

Introduction

Phenomenological Motivation

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

$\rightarrow\,$ Theoretical uncertainties are already larger than experimental ones, ATLAS Collab. '24

$\rightarrow\,$ NNLO is essential for agreement with experiment, Mazzitelli,

Sotnikov,

Wiesemann '24

Other studies at NNLO only for $q\bar q'\rightarrow Wb\bar b, \; \text{e.g. no} \; gg\rightarrow Wq\bar q'$ despite available amps

$\,$Buonocore, Devoto, Kallweit, Mazzitelli, Rottoli, Savoini '22; Hartanto, Poncelet, Popescu, Zoia '22;$\,$

$\circ\,$ $pp\rightarrow t\bar{t}H$ of interest primarily for direct access to top Yukawa $y_t$ (but also CP, EFTs, 2HDM, etc.)
$\phantom{\circ}\,$ current N$^2$LO pheno. relies on approx. amplitudes Catani, Devoto, Grazzini, Kallweit, Mazzitelli, Savoini '22;$\,$ Devoto, Grazzini, Kallweit, Mazzitelli, Savoini '24;$\,$

Theoretical Motivation

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

$\;\star\,$ Limited knowledge at higher loops/points;
$\;\star\,$ All amplitudes in the lower triangle contribute
$\;\phantom{\star}\,$ at a given perturbatifve order;
$\;\star\,$ Pheno can be hindered by complexity of results,
$\;\phantom{\star}\,$ especially if IR cancellations are needed;
$\;\star\,$ E.g. the two-loop amps of [5] were >1GB of files.

$\circ\,$ Goal: reduce complexity of [5] by manifesting the analytic structure to facilitate future computations

Loops ↑ Jets →	$1$	$2$	$\geq3$
3	2023 [6]	?	?
2	2007 [4]	2021 [5]	?
1	1981 [1]	1997 [2]	2008 [3]

Analytic Numeric Analytic (LCA) Unknown

[1] Ellis, Ross, Terrano; [2] Bern, Dixon, Kosower; [3] BlackHat; OpenLoops; [4] Gehrmann-De Ridder, Gehrmann, Glover, Heinrich; [5] Abreu, Febres Cordero, Ita, Klinkert, Page, Sotnikov + This work; [6] Gehrmann, Jakubčík, Mella, Syrrakos, Tancredi

$\circ\,$ 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

Numerical Computation

Partial Amplitudes & Finite Remainders

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

$\circ$ $\Gamma$: 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)

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 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 $\mathbb{F}$

$\mathbb{F}_p$: von Manteuffel, Schabinger ('14); $\phantom{\mathbb{F}_p}$ Peraro ('16)
$\mathbb{C}$: GDL, Maitre ('19); $\mathbb{Q}_p$: GDL, Page ('22)

Setting up the Calculation

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

C++ code

Abreu, Dormans,

Febres Cordero, Ita

Kraus, Page, Pascual,

Ruf, Sotnikov ('20)

$\rightarrow$ Numerical Berends-Giele recursion for LHS, solve for coeffs. in RHS.
$\rightarrow$ IBP reduction = decomposition on RHS, $\; m_{\Gamma,i} \in M_\Gamma \cup S_\Gamma$

$\circ$ 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

Buccioni, Lang, Lindert, Maierhöfer, Pozzorini, Zhang, Zoller

Analytic & Geometric Structure

algebro-geometric formulation for physicists in:
GDL, Page (JHEP 12 (2022) 140)

see also Sturmfeld et al. “Spinor-Helicity Varieties”:
arXiv:2406.17331

Guiding Principles

$\circ$ 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)

Trade-offs and Challenges

$\circ$ 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.

Massless Scattering

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

$\circ$ 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. $\mathbb{F}[s,t]$