Mathematical and Physical Structures
of Rational Functions in Scattering Amplitudes
Title for Mathematicians: Vector Spaces over Fraction Fields of Polynomial Quotient Rings
Giuseppe De Laurentis
arXiv:2203.04269
Paul Scherrer Institut / University of Edinburgh
arXiv:2203.04269
(GDL, B. Page)
arXiv:2305.17056 (S. Abreu, GDL, H. Ita, M. Klinkert, B. Page, V. Sotnikov)
MathemAmplitudes 2023 - Padova
Find these slides at gdelaurentis.github.io/slides/mathemamplitudes_sept2023
Introduction
Scattering Amplitudes
$\circ$ Amplitude (integrands) can be written as
$$
\require{color}
\require{amsmath}
\displaystyle A(\lambda, \tilde\lambda, \ell) =
\sum_{\substack{\Gamma,\\ i \in M_\Gamma \cup S_\Gamma}} \, c_{\,\Gamma,i}(\lambda, \tilde\lambda, \epsilon) \, \frac{m_{\Gamma,i}(\lambda\tilde\lambda, \ell)}{\textstyle \prod_{j} \rho_{\,\Gamma,j}(\lambda\tilde\lambda, \ell)} \;\; \xrightarrow[]{\int d^D\ell} \;\; \sum_{\substack{\Gamma,\\ i \in M_\Gamma}} {\color{red}c_{\,\Gamma, i}}(\lambda, \tilde\lambda, \epsilon) \, {\color{orange}I_{\Gamma, i}}(\lambda\tilde\lambda, \epsilon)
$$
$\circ$ For a suitable choice of integrands, we get:
$$
\displaystyle
{\color{red}c_{\Gamma, i}}(\lambda, \tilde\lambda, \epsilon) = \frac{ \sum_{k=0}^{\text{finite}} \, {\color{red}c^{(k)}_{\,\Gamma, i}}(\lambda, \tilde\lambda) \, \epsilon^k}{\prod_j (\epsilon - a_{ij})} \;, \;\;\text{with} \quad a_{ij} \in \mathbb{Q}
$$
Some notation:
$\circ$ $\Gamma$: topologies $\quad\circ$ $M_\Gamma$: masters $\quad\circ$ $S_\Gamma$: surface terms
$\circ$ Spinors: $\lambda_i = |i\rangle, \tilde\lambda_i =[i|$
$\quad\circ$ 4-momenta: $\lambda\tilde\lambda=p\kern-3mm/$
$\quad\circ$ Loop $D$-momenta: $\ell $
Outline
Disclaimer: I will focus on the $c_{\,\Gamma, i}^{(k)}(\lambda, \tilde\lambda)$ $-$ call them $c_i(\lambda, \tilde\lambda)$ for short,
nevertheless some concepts can be extended to whole amplitudes.
nevertheless some concepts can be extended to whole amplitudes.
We will discuss:
$1.$ Where are the $\color{red}\text{poles}$? What is their order?
$2.$ Relation between pole structure, and $\color{red}\text{analytic constraints}$ (e.g. partial fraction decomposition)
$3.$ Reconstructing $\color{red}\text{sets of functions}$: Why is it easier than reconstructing individual ones?
$4.$ Example process @ 2-loop: $\;\;pp \rightarrow \gamma\gamma\gamma\;\;$ and $\;\;pp \rightarrow Wjj\;\;$ (preliminary)
Key take away:
What is important is not the number of variables,
but the size of the parametrization (a.k.a. ansatz), and our ability to constrain it.
but the size of the parametrization (a.k.a. ansatz), and our ability to constrain it.
