Analytic One-Loop Amplitudes for $gg \rightarrow HHH$

Giuseppe De Laurentis

University of Edinburgh

arXiv:2507.19313

with J. M. Campbell and R. K. Ellis

See also:
$q\bar{q}\rightarrow t\bar{t}H$ (arXiv:2504.19909, JHEP07(2025)147)
$pp\rightarrow HHj$ (arXiv:2408.12686, JHEP10(2024)230)
$pp\rightarrow Hjj$ (arXiv:2002.04018, JHEP05(2020)079)

HHH Workshop

Dubrovnik, HR

Find these slides at gdelaurentis.github.io/slides/hhhsep2025

Introduction

Theoretical Motivation

$\circ\,$ Direct probe of triple and quartic Higgs self-couplings at current and future colliders.
$\phantom{\circ}\,$ We write the potential in the kappa framwork (SM: $\kappa_3 = \kappa_4 = 1$)

$$ V(H) = \frac{1}{2} m_h^2 H^2 + \kappa_3 \lambda v H^3 + \kappa_4 \frac{\lambda}{4} H^4 $$

$\phantom{\circ}\,$ There are contributions proportional to $\kappa_4$, $\kappa_3^2$ ($A_3$), $\kappa_3$ ($A_4$), and no $\kappa$ ($A_5$).

$$ A_{\rm tot} = \delta^{AB} \frac{g_s^2}{16\pi^2} \, \frac{m_t^4}{v^3} \left( A_3 + A_4 + A_5 \right)\, . $$

$\circ\,$ Facilitate phenomenological studies through faster and more stable evaluations:
$\phantom{\circ}\,$ we observe an order of magnitude speed up compared to Recola2 and OpenLoops2.

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

$\circ\,$ Improve understanding of the analytical structure:

$\qquad\star\,$ Stepping stone towards real-radiation processes and, eventually, multi-loop corrections.

$\qquad\star\,$ Provide necessary input to understand cancellation of spurious kinematic singularities.

$\phantom{\circ}\,$ In this context full control over the leading order result is an essential baseline.

Feynman Diagrams for $A_3$: $\kappa_4$ & $\kappa_3^2$

$\circ\,$ The $\kappa_4$ and $\kappa_3^2$ diagrams are triangles (no contribution from pinch bubbles)

$\phantom{\circ}\,$ This sub-amplitude is easily stated as (for the two indep. helicity configurations)

$$ \def\mt{m} \def\mh{M_H} \def\spa#1.#2{\left\langle#1\,#2\right\rangle} \def\spb#1.#2{\left[#1\,#2\right]} \begin{eqnarray} A_3^{++} &=& \frac{\spb1.2}{\spa1.2} \, \frac{6\mh^2}{\mt^2(s_{12}-\mh^2)} \Bigl[(4\mt^2-s_{12}) C_0(p_1,p_2; \mt)+2\Bigr]\times \left(\kappa_4+ \frac{3\kappa_3^2 \mh^2}{s_{34}-\mh^2} + \text{perms.} \right) \, , \\ A_3^{-+} &=& 0 \, . \end{eqnarray} $$

$\phantom{\circ}\,$ Where $C_0(p_1,p_2; \mt)$ is the scalar triangle Feynman integral: $\frac{1}{i \pi^{2}} \int \, \frac{d^4 l}{d_0 \; d_1 \; d_2} $

Feynman Diagrams for $A_4$ and $A_5$: $\kappa_3$ & no $\kappa$

$\circ\,$ The $\kappa_3$ diagrams are boxes (and triangle pinches, but no bubble contribution)

$\phantom{\circ}\,$ Their contribution is also fairly simple, it can be written in 4 or 5 lines.

$\circ\,$ The background diagrams are by far the most complicated,

$\phantom{\circ}\,$ We require a different approach to tackle them analytically.

Computation Setup

Setting up the Calculation

$\circ$ We perform a first analytic computation in two ways

1. A standard computation directly from Feynman diagrams
2. A generalized-unitarity computation from cut-diagrams (i.e. products of trees)
$\kern2mm$ In this approach the amplitude is constructed as (schematically)

$$ \require{color} \displaystyle \sum_{\text{states}} \, \prod_{\text{trees}} A^{\text{tree}}(\lambda, \tilde\lambda, \ell)\big|_{\text{cut}_{\Gamma}} = \sum_{\Gamma' \ge \Gamma} \kern0mm {\color{black}{c_{\,\Gamma',i}(\lambda, \tilde\lambda)}} \, \frac{m_{\Gamma',i}(\lambda\tilde\lambda, \ell)}{\displaystyle \prod_{j\in P_{\Gamma'} / P_{\Gamma}} \rho_{j}(\lambda\tilde\lambda, \ell)}\Bigg|_{\text{cut}_\Gamma} $$

$\kern2mm$ The sum in the RHS is over all topologies $\Gamma'$ that have at least the cut propagators $\Gamma$,
$\kern2mm$ and the product is over propagators that have not been cut.

