Non-planar two-loop QCD
corrections to $\boldsymbol{q\bar q \rightarrow \gamma\gamma\gamma}$:
finite remainders in the spinor-helicity formalism

Giuseppe De Laurentis

in collaboration with:
S. Abreu, H. Ita, M. Klinkert, B. Page, V. Sotnikov

based on: arXiv:2305.17056

LoopFest XXI



Find these slides at gdelaurentis.github.io/slides/loopfestxxi_june2023

Introduction

State-of-the-Art of $\boldsymbol{\mathcal{A}^{(2-\textbf{loop})}_{,n}}$

Five-point massless amplitudes in full color:
$\circ\,$ $pp\rightarrow \gamma jj$ Badger, Czakon, Hartanto, Moodie, Peraro, Poncelet, Zoia ('23)
$\circ\,$ $pp\rightarrow \gamma\gamma j$ Agarwal, Buccioni, von Manteuffel, Tancredi ('21) Badger, Brønnum-Hansen, Chicherin, Gehrmann, Hartanto, Henn, Marcoli, Moodie, Peraro, Zoia ('21)

$\circ\,$ $pp\rightarrow \gamma\gamma\gamma$ This talk! Abreu, GDL, Ita, Klinkert, Page, Sotnikov ('23); 
$\circ\,$ $pp\rightarrow jjj \quad (?)$ Next talk by Federico

Five-point one-mass amplitudes at leading color:
$\circ\,$ $pp\rightarrow Wb\bar b$ Badger, Hartanto, Zoia ('21)
$\circ\,$ $pp\rightarrow Hb\bar b$ Badger, Hartanto, Kryś, Zoia ('21)
$\circ\,$ $pp\rightarrow Wjj$ A bit about this towards the end of this talk! Abreu, Febres Cordero, Ita, Klinkert, Page, Sotnikov ('21); 
$\circ\,$ $pp\rightarrow W\gamma j$ Badger, Hartanto, Kryś, Zoia ('22)
Two-loop five-point amplitudes remain a challenge, but are now very much feasible.

Sizable NNLO Corrections to $\boldsymbol{q\bar q \rightarrow \gamma\gamma\gamma}$

$\circ\,$ NNLO cross-sections computed with leading-color double-virtual amplitudes
$\circ\,$ Analytic two-loop amplitudes in limit $N_c \rightarrow \infty \, , \; N_c/N_f = \text{const}.$
Question: are the subleading-color contributions trully negliegible?

Gauge-Invariant Subamplitudes

$$\require{color} \displaystyle \mathcal{A}^{(2)}_{\,2q3\gamma} = \frac{N_c^2}{4}\left( {\color{green} A^{(2,0)}_{\,2q3\gamma} } - \frac{1}{N_c^2}(A^{(2,0)}_{\,2q3\gamma}+A^{(2,1)}_{\,2q3\gamma}) + \frac{1}{N_c^4} {\color{red} A^{(2,1)}_{\,2q3\gamma} } \right) \\[2mm] \qquad + C_F T_F N_f {\color{green} A^{(2,N_f)}_{\,2q3\gamma} } + \underbrace{C_F T_F \left(\sum_{f=1}^{N_f} Q_f^2\right)}_{\text{trully suppressed?}} \, {\color{red} A^{(2,\tilde{N}_f)}_{\,2q3\gamma} } \, , $$
$\circ\,$ Example diagram for each amplitude:
${\color{green} A^{(2, 0)}_{\,2q3\gamma} }$: ${\color{green} A^{(2, N_f)}_{\,2q3\gamma} }$: Previously known
${\color{red} A^{(2, 1)}_{\,2q3\gamma} }$: ${\color{red} A^{(2, \tilde{N}_f)}_{\,2q3\gamma} }$: New in this work