Polynomial Quotient Rings
$\circ$ Let us start from the polynomial ring of spinor components
$$\displaystyle \kern-50mm S_n = \mathbb{F}\left[|1⟩, [1|, \dots, |n⟩, [n|\right]$$
$\phantom{\circ}$ the field $\mathbb{F}$ can be any of $\mathbb{Q},\mathbb{R},\mathbb{C},\mathbb{F}_p,\mathbb{Q}_p,\dots$
$\circ$ Define the momentum-conservation ideal as
$$
\displaystyle J_{\Lambda_n} = \Big\langle \sum_i |i⟩[i| \Big\rangle_{S_n}
$$
Artist's Impression of $V(J_{\Lambda_n})$
I can't draw in $4n$ dims!
I can't draw in $4n$ dims!
$\phantom{\circ}$ physically, two polynomials $p$ and $q$ are equivalent if $p-q\in J_{\Lambda_n}$
$\circ$ This defines the needed polynomial quotient ring$\kern-4mm\phantom{x}^{\star}$: $\;R_n = S_n / J_{\Lambda_n} $
$c_i(\lambda, \tilde\lambda)$ at $n$-point belong to the Field of Fractions$\kern-4mm\phantom{x}^{\dagger}$ of $R_n$
$\kern-4mm\phantom{x}^\star R_4$ is "weird" (not a UFD), but it proves that polynomial rings are insufficient;
$\;\kern-4mm\phantom{x}^\dagger$ The field of fractions of $R_3$ does not exist.
The Pole Structure
Prime Ideals & Irreducible Varieties
$\circ$ Let us consider a very simple example
$\displaystyle \kern-50mm iA_{g^-g^-g^+g^+}^{\text{tree}} = \frac{\langle 12 \rangle^3}{\langle 23 \rangle \langle 34 \rangle \langle 41 \rangle} = \frac{[34]^3}{[12][23][41]} $
$\phantom{\circ}$ is, say, $\langle 23 \rangle$ a pole of this amplitude?
Artist's Impression of $V(\big\langle \langle 23 \rangle\big\rangle_{R_4})$
as the union of two irreducibles
as the union of two irreducibles
$\circ$ The question is ill posed!
$\phantom{\circ} \langle 23 \rangle$ does not identify an irreducible variety in $R_4$.
$\phantom{\circ}$ Compute $\color{green}\text{primary decompositions}$, such as
$\displaystyle \big\langle \langle 23\rangle \big\rangle_{R_4} = {\color{orange} \big\langle \langle 23\rangle, [14] \big\rangle_{R_4}} \cap {\color{blue} \big\langle \langle 12\rangle, \langle 13 \rangle, \langle 14\rangle, \langle 23\rangle, \langle 24 \rangle, \langle 34 \rangle \big\rangle_{R_4}} $
$\phantom{\circ}$ On the first branch there is a simple pole, on the latter branch the amplitude is regular.
Poles & Zeros $\;\Leftrightarrow\;$ Irreducible Varieties $\;\Leftrightarrow\;$ Prime Ideals
Physics $\kern38mm$ Geometry $\kern38mm$ Algebra
Physics $\kern38mm$ Geometry $\kern38mm$ Algebra
Constraints from Poles
Bootstrapping trees (?)
$\circ$ The degree of divergence / vanishing on various surfaces imposes strong constraints, e.g.
$ A^{\text{tree}}_{q^+g^+g^+\bar q^-g^-g^-} = \frac{\mathcal{N(\text{m.d.} = 6\,,\; \text{p.w.} = [-1, 0, 0, 1, 0, 0])}}{\langle 12\rangle\langle 23\rangle\langle 34\rangle [45][56][61]s_{345}}$
$\circ$ Pretend this is un unknown integral coefficient, $\mathcal{N}$ has 143 free parameters.
$\circ$ List the various prime ideal, such as
$ \big\langle \langle 12\rangle, \langle 23\rangle, \langle 13\rangle \big\rangle, \; \big\langle |1\rangle \big\rangle, \; \big\langle \langle 12\rangle, |1+2|3]\big\rangle, \dots$
$\phantom{\circ}$ and impose that $\mathcal{N}$ vanishes to the correct order. We determine it up to an overall constant.