$\circ$ Pentagons are reducible to linear combination of boxes, and we observe all bubbles vanish, leaving:

$$ A_5^{h_1h_2} = \sum_{a,b,c} d^{h_1h_2}_{p_a\times p_b \times p_c } D_0(p_a, p_b, p_c; m_t) + \sum_{a,b} c^{h_1h_2}_{p_a\times p_b} C_0(p_a, p_b; m_t) $$

$\circ$ This yields a few MBs of optimized FORM routines for the integral coefficients, which we simplify.

$\circ$ In principle, a numerical program for $d^{h_1h_2}_{p_a\times p_b \times p_c }$ and $c^{h_1h_2}_{p_a\times p_b}$ would suffice for what follows.

Overview of the Approach

$\circ$ Goal is to obtain simple forms for $d^{h_1h_2}_{p_a\times p_b \times p_c }$ and $c^{h_1h_2}_{p_a\times p_b}$

$\star$ We will use only numerical evaluations to study their analytic structure
$\star$ We will parametrize the possible functional form (Ansatz) and solve for free coefficients

Think of this as a bootstrap approach, helped by additional numerical information.

$\circ$ The analytic structure should be clear with $p^\mu \in \mathbb{C}^{4}$ (good $\mathbb{R}^{4}$ behaviour will follow)

$\phantom{\star}$ In practice, take $p^{\mu=y}\in i\mathbb{Q} \; \Rightarrow \; E\pm p^z \text{ and } p^x\pm ip^y \in \mathbb{R} \; \Rightarrow \; \lambda_{\alpha} \in \mathbb{R} \text{ or } \mathbb{Q}$
$\phantom{\star}$ This allows us to generate phase space points in a finite field (modular arithmetics)

from syngular import Field
from lips import Particles
Particles(5, field=Field("finite field", 2 ** 31 - 1, 1), seed=0)  # Fp
Particles(5, field=Field("padic", 2 ** 31 - 1, 5), seed=0)  # Qp
Particles(5, field=Field("mpc", 0, 300), seed=0)  # C (examples for massless PSPs)

$\circ$ Analytic manipulations are too complex, we bypass this complexity by letting cancellations
$\phantom{\circ}$ happen numerically. Modular arithmetic will ensure we do not lose precision.

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

Variables Subject to Constraints

$\circ$ Consider polynomials $f, g, h$ in two variables $x, y$. They live in a polynomial ring:

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

$\circ$ We may want to consider e.g. funcitons on the unit circle, $(x^2+y^2-1)$. If we have

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

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

$\circ$ This 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, that is the infinite set of polynomials
$\phantom{\circ}$ $h(x, y) (x^2+y^2-1)$, for any $h(x,y)$, that vanishes on the unit circle.

Massless Scattering

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

GDL, Page ('22)

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

$\phantom{\circ}$ Recall the simple relation $p_i^\mu \sigma^\mu_{\alpha\dot\alpha} = |i\rangle_\alpha [i|_{\dot\alpha}$.

$\circ$ 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

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

$\circ$ Integral coefficients are rational functions $r_i$ in the field of fractions of $R_n$,

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

Covariant Q-Ring for $\text{ggHHH}$

$\circ$ 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.

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

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

$\phantom{\circ}$ 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).

$\circ$ It is sometimes useful to map to a set of all massless momenta / spinors,

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!

$\circ$ Our coefficients $d^{h_1h_2}_{p_a\times p_b \times p_c }$ and $c^{h_1h_2}_{p_a\times p_b}$ belong to the field of fractions over $R_{HHH}$.

Least Common Denominator

(i.e. geometry at codimension one)

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

$$ \displaystyle \mathcal{D} = \prod_j \mathcal{D}_j^{q_{ij}} \in R_{HHH} \; , \; \mathcal{D}_j \text{ irreducible} \, , $$

$\phantom{\circ}\,$ from a univariate slice $\vec\lambda(t)$ giving us $\mathcal{D}(t)$,
$\phantom{\circ}\,$ if we know the possible $\mathcal{D}_j$.