Organization
of the Computation

Generalized Unitarity

$\circ$ Loop integrands can be written as ($\lambda = |\bullet\rangle, \tilde\lambda=[\bullet|, \lambda\tilde\lambda=p\kern-3mm/$)

$$ \require{color} \displaystyle A(\lambda, \tilde\lambda, \ell) = \sum_{\Gamma} \, \sum_{i \in M_\Gamma \cup S_\Gamma} \, c_{\,\Gamma,i}(\lambda, \tilde\lambda) \, \frac{m_{\Gamma,i}(\lambda\tilde\lambda, \ell)}{\textstyle \prod_{j} \rho_{\,\Gamma,j}(\lambda\tilde\lambda, \ell)} $$
$\circ$ Generalized unitarity relates cuts of loop amplitudes to products of trees
$$ \require{color} \displaystyle \sum_{\text{states}} \, \prod_{\text{trees}} A^{\text{tree}}(\lambda, \tilde\lambda, \ell)\big|_{\text{cut}} = \sum_{\substack{\Gamma' \ge \Gamma, \\ i \in M_\Gamma' \cup S_\Gamma'}} \kern-2mm c_{\,\Gamma',i}(\lambda, \tilde\lambda) \, \frac{m_{\Gamma',i}(\lambda\tilde\lambda, \ell)}{\displaystyle \prod_{j\in P_{\Gamma'} / P_{\Gamma}} \rho_{j}(\lambda\tilde\lambda, \ell)}\Bigg|_{\text{cut}} $$
$$ \left. \begin{aligned} \underline{\text{Master integrands}}: \; & \int d^{D} \ell \; \frac{m_{i\in M_\Gamma}}{\small \prod_j \rho_j} \neq 0 \\ \underline{\text{Surface terms}}: \; & \int d^{D} \ell \; \frac{m_{i\in S_\Gamma}}{\small \prod_j \rho_j} = 0 \\ \end{aligned} \right\} \; \begin{aligned} & \text{Equivalent to} \\ & \text{IBP reduction} \end{aligned} $$ Ita ('15)

New Features of the Reduction

$\circ$ Master / surface decomposition for non-planar topologies
$$ \require{color} \begin{align} \kern-25mm \text{IBP-generating vectors: } & \quad \displaystyle \int d^D \ell \frac{\partial }{\partial \ell^\mu_a} \frac{v^\mu_a(\ell)}{\rho_1 \dots \rho_N} = 0 \quad (\text{in dim. reg.}) \\[2mm] \kern-25mm \text{No propagator doubling: } & \quad \displaystyle \sum_{a, \mu} v^\mu_a(\ell) \frac{\partial \rho_i}{\partial \ell^\mu_a} - f_i(\ell)\rho_i = 0 \end{align} $$
$(v^\mu_a, f_i)$ form a syzygy module, solved for in embedding space using Singular + linear algebra.

$\circ$ Semi-numerical surface terms: $\quad m_{i\in S_\Gamma}(\ell \leftarrow \text{analytical}, s_{ij} \leftarrow \text{numerical})$
$\kern20mm\star$ dependance on external kinematics ($s_{ij}$) obtained from sparse linear systems

$\circ$ Little group information retained throughout the computation
$\kern20mm\star$ genuine $c_{\Gamma,i}(\lambda, \tilde\lambda)$ instead of $c_{\Gamma,i}(\lambda\tilde\lambda)$ + conventions for the polarization states.

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)

Analytic Reconstruction

The Least Common Denominator

$\circ\,$ The $r_i(\lambda,\tilde\lambda)$ belong to the field of fractions over a poly. quotient ring, $FF(R_5)$
GDL, Page ('22);  Campbell, GDL, Ellis ('22)$\phantom{;\,}$
$\displaystyle r_i(\lambda,\tilde\lambda) = \frac{\text{Num. poly}(\lambda,\tilde\lambda)}{\text{Denom. poly}(\lambda,\tilde\lambda)} = \frac{\text{Num. poly}(\lambda,\tilde\lambda)}{\prod_j W_j^{q_{ij}}(\lambda,\tilde\lambda)}$
$\circ\,$ The denominator factors $W_j$ are conjectured to be restricted to the letters of the symbol alphabet
Abreu, Dormans, Febres Cordero, Ita, Page ('18)