GDL, Page ('22)
$\circ$ Likewise, the ansatz for $A^{\text{tree}}_{g^+g^+g^+ g^-g^-g^-}$ shrinks $1326 \rightarrow 1$, etc..
Effectively, we can compute trees, just from their poles orders.
Note: compared to BCFW there is no information about residues.
Note: compared to BCFW there is no information about residues.
Partial Fraction Decompositions
$\circ$ For true integral coefficients, we can't rely on the Ansatz to shrinks to an overall constant.
$\circ$ Partial fraction decompositions (PFDs) are a popular method to tame algebraic complexity.
$\circ$ In my opinion, a PFD algorithm needs
$1.$ to say if two poles $W_a$ and $W_b$ are separable into different fractions;
$2.$ ideally, to answer $(1.)$ without having access to an analytic expression.
$2.$ ideally, to answer $(1.)$ without having access to an analytic expression.
$\circ$ Hilbert's nullstellensatz: if $\mathcal{N}$ vanishes on all branches of $\langle W_a, W_b \rangle$, then the PFD is possible$\kern-3mm\phantom{x}^\dagger$.
$\circ$ Generalizing to powers $>\kern-1mm 1$ can be done via symbolic powers and the Zariski-Nagata Theorem.
GDL, Page ('22)
$\circ$ Similarly, generalizing to non-radical ideals requires ring extensions.
Campbell, GDL, Ellis ('22)
Issue: evaluations on singular surfaces are expensive, but are not always needed!
Opportunity: we get more than partial fraction decompositions.
$\kern-4mm\phantom{x}^\dagger$ $\langle W_a, W_b\rangle$ needs to be radical.
Beyond Partial Fractions
$\circ$ $\color{red}\text{Case 0}$: the ideal does $\color{green}\text{not involve denominator factors}$.
E.g. a 6-point function $c_i$ has a pole at $⟨1|2+3|4]$ but not at $⟨4|2+3|1]$,
yet it is regular on the irreducible surface $V(\big\langle ⟨1|2+3|4], ⟨4|2+3|1] \big\rangle)$. Then
$\displaystyle c_i \sim \frac{⟨4|2+3|1]}{⟨1|2+3|4]} + \mathcal{O}(⟨1|2+3|4]^0) \; \text{ instead of } \; c_i \sim \frac{1}{⟨1|2+3|4]} + \mathcal{O}(⟨1|2+3|4]^0)$
$\circ$ $\color{red}\text{Case 1}$: the $\color{green}\text{degree of vanishing is non-uniform}$ across branches, for example:
$\displaystyle \frac{s_{14}-s_{23}}{⟨1|3+4|2]⟨3|1+2|4]}$
has a double pole on the first branch, and a simple pole on the second branch of
$\big\langle⟨1|3+4|2], ⟨3|1+2|4]\big\rangle_{R_6} = \big\langle ⟨13⟩, [24] \big\rangle_{R_6} \cap \big\langle ⟨1|3+4|2], ⟨3|1+2|4], (s_{14}-s_{23})\big\rangle_{R_6}$
$\circ$ $\color{red}\text{Case 2}$: ideal is $\color{green}\text{non-radical}$ (example on last slide)
$\displaystyle \small \kern0mm \sqrt{\big\langle {\color{black}⟨3|1+4|2]}, {\color{black}Δ_{23|14|56}} \big\rangle_{R_6}} = \big\langle {\color{black}⟨3|1+4|2]}, {\color{black}s_{124}-s_{134}} \big\rangle_{R_6} $
Analytic Reconstruction
Choosing Independent Functions
$\circ$ The set $c_i$ can be very large, so pick a set of independent ones, and write:
$\displaystyle c_i = \tilde{c}_j M_{ji} \quad \text{with} \quad M_{ji} \in \mathbb{Q}$
$\phantom{\circ}$ with $\tilde{c}$ an independent subset of $c$.
