aainstitutetext: bbinstitutetext: ccinstitutetext: ddinstitutetext:

# Feynman Integral Reductions by Intersection Theory with Orthogonal Bases and
Closed Formulae

Giulio Crisanti a,b,d Sid Smith<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present a prescription for choosing orthogonal bases of differential
$n$-forms belonging to quadratic twisted period integrals, with respect to the
intersection number inner product. To evaluate these inner products, we
additionally propose a new closed formula for intersection numbers beyond
$\differential\log$ forms. These findings allow us to systematically construct
orthonormal bases between twisted period integrals of this type. In the
context of Feynman integrals, this represents all diagrams at one-loop.

††preprint: MPP-2024-104

feyndiagrams

##  1 Introduction

It has recently become clear that the study and computation of twisted period
integrals GOTO_2013 ; goto2013twisted ; goto2014monodromy  is of the utmost
importance to fundamental theories in many areas of physics. Generally
speaking, these integrals can be viewed as a pairing between a differential
form (cocycle) and a cycle. The pairing is simply the integration of the form
over the contour, but with the addition of a multivalued function in the
integrand known as the twist. Integrals of this form belong to a vector space,
which is isomorphic to a twisted de Rham cohomology group. The (cohomology)
intersection number Matsumoto_1994 ; Cho_Matsumoto_1995 ; Matsumoto1998-2 ;
ojm/1200788347 ; Mimachi_2003 ; Mimachi:2004ez ; matsubaraheo2020computing ;
goto2020homology ; matsubaraheo2021algorithm ; matsubaraheo2022localization ;
mizerascatt ; Mastrolia_2019 ; Frellesvig_2019  between differential $n$-forms
is an elementary quantity that defines a pairing between forms of this group.

In the context of scattering amplitudes, Feynman integrals admit a vector
space structure Mastrolia_2019 ; Frellesvig_2019 ; PhysRevLett.123.201602 ;
Smirnov:2010hn ; Lee_2013 , and can be viewed as twisted period integrals by
transforming them to a parametric form such as Baikov or Lee-Pomeransky
representation Baikov:1996iu ; Lee_2013 . Thus, through the usage of
intersection numbers, interpreted as inner products, any Feynman integral can
be decomposed onto a master integral basis Mastrolia_2019 ;
PhysRevLett.123.201602 ; Frellesvig_2019 ; Frellesvig_2021 . Therefore,
integral family reductions and the construction of differential equations
KOTIKOV1991158 ; GhermannRemiddi2000 ; Henn2013  can be performed in a
completely complementary way to Integration by Parts (IBP) methods
Chetyrkin:1981qh ; Laporta:2000dsw .

In this formalism, reductions to master integrals are given by combinations of
intersection numbers, given by the master decomposition formula Mastrolia_2019
. Any reduction can be divided into two main parts. The first is the
computation of the so called $\mathbf{C}$-matrix, or metric, which is the
matrix of intersection numbers between a chosen basis and its dual
counterpart. The second stage is the computation of projections, namely the
intersection numbers between any target integral, and the basis on which one
wants to reduce it.

Beyond Feynman integrals, many other problems admit a twisted period integral
structure, and can be studied with intersection theory. Recently, canonical
bases for differential equations Chen_2021 ; Chen_2022 ; Chen_2024 , lattice
correlation functions Gasparotto_2023 ; Gasparotto_2023_2 , quantum mechanics
cacciatori2022intersection , cosmological correlators de2023cosmology ,
Fourier integrals brunello2023fourier , general relativity amplitudes
brunello2024improved ; frellesvig2024general , string amplitudes
Bhardwaj:2023vvm , as well as much more Mizera_2020 ; Weinzierl_2020 ;
Chestnov_2022 ; Giroux_2023 ; jiang2023recursive , have all been tackled using
intersection theory.

Intersection numbers have also been studied in their own right, with many
different strategies developed for their computation over the last few years
mizerascatt ; matsumoto2019relative ; Weinzierl_2021 ; Caron_Huot_2021 ;
Caron_Huot_2022 ; Fontana_2023 ; Chestnov_2022 ; Chestnov_2023 ;
brunello2023intersection . Recent developments include the removal of analytic
regulators in intersection computations for Feynman integrals. This was done
through the introduction of relative twisted intersection numbers
matsumoto2019relative ; Caron_Huot_2021 ; Caron_Huot_2022 , as well as a novel
independent improvement to the regularisation approach presented in
Fontana_2023 . These two strategies were shown to be connected in
brunello2023intersection .