$\circ$ 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. For $HHH$,
$\phantom{\circ}\,$ we can use the massless algorithm at 8 point (or shift the $p_{\alpha,\dot\alpha}$).

The space has dimension $20-6=14$,

$\mathcal{D}_j = 0$ have dimension $14-1=13$,

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

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

$HHH$ LCD Factors

$\circ\,$ The irreducible denominator factors $\mathcal{D}_j$ for $HHH$ are

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

$\phantom{\circ}$ plus closure under permutations, where

$$ \Delta_{12|3|4|5} \;=\; \det\begin{pmatrix} p_{12}\!\cdot\! p_{12} & p_{12}\!\cdot\! p_{3} & p_{12}\!\cdot\! p_{4} \\ p_{3}\!\cdot\! p_{12} & p_{3}\!\cdot\! p_{3} & p_{3}\!\cdot\! p_{4} \\ p_{4}\!\cdot\! p_{12} & p_{4}\!\cdot\! p_{3} & p_{4}\!\cdot\! p_{4} \end{pmatrix} \quad \text{ and } \quad\quad \begin{aligned} \text{tr}_5(1|2|3|4) &= \text{tr}(\gamma^5 p_1 p_2 p_3 p_4) \\ &= [1|2|𝟑|𝟒|1⟩ - ⟨1|2|𝟑|𝟒|1] \end{aligned} $$

$\phantom{\circ}$ The two poles mixing kinematics with the top mass are what is left overs of the pentagons.

$\circ$ For example, for an integral coefficient at this stage we see

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

$\phantom{\circ}$ For some unknown $\mathcal{N}$ which would be fairly complicated in this LCD form.

A Concrete Example

$\circ$ For instance, we aim to find a form like

$$ \begin{gathered} \hat d^{++}_{12\times 3 \times 4}=\Bigg\{\frac{[2|𝟒|𝟑-𝟓|2]\text{tr}(𝟓|𝟒|𝟑|1-2)}{4⟨12⟩⟨1|𝟓|𝟒|𝟑|2]} - \frac{(s_{𝟑𝟒}-2m_h²)(s_{𝟑𝟓}+m_h²-2s_{2𝟒})}{2⟨12⟩²} - \frac{(\text{tr}(1-2|𝟑)m_h²+⟨1|𝟑|𝟒|2⟩[12])}{⟨12⟩²} +\\ -\frac{(s_{𝟒𝟓}-s_{𝟑𝟒})²(s_{1𝟑}-s_{2𝟑})(s_{1𝟑}+s_{2𝟑})(\text{tr}(1+2|𝟒)+4s_{𝟑𝟒}-8m_t²)}{32⟨12⟩²Δ_{12|𝟑|𝟒|𝟓}} +\\ -\frac{(s_{𝟒𝟓}-s_{𝟑𝟒})(s_{1𝟑}-s_{2𝟑})(s_{𝟑𝟒}-m_h²)((s_{𝟒𝟓}-s_{𝟑𝟒})\text{tr}(1+2|𝟒)+s_{𝟑4}(s_{1𝟑}+s_{2𝟑})-s_{𝟑4}(s_{𝟑𝟒}-2m_h²)-8s_{123}m_t²)}{8⟨12⟩²Δ_{12|𝟑|𝟒|𝟓}} +\\ \frac{Δ_{12|𝟒|𝟑5}(s_{1𝟑}-s_{2𝟑})(s_{1𝟑}+s_{2𝟑})(\text{tr}(1+2|𝟒)-8m_t²)}{8⟨12⟩²Δ_{12|𝟑|𝟒|𝟓}} \Bigg\} + \Bigg\{12𝟑𝟒𝟓\rightarrow21𝟓𝟒𝟑\Bigg\} \end{gathered} $$

$\circ\,$ Challenge 1: how do we parametrize the numerators?

$\circ\,$ Challenge 2: in LCD form the numerators are often too complicated.
$\kern18mm$ How do we identify allowed partial fraction decompositions?