$\phantom{\circ}\Longrightarrow$ $M_{ji}$ is, up to a permutation of columns, in row reduced echelon form.
$\phantom{\circ}\Longrightarrow$ $M_{ji}$ is, up to a permutation of columns, in row reduced echelon form.
$\circ$ We might as well use a set $\tilde{c}$ which is not a subset of $c$, at the trivial cost of changing $M_{ji}$.
$\circ$ Consider a PFD of one of the $\tilde{c}$'s:
$\displaystyle \tilde{c}_i(\lambda,\tilde\lambda) = \frac{\mathcal{N}_i(\lambda,\tilde\lambda)}{\prod_j W_j^{q_{ij}}(\lambda,\tilde\lambda)} = \sum_k \frac{\mathcal{N}_{ik}(\lambda,\tilde\lambda)}{\prod_j W_j^{q_{ijk}}(\lambda,\tilde\lambda)} = \sum_k \tilde{c}_{ik}(\lambda,\tilde\lambda)$
We $\color{red}\text{cannot have } \tilde{c}_i \in \text{span}(\tilde{c}_{j\neq i})$, but we $\color{green} \text{can have }\tilde{c}_{ik} \in \text{span}(\tilde{c}_{j\neq i})$, for some, but not all, $k$.
Least Least-Common-Denominator
$\circ$ In other words, the $\tilde{c}$ span a vector space, and we should $\color{green}\text{consider one modulo the others}$
$\displaystyle \tilde{c}_i = \sum_{j\neq i} q_j \tilde{c}_j + \tilde{c}'_{i}$
$\phantom{\circ}$ the basis function $\tilde{c}_i$ can be replaced by $\tilde{c}'_i$ without changing the vector space.
$\circ$ In particular, $\tilde{c}'_i$ needs not have all the poles of $\tilde{c}_i$, thus it can be much simpler.
$\phantom{\circ}$ In other words, $\color{red}\text{the LCD of }\tilde{c}'_i\text{ can be smaller than that of }\tilde{c}_i$.
$\phantom{\circ}$ In other words, $\color{red}\text{the LCD of }\tilde{c}'_i\text{ can be smaller than that of }\tilde{c}_i$.
$\circ$ Brute-force search works well when an analytic expressions is available.
$\circ$ In a future publication we will provide an algorithm based on finite field evaluations.
Reconstructing a set of $c_i$ is not as bad as reconstructing the most complex function in the set.
Example
processes
Three-photon production at two loops
$\circ\,$ The denominator factors $W_j$ are conjectured to be restricted to the letters of the symbol alphabet
Abreu, Dormans, Febres Cordero, Ita, Page ('18)
$\displaystyle \{W_j\} = \bigcup_{\sigma \; \in \; \text{Aut}(R_5)} \sigma \circ \big\{ \langle 12 \rangle, \langle 1|2+3|1] \big\} {\quad\color{green}\text{Identical to 1-loop!}}$
$\circ\,$ Advantage of spinor variables due to:
$1.$ little group covariant LCD (no spurious poles); $\;\;2.$ avoiding parity even/odd split;
$\Rightarrow\;$ in LCD form we would need $\color{green}29\,059$ evaluations instead of $\color{red}117\,810$ (with $s_{ij}$) for $\mathcal{R}^{(2)}_{2q3\gamma}$ .
$\Rightarrow\;$ in LCD form we would need $\color{green}29\,059$ evaluations instead of $\color{red}117\,810$ (with $s_{ij}$) for $\mathcal{R}^{(2)}_{2q3\gamma}$ .
$\circ\,$ To$\color{green}\text{ avoid evaluations on singular surfaces}$, use insights from physics (locality).
$\phantom{\circ}\,$ E.g. conjecture that no 5-point denominator has pairs of $\langle i |j + k | i]$, like at 1 loop.
$\phantom{\circ}\,$ E.g. conjecture that no 5-point denominator has pairs of $\langle i |j + k | i]$, like at 1 loop.