$\displaystyle \{W_j\} = \bigcup_{\sigma \; \in \; \text{Aut}(R_5)} \sigma \circ \big\{ \langle 12 \rangle, \langle 1|2+3|1] \big\} {\quad\color{green}\text{Identical to 1-loop!}}$
$\circ\,$ Why bother with (redundant) spinor variables:


$\star$ the LCD is not little group invariant: the degree is lower in spinors;
$\star$ no (arbitrary) split into parity even and odd: half sampling requirement;
$\star$ in LCD form we would need $\color{green}29\,059$ evaluations instead of $\color{red}117\,810$ (with $s_{ij}$) for $\mathcal{R}^{(2)}_{2q3\gamma}$ .

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)





$\circ\,$ All linear systems solved with CUDA over $\mathbb{F}_{p\leq 2^{31}-1}$ on a laptop ($t_{\text{solving}} \ll t_{\text{sampling}}$)

Taming the Algebraic Complexity

$\circ\,$ Instead of the common denominator form, perform a partial fraction decomposition
$\displaystyle r_i(\lambda,\tilde\lambda) = \frac{\mathcal{N}(\lambda,\tilde\lambda)}{\prod_j W_j^{q_{ij}}(\lambda,\tilde\lambda)} = \sum_k \frac{\mathcal{N}_k(\lambda,\tilde\lambda)}{\prod_j W_j^{q_{ijk}}(\lambda,\tilde\lambda)} = \sum_k r_{ik} \quad \text{with} \quad q_{ijk} \le q_{ij}$
$\circ\,$ Use insights from physics, e.g. no denominator in $\mathcal{R}^{(2)}_{2q3\gamma}$ contains more than a single $\langle i |j + k | i]$
$\circ\,$ As by now standard, we pick a set of independent $r_i$ to reconstruct: $r_i \not\in \text{span}(r_{j\neq i})$.
$\phantom{\circ\,}$ However, generally $r_{ik} \in \text{span}(r_{j\neq i})$ for some, but not all, $k$. Thus, write:
$\displaystyle $
$r_i = \sum_{j\neq i} c_j r_j + \sum_{k' \subset \{k\}} r_{ik'}$

Sampling requirement reduced from $\color{red}29\,059$ to $\color{green}4\,003$ points.
$\circ$ For example, a posteriori, we find that for the most complicated $r_i$, we only needed
$\displaystyle \sum_{k' \subset \{k\}} r_{ik'} = \frac{⟨13⟩[14]^2⟨24⟩⟨34⟩[45]}{⟨45⟩⟨4|1+3|4]^3}-\frac{[14]⟨25⟩⟨34⟩^2[45]}{⟨45⟩^2⟨4|1+3|4]^2}-\frac{[14]⟨24⟩⟨34⟩⟨35⟩}{⟨45⟩^3⟨4|1+3|4]}$

Towards
Phenomenology

SLC Corrections to the Hard Functions
$\circ\,$ Full-color 2-loop remainders & 1-loop amplitudes implemented in an open-source C++ Program
$\circ\,$ To estimate the impact of the subleading-color contributions, consider the 2-loop hard functions
$\displaystyle \small \mathcal{H}^{(2)} = \sum_h |\mathcal{R}_h|^2 \Big/ \sum_h |\mathcal{A}^{(0)}_h|$
About $25\%-35\%$ correction to $\mathcal{H}^{(2)}_{\text{l.c.}}$. The correction to $\sigma^{\text{NNLO}}_{q\bar q \rightarrow \gamma\gamma\gamma}$ will be much smaller.