Aside from computational algorithms, some properties of intersection numbers
are well understood, particularly for $\mathrm{d}\log$ forms. In the case of
hyperplanar arrangements ojm/1200788347 ; Mizera_2020_b , but also beyond
mizerascatt , closed formulae are known for intersection numbers of this kind.

In this paper, we explore new properties of intersection numbers, specifically
for twists originating from quadratic multivariate polynomials. Given a
Laporta basis, we provide an algorithm for choosing a dual basis that produces
a diagonal $\mathbf{C}$-matrix, for relative intersection numbers. For
$1$-form intersection numbers, such a procedure already exists Frellesvig_2019
, and similar results have been found for canonical bases Caron_Huot_2021 . In
this paper however we focus our attention on Feynman integrals in Baikov
representation. We observe that our procedure works best when the coefficients
of the quadratic polynomial are “generic”. In practice, this requirement
translates to one-loop Feynman diagrams with massive propagators. For massless
diagrams our prescription also often works, with some minor modifications.
Beyond one-loop, we also test our results for a simple two loop diagram, and
find success in producing a block diagonal $\mathbf{C}$-matrix.

Our second result is a new, closed formula for intersection numbers with
quadratic twists. The expression is remarkably simple, is the first of its
kind beyond $\differential\log$ forms, and provides evidence of deep
mathematical structures for objects of this kind. The main building blocks of
the expression are two distinct multivariate discriminants, expressed as
determinants of Hessian matrices. The appearance of repeated univariate
discriminants had previously been noticed in Caron_Huot_2021 ; Caron_Huot_2022
, and our expression connects to this but considers a different class of
intersection numbers.

Combined with the diagonal prescription above, this result makes the
computation of the $\mathbf{C}$-matrix extremely efficient for one-loop
Feynman integrals, by requiring no explicit algorithmic computations of
intersection numbers. Interpreted differently, these two findings combined
allow us to construct not just diagonal, but identity $\mathbf{C}$-matrices.

The paper is structured as follows: In Section 2 we will give a brief review
of intersection theory, explaining how intersection numbers are computed, and
how they can be used to compute integral reductions. We also review relative
intersection numbers, as well as outline how one-loop Feynman Integrals can be
seen as twisted integrals from the Baikov representation. In Section 3 we
describe the algorithm to choose orthogonal bases of differential forms, as
well as provide a few initial examples of its usage. In Section 4 we present
the closed form formula for intersection numbers, and show how this can be
applied to compute every element in a diagonal $\mathbf{C}$-matrix. In Section
5 we provide an extensive list of examples where we verify the results of
Sections 3 and 4 to be working correctly. Finally, we provide concluding
remarks and discuss possible extensions to this work in Section 6.

During our research the following software was used: Mathematica, LiteRed
lee2012presenting ; Lee_2014 , FiniteFlow Peraro_2019 , JaxoDraw Binosi_2009 ;
VERMASEREN199445 .

##  2 Review of Intersection Theory

The central object of study of this work is the computation of intersection
numbers for twisted integrals, with the ultimate goal of integral reductions
onto bases. To this end in this section we provide a review of twisted
integrals, intersection theory, and how intersection numbers can be used to
compute IBP-like reductions for Feynman integrals.

###  2.1 Twisted Cohomology

We are concerned with twisted period integrals of the form

$I=\int_{C_{R}}u(\mathbf{z})\
\varphi_{L}(\mathbf{z})=:\bra{\varphi_{L}}C_{R}]\,.$ (1)

In this expression, $u(\mathbf{z})$ is a multi-valued function called the
twist, which takes the form

$u(\mathbf{z})=\prod_{i}b(\mathbf{z})_{i}^{\gamma_{i}}\,,$ (2)

where each $b_{i}(\mathbf{z})$ is a polynomial in $n$ variables
$\mathbf{z}=\\{z_{1}\cdots z_{n}\\}$, and $\gamma_{i}$ is a generic exponent.
$\varphi_{L}$ is a meromorphic $n$-form
$\varphi_{L}(\mathbf{z})=\hat{\varphi}_{L}(\mathbf{z})\differential^{n}\mathbf{z}\equiv\hat{\varphi}_{L}(\mathbf{z})\differential
z_{1}\wedge\dots\wedge\differential z_{n}$. We impose that the integration
contour $C_{R}$ is defined such that $u$ vanishes on the boundary $u(\partial
C_{R})=0$. This gives rise to integration-by-parts (IBP) identities of the
form

$0=\int_{C_{R}}\differential(u\,\xi_{L})=\int_{C_{R}}u\,\gradient_{\omega}\,\xi_{L}\,,$
(3)

where $\xi_{L}$ is some arbitrary differential $(n-1)$-form, and we define the
covariant derivative $\gradient_{\omega}$ as