Analytic Reconstruction

Invariant Quotient Sub-Rings

(see also 2509.14350, Some remarks on invariants)

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

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

$\quad\star$ Build a ring with both covariant and invariant variables (here showing massless case)

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

$\quad\star$ Define relations among variables (on top of existing constraints, e.g. $p_3^2=p_4^2$)

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

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

$\circ$ For $HHH$, this yields the following quotient ring for the invariants

$$ \displaystyle \mathcal{R}_{HHH} = \frac{\underbrace{\substack{\normalsize\kern-30mm\mathbb{F}\big[ ⟨1|2⟩, [1|2], ⟨1|𝟑|1], ⟨1|𝟑|2], ⟨2|𝟑|1], ⟨2|𝟑|2], ⟨1|𝟒|1], ⟨1|𝟒|2], ⟨2|𝟒|1], ⟨2|𝟒|2],\\[2mm] \normalsize \kern10mm ⟨1|𝟑|𝟒|1⟩, ⟨1|𝟑|𝟒|2⟩, ⟨2|𝟑|𝟒|2⟩, [1|𝟑|𝟒|1], [1|𝟑|𝟒|2], [2|𝟑|𝟒|2], \text{tr}(𝟑|𝟑), \text{tr}(𝟑|𝟒), \text{tr}(𝟒|𝟒), m_h^2 \big]}}_{20 \text{ variables}}}{\big\langle \underbrace{\text{tr}(𝟒|𝟒)-2m_h^2, \text{tr}(𝟑|𝟑)-2m_h^2, ⟨2|\boldsymbol{3}|2]⟨2|\boldsymbol{4}|1]-⟨2|\boldsymbol{3}|1]⟨2|\boldsymbol{4}|2]-[1|2]⟨2|\boldsymbol{3}|\boldsymbol{4}|2⟩, ...}_{\text{subject to } 122 \; \text{redundancy relations / Schouten identities (only first 2 are trivial rewritings)}} \big\rangle} $$

The Numerator Ansatz

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

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.
$\phantom{\circ}$ For HHH we have polynomials in the 20 invariants from the previous slide.

$\circ\,$ Construct the Ansatz via the algorithm from Section 2.2 of GDL, Page ('22)

Linear independence = irreducibility by the Gröbner basis of the ideal of the redundancies.

$\circ\,$ 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)

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 the irreducible polynomials $\mathcal{D}_1, \mathcal{D}_2$.

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

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

$\cap$

$=$

$$ \langle {\color{orange}xy^2 + y^3 - z^2} \rangle + \langle {\color{blue}x^3 + y^3 - z^2} \rangle = \langle xy^2 + y^3 - z^2, x^3 + y^3 - z^2 \rangle = \langle {\color{red}2y^3-z^2, x-y} \rangle \cap \langle {\color{green}y^3-z^2, x} \rangle \cap \langle {\color{blue}z^2, x+y} \rangle $$

$\phantom{\circ}$ This is a primary decomposition, it is the equivalent for polynomials of say: $12 = 2^2 \times 3$
$\phantom{\circ}$ 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)

$\circ$ Let's go back to our example

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

$\circ$ 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 $⟨1|𝟓|𝟒|𝟑|2] \rightarrow 0$

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

Conclusions
&
Outlook

Analytic Results for Theory and Phenomenology

$\circ$ Analytic expressions implemented in MCFM, for phenomenology use this efficient Fortran implementation

Campbell, Neumann Campbell, Ellis, Giele; Campbell, Ellis, Williams;

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

Challenges

$\circ\,$ Can we always verify constraints numericaly? Alternatively, can we predict/guess them?
$\phantom{\circ}\,$ $p$-adic evaluations can be costly (especially with multi-loop amplitudes).

$\circ\,$ Imposing multiple constraints at ones means computing ideal intersections, which 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.

$\phantom{\circ}\,$ 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 (trial and error).

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

$\circ\,$ Even constructing the ansatz requires a Groebner Basis, which in some cases Singular doesn't easily give.
$\phantom{\circ}\,$ For $pp\rightarrow HHHj$ we don't have the full GB, we need to remove redundancies through linear algebra.

$\circ\,$ The reduction to master integrals of the amplitude is often not easy in the first place.

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Effective Pentagons

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

$$ \begin{eqnarray} &&E_0(p_1,p_2,p_3,p_4;\mt)= c^{(1)} D_0(p_2,p_3,p_4;\mt) +c^{(2)} D_0(p_{12},p_3,p_4;\mt) \\ &+&c^{(3)} D_0(p_1,p_{23},p_4;\mt) +c^{(4)} D_0(p_1,p_2,p_{34};\mt) +c^{(5)} D_0(p_1,p_2,p_3;\mt)\, . \end{eqnarray} $$

$\circ$ 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

$\circ$ 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.

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)