Preview:
$pp\rightarrow Wjj$ Revisited

in collaboration with:
H. Ita, B. Page, V. Sotnikov
Bottlneck for $pp\rightarrow Wjj$ at NNLO
$\;\circ\,$ No pheno study yet, despite the amplitudes have been available for almost 2 years!
$\circ$ The algebraic complexity$-$think Ansatz size$-$grows quickly with multiplicity (m)
and mass dimension (d):

$\displaystyle \left(\mkern -9mu \begin{pmatrix}\, m(m-3)/2 \, \\ \, d/2 \, \end{pmatrix} \mkern -9mu \right)$

is a lower bound. GDL, Maître ('20)
$\circ\,$ The anlytic expressions of Abreu, Febres Cordero, Ita, Klinkert, Page, Sotnikov ('21) are 1.2GB.
Having more control on the analytic structure starts to become important!
Simplification strategy
$1.\,$ Script to split up the expressions, and compile them ($\sim 20$GB of C++) 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 $|5\rangle,[5|,|6\rangle,[6|$);
$3.\,$ Perform partial fraction decompositions$^{*}$ 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\} $

$\phantom{x}^{*}$ sometimes it's actually a bit more than a partial fraction decomposition, see next slide.
Analytic Structures of 2-loop 5-point 1-mass Amplitudes
$\circ\,$ 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.:
$$ r^{(73 \text{ of } 120)}_{\bar{u}^+g^-g^+d^-(V\rightarrow \ell^+ \ell^-)} = \frac{105}{128}\frac{⟨2|1+4|3]⟨4|2+3|1]⟨6|1+4|5]s_{14}s_{23}s_{56}{\color{green}(s_{124}-s_{134})}(s_{123}-s_{234})(s_{25}+s_{26}+s_{35}+s_{36})}{{\color{orange}⟨3|1+4|2]}{\color{red}Δ_{23|14|56}^4}} + \\ \Bigg[-6\frac{[12]^2⟨13⟩[25]⟨34⟩⟨36⟩⟨56⟩[56]{\color{green}(s_{124}-s_{134})}}{{\color{orange}⟨3|1+4|2]^5}}\Bigg] + \Bigg[ \; \Bigg]_{1234\rightarrow \overline{4321}}+ \mathcal{O}\left(\frac{1}{⟨3|1+4|2]^{4}Δ_{23|14|56}^{3}}\right)$$
$\phantom{\circ\,}$ but the pole orders have been doubled, see Bern, Dixon, Kosower ('97)
$\circ\,$ $\small Δ_{23|14|56}$ behaves as a perfect square on the surface where $\small ⟨3|1+4|2]$ vanishes: $$\small \kern-30mm \sqrt{\big\langle {\color{orange}⟨3|1+4|2]}, {\color{red}Δ_{23|14|56}} \big\rangle_{R_6}} = \big\langle {\color{orange}⟨3|1+4|2]}, {\color{green}s_{124}-s_{134}} \big\rangle_{R_6} $$
Thank you
for your attention!

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Backup Slides

Absolute Values
on the Rationals

Finite Fields

$\circ\,$ Any rational number, other than multiples of $1/p$, has an equivalent in the finite field $\mathbb{F}_p$.

$\circ\,$ For example, let's work with $p=7$, i.e. with $\mathbb{F}_7 = \{0, 1, 2, 3, 4, 5, 6\}$:

$-1$ is the additive inverse of 1
$\Rightarrow \quad -1=6$ in $\mathbb{F}_7$, because $1+6 = 7 \, \% \, 7 = 0$
$\frac{1}{3}$ is the multiplicative inverse of 3
$\Rightarrow \quad \frac{1}{3}=5$ in $\mathbb{F}_7$, because $3 \times 5 = 15 \, \% \, 7= 1$
$\phantom{\circ}\,$ The Euclidean algorithm allows to compute inverses without checking every entry.
$\circ\,$ Numbers cannot grow out of control!
$\frac{14611884945785561885978841755360860231120837652831038320107}{1853742276676202006476394341472012983521981235200}=1251868773$ in $\mathbb{F}_{2147483647}$
$\phantom{\circ}\,$ $2147483647$ is $(2^{31}-1)$ which is the largest possible value $p$ working with 32-bits.