$\circ$ Remove some overlap with other $\tilde{c}$'s, obtain the $\tilde{c}_i$ with higher degree LCD with $\color{green}4\,003$ points.
$\circ$ A posteriori, we find that for the $c_i$ with highest degree LCD the following would have sufficied
$\displaystyle \tilde{c}_i = \frac{⟨13⟩[14]^2⟨24⟩⟨34⟩[45]}{⟨45⟩⟨4|1+3|4]^3}-\frac{[14]⟨25⟩⟨34⟩^2[45]}{⟨45⟩^2⟨4|1+3|4]^2}-\frac{[14]⟨24⟩⟨34⟩⟨35⟩}{⟨45⟩^3⟨4|1+3|4]}$
$0.\,$ Start from analytics of Abreu, Febres Cordero, Ita, Klinkert, Page, Sotnikov ('21) - 1.2GB of
C++ source code.
$1.\,$ Script to split up the expressions, and compile them ($\sim 20$GB binaries) for evaluation over $\mathbb{F}_p$;
$2.\,$ Recombine the 3 projections $p_V \parallel p_1, p_V \parallel p_2, p_V \parallel p_3$ and reintroduce the little group factors
to build 6-point spinor-helicity amplitudes (subject to degree bounds on $|5\rangle,[5|,|6\rangle,[6|$);
to build 6-point spinor-helicity amplitudes (subject to degree bounds on $|5\rangle,[5|,|6\rangle,[6|$);
$3.\,$ Perform (rough) PFDs based on expected structures and fit the Ansatze.
Comparison of $q\bar q \rightarrow \gamma \gamma \gamma$ (in full color) to $pp \rightarrow Wjj$ (at leading color):
| Kinematics | # Poles ($W$) | LCD Ansatz | Partial-Fraction Ansatz | Rational Functions |
| 5-point massless | 30 | 29k | 4k | $\sim$300 KB |
| 5-point 1-mass | >200 | >5M | $\sim$40k | $\sim$25 MB |
$\displaystyle \kern-10mm \{W_j\} = \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\} $
$\circ$ The Ansatz size grows quickly with
multiplicity (m) and mass dimension (d):
$\displaystyle \small \left(\mkern -9mu \begin{pmatrix}\, m(m-3)/2 \, \\ \, d/2 \, \end{pmatrix} \mkern -9mu \right)$
is a lower bound. GDL, Maître ('20)
multiplicity (m) and mass dimension (d):
$\displaystyle \small \left(\mkern -9mu \begin{pmatrix}\, m(m-3)/2 \, \\ \, d/2 \, \end{pmatrix} \mkern -9mu \right)$
is a lower bound. GDL, Maître ('20)
$\circ\,$ Compact residues for the new 2-loop (spurious?) pole, $⟨k|j|p\mkern-7.5mu/_V|l|k]-⟨j|i|p\mkern-7.5mu/_V|l|j]$, e.g.:
$$r^{(5 \text{ of } 54)}_{\bar{u}^+g^+g^+d^-(V\rightarrow \ell^+ \ell^-)} = \frac{[12][23]⟨24⟩⟨46⟩^2⟨1|2+3|4]⟨2|1+3|4]}{⟨12⟩⟨23⟩⟨56⟩(⟨3|2|5+6|4|3]-⟨2|1|5+6|4|2])^2}$$
$\circ\,$ The three mass Grams, $\Delta_{12|34|p_V}, \Delta_{14|23|p_V}$, behave analogously to one-loop amplitudes, e.g.:
$$ r^{(73 \text{ of } 120)}_{\bar{u}^+g^-g^+d^-(V\rightarrow \ell^+ \ell^-)} = \frac{105}{128}\frac{⟨2|1+4|3]⟨4|2+3|1]⟨6|1+4|5]s_{14}s_{23}s_{56}{\color{green}(s_{124}-s_{134})}(s_{123}-s_{234})(s_{25}+s_{26}+s_{35}+s_{36})}{{\color{orange}⟨3|1+4|2]}{\color{red}Δ_{23|14|56}^4}} + \\