$\boldsymbol p,$-adic Numbers

$\circ\,$ We have again a problem, in a finite field 1 is not smaller than 2. In fact:

$|x = 0|_{\mathbb{F}_p} = 0 \quad \text{and} \quad |x \neq 0|_{\mathbb{F}_p} = 1$
$\phantom{\circ}\,$ Can't easily take limits, without dividing by zero.

$\circ\,$ A $p$-adic number $x \in \mathbb{Q}_p$ is Laurent expansion in powers of the prime $p$
$x = a_{\nu_p} p^{\nu_p} + \dots + a_{-1}p^{-1} + a_{0} p^{0} + a_1 p^1 + \dots $
$\circ\,$ The $p$-adic absolute value is defined as (note the minus sign!)
$|x|_{\mathbb{Q}_p} = p^{-\nu_p} \quad \Rightarrow \quad |p|_{\mathbb{Q}_p} < |1|_{\mathbb{Q}_p} < |\frac{1}{p}|_{\mathbb{Q}_p}$
Retain integer arithmetics, while restoring the ability to take limits!

Python Packages

pyAdic

$\circ\,$ Pyadic provides flexible number types for finite fields and $p$-adic numbers in Python.
Related algorithms, such as rational reconstruction are also implemented. 
 from pyadic import ModP
 from fractions import Fraction as Q
 ModP(Q(7, 13), 2147483647)
 <<< 1817101548 % 2147483647
 # Can also go back to rationals
 from pyadic.finite_field import rationalise
 rationalise(ModP(Q(7, 13), 2147483647))
 <<< Fraction(7, 13)

Lips

$\circ\,$ Lips is a phase-space generator and manipulator for 4-dimensional kinematics in any field, $\mathbb{C}, \mathbb{F}_p, \mathbb{Q}_p, \mathbb{Q}[i]$. It is particularly useful for spinor-helicity computations. 
 from lips import Particles
 from lips.fields.field import Field
 # Random finite field phase space point, arbitrary multiplicity
 multiplicity = 5
 PSP = Particles(multiplicity, field=Field("finite field", 2 ** 31 - 1, 1), seed=0)
 # Evaluate an arbitrary complicated expression
 PSP("(8/3s23⟨24⟩[34])/(⟨15⟩⟨34⟩⟨45⟩⟨4|1+5|4])")
 <<< 683666045 % 2147483647
$\circ\,$ It can also be used to generate points in singular configuration.

Spinor Helicity

Representations of the Lorentz group

(Recall: $\mathfrak{so}(1, 3)_\mathbb{C} \sim \mathfrak{su}(2) \times \mathfrak{su}(2)$)
$(j_{-},j_{+})$ dim. name quantum field kinematic variable
(0,0) 1 scalar $h$ m
(0,1/2) 2 right-handed Weyl spinor $\chi_{R\,\alpha}$ $\lambda_\alpha$
(1/2,0) 2 left-handed Weyl spinor $\chi_L^{\,\dot\alpha}$ $\bar{\lambda}^{\dot\alpha}$
(1/2,1/2) 4 rank-two spinor/four vector $A^\mu/A^{\dot\alpha\alpha}$ $P^\mu/P^{\dot\alpha\alpha}$
(1/2,0)$\oplus$(0,1/2) 4 bispinor (Dirac spinor) $\Psi$ $u, v$

Spinor Covariants

Weyl spinors are sufficient for massless particles:

$\text{det}(P^{\dot\alpha\alpha})=m^2 \rightarrow 0 \quad \Longrightarrow \quad P^{\dot\alpha\alpha} = \bar\lambda^{\dot\alpha}\lambda^\alpha$.