\Bigg[-6\frac{[12]^2⟨13⟩[25]⟨34⟩⟨36⟩⟨56⟩[56]{\color{green}(s_{124}-s_{134})}}{{\color{orange}⟨3|1+4|2]^5}}\Bigg] + \Bigg[ \; \Bigg]_{1234\rightarrow \overline{4321}}+ \mathcal{O}\left(\frac{1}{⟨3|1+4|2]^{4}Δ_{23|14|56}^{3}}\right)$$
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Backup Slides
Finite remainders & the
Rational / Transcendental split
$\circ$ In general, in $D= 4- 2 \epsilon$, with pure master integrals $I_{\Gamma, i}$ we have
$$ A^{\ell-loop}_n(\lambda, \tilde\lambda) = \sum_\Gamma \sum_{i \in M_\Gamma} \frac{\color{orange}{c_{\,\Gamma, i}}(\lambda, \tilde\lambda, \epsilon) \, \color{red}{I_{\Gamma, i}}(\lambda\tilde\lambda, \epsilon)}{\prod_j (\epsilon - a_{ij})}\;, \quad \text{with} \quad a_{ij} \in \mathbb{Q}$$
$\circ$ For NNLO applications, we are interested in the finite remainder
$$
\mathcal{A}^{(2)}_R = \underbrace{\mathcal{R}}_{\text{finite remainder}} + \underbrace{I^{(1)}\mathcal{A}^{(1)}_R \quad + \quad I^{(2)}\mathcal{A}^{(0)}_R}_{\text{divergent + convention-dependent finite part}} + \mathcal{O}(\epsilon)
$$
$\circ$ Finite remainder as a weighted sum of pentagon functions Chicherin, Sotnikov ('20);
$$
\textstyle \mathcal{R}(\lambda, \tilde\lambda) = \sum_i \color{orange}{r_{i}(\lambda,\tilde\lambda)} \, \color{red}{h_i(\lambda\tilde\lambda)}
$$
Reconstruct $\color{orange}{r_{i}(\lambda,\tilde\lambda)}$ from $\mathbb{F}_p$ samples
Peraro ('16)
von Manteuffel, Schabinger ('14)
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
Integer programming $\rightarrow$ enumerate sols. $\vec\alpha,\vec\beta$
Perron and Furnon (Google optimization team)
$\circ\,$ All linear systems solved with CUDA over $\mathbb{F}_{p\leq 2^{31}-1}$ on a laptop ($t_{\text{solving}} \ll t_{\text{sampling}}$)
Polynomial Quotient Rings
vs.
Polynomial Rings
$\circ$ Can we get rid of equivalence relations (redundancies) by changing variables?
$\phantom{\circ}$ In other words, can we solve the redundancies and turn the quotient ring in a ring?
$\phantom{\circ}$ In other words, can we solve the redundancies and turn the quotient ring in a ring?
With four-point massless kinematics, you cannot. In $R_4$ we have
$ \langle 12\rangle [12] = \langle 34\rangle [34]$
this means that $R_4$ is not a unique factorization domain (UFD).
All polynomial rings are UFDs, so $R_4$ cannot be isomorphic to one.
All polynomial rings are UFDs, so $R_4$ cannot be isomorphic to one.
The Field of Fractions of $R_3$ does not exists
$\circ$ $R_3$ is not an integral domain, i.e. there exists products of non-zero elements which are zero.
$\langle 12\rangle [23] = \langle 1|2|3] = -\langle 1| 1+3 |3] = 0 \;\text{but}\; \langle 12\rangle \neq 0 \;\text{and}\; [23] \neq 0$
Hence, one cannot define a field of fractions, as this would not be closed under multiplication.