In terms of 4-momentum components we have:

$$ \lambda\_\alpha=\frac{1}{\sqrt{p^0+p^3}}\begin{pmatrix}p^0+p^3 \\\ p^1+ip^2\end{pmatrix} \, , \;\;\; \lambda^\alpha=\epsilon^{\alpha\beta} \lambda_\beta =\frac{1}{\sqrt{p^0+p^3}}\begin{pmatrix}p^1+ip^2 \\\ -p^0+p^3\end{pmatrix} $$ $\bar\lambda\_{\dot\alpha}=\frac{1}{\sqrt{p^0+p^3}}\begin{pmatrix}p^0+p^3 \\\ p^1-ip^2\end{pmatrix} \, , \;\;\; \bar\lambda^{\dot\alpha}=\epsilon^{\dot\alpha\dot\beta}\bar\lambda_{\dot\beta}=\frac{1}{\sqrt{p^0+p^3}}\begin{pmatrix}p^1-ip^2 \\\ \-p^0+p^3\end{pmatrix}$
$$ \bar\lambda\_{\dot\alpha} = (\lambda\_\alpha)^* \quad if \quad p^i \in \mathbb{R}; \quad \quad \bar\lambda\_{\dot\alpha} \neq (\lambda\_\alpha)^* \quad if \quad p^i \in \mathbb{C} $$

Spinor Invariants

$$ ⟨ij⟩ = λ_iλ_j = (λ_i)^α(λ_j)_α \quad \quad \quad [ij] = \barλ_i\barλ_j = (\barλ_i)\_\dotα(\barλ_j)^\dotα $$ $$ s_{ij} = ⟨ij⟩[ji] $$ $$ ⟨i\;|\;(j+k)\;|\;l] = (λ_i)^α (\not P_j + \not P_k )\_{α\dotα} \barλ_l^\dotα $$ $$ ⟨i\;|\;(j+k)\;|\;(l+m)\;|\;n⟩ = (λ_i)^α (\not P_j + \not P_k )\_{α \dot α} (\bar{\not P_l} + \bar{\not P_m} )^{\dot α α} (λ_n)_α $$ $$ tr_5(ijkl) = tr(\gamma^5 \not P_i \not P_j \not P_k \not P_l) = [i\,|\,j\,|\,k\,|\,l\,|\,i⟩ - ⟨i\,|\,j\,|\,k\,|\,l\,|\,i] $$





The Geometry of Phase Space





based on: GDL, Page (JHEP 12 (2022) 140)

Least Common Denominator Redux

$\circ\,$ Can't draw pictures in high (complex) dimensions, so let's consider the simplified case $\mathbb{R}[x, y, z]$.
$\circ\,$ Denominator factors $W_j$ correspond to singular surfaces .

${\color{orange}W_1 = (xy^2 + y^3 - z^2)}$
Say we have:
$W_1 = xy^2 + y^3 - z^2$
A function $c_i(x,y,z)$ may or may not have $W_1$ as a pole, depending on what happens on the orange surface
$\displaystyle \lim_{W_j \rightarrow \epsilon} c_i(x,y,z) \sim \frac{1}{\epsilon^{q_{ij}}} $









The LCD tells us about what happens on surfaces with one less dimension than the full space.

Multivariate Partial Fractions

$\circ\,$ To distinguish $\displaystyle \frac{1}{W_1W_2}$ from $\displaystyle \frac{1}{W_1} + \frac{1}{W_2}$, look at $W_1 \sim W_2 \rightarrow \epsilon \ll 1$. Geometrically:

${\color{orange}W_1 = (xy^2 + y^3 - z^2)}$

${\color{blue}W_2 = (x^3 + y^3 - z^2)}$

$V(W_1) \cap V(W_2)$
$\circ\,$ Primary decompositions of sets of polynomials ( ideals ), anogous to integers:
$60 = 5 \times 3 \times 2^2$
$({\color{orange}xy^2 + y^3 - z^2}, {\color{blue}x^3 + y^3 - z^2}) = \\ {\color{magenta}(z^2,x+y)} \cup {\color{green}(y^3-z^2,x)} \cup {\color{red}(2y^3-z^2,x-y)}$
Partial-fraction decompositions tell us about the relations between poles.