$\omega:=\differential\log{u}=\sum_{i=1}^{n}\hat{\omega}_{i}dz_{i}\,,\quad\hat{\omega}_{i}=\partialderivative{\log{u}}{z_{i}}\,,\quad\gradient_{\omega}=\differential+\omega\wedge\,.$
(4)

This allows us to define an equivalence relation between differential
$n$-forms inside twisted integrals
$\bra{\varphi_{L}}\sim\bra{\varphi_{L}+\gradient_{\omega}\xi_{L}}$. The
twisted cohomology group $H_{\omega}^{n}$ is defined as the vector space of
closed $n$-forms modulo this equivalence relation. This space identifies
integrands that give the same $I$, upon integration with $C_{R}$. This vector
space has finite dimension $\nu:=\text{dim}H_{\omega}^{n}$. We denote elements
of this group $\bra{\varphi_{L}}\in H_{\omega}^{n}$.

There additionally exists a dual vector space, composed of dual integrals,
defined as

$\tilde{I}=\int_{C_{L}}u^{-1}\varphi_{R}=[C_{L}\ket{\varphi_{R}}\,.$ (5)

Its respective dual twisted cohomologoy group is given by $H_{-\omega}^{n}$,
and we denote elements of this group by $\ket{\varphi_{R}}\in
H_{-\omega}^{n}$.

The dimension $\nu$ of the twisted cohomology group counts the number of
independent integrals that can be built from $n$-forms satisfying the
conditions given above. For the case of Feynman integrals, this corresponds to
the number of MIs. The dimension of the group is given directly by Lee_2013 ;
Frellesvig_2021

$\nu=\text{dim}\ H_{\pm\omega}^{n}=\\{\\#\text{ of solutions to
}\omega=0\\}\,.$ (6)

###  2.2 Basis of Master Integrals

Given a twist $u$ one can formally define a set of bases that span the (dual)
cohomology group

$\bra{e_{i}}\in H_{\omega}^{n}\,,\qquad\ket{h_{i}}\in H_{-\omega}^{n}\,,\qquad
i=1,\dots,\nu\,.$ (7)

Any arbitrary $n$-form $\bra{\varphi_{L}}$ can then be decomposed in terms of
this basis schematically as

$\bra{\varphi_{L}}=\sum_{i=1}^{\nu}c_{i}\bra{e_{i}}\,.$ (8)

This is equivalent to decomposing our original integral
$I=\bra{\varphi_{L}}C_{R}]$ in terms of a set of Master Integrals
$J_{i}=\bra{e_{i}}C_{R}]$:

$I=\bra{\varphi_{L}}C_{R}]=\sum_{i=1}^{\nu}c_{i}\bra{e_{i}}C_{R}]=\sum_{i=1}^{\nu}c_{i}J_{i}\,.$
(9)

The coefficients $c_{i}$ of this decomposition can be found by using the
master decomposition formula Mastrolia_2019

$c_{i}=\sum_{j=1}^{\nu}\innerproduct{\varphi_{L}}{h_{j}}(\mathbf{C}^{-1})_{ji}\,,$
(10)

where we define

$\mathbf{C}_{ij}=\innerproduct{e_{i}}{h_{j}}\,,$ (11)

which will be denoted as the $\mathbf{C}$-matrix, or metric in the rest of
this work. Equation (10) assumes that there exists an inner product between
forms and dual forms, $\innerproduct{\bullet}{\bullet}$. In the context of
twisted period integrals, this object exists, is well defined, and is denoted
as the intersection number Cho_Matsumoto_1995 ; Matsumoto1998-2 ; Mimachi_2003
; Mimachi:2004ez ; ojm/1200788347 ; matsubaraheo2020computing ;
goto2020homology ; matsubaraheo2021algorithm ; matsubaraheo2022localization ;
mizerascatt ; Mastrolia_2019 ; Frellesvig_2019 .

###  2.3 Intersection Numbers

The intersection number is an inner product between $n$-forms of a twisted
cohomology group and its dual counterpart. Formally, it is defined as an
integral over these two forms:

$\innerproduct{\varphi_{L}}{\varphi_{R}}:=\frac{1}{2\pi
i}\int_{X}\iota(\varphi_{L})\wedge\varphi_{R}\,.$ (12)

Here $\iota$ defines a regularisation operator, which is necessary for the
computation to be well defined.

####  2.3.1 Univariate Intersection Numbers

Specifying for the moment to the case $n=1$, $\iota$ is defined as

$\iota(\varphi_{L}):=\varphi_{L}-\gradient_{\omega}(h\psi_{L})\,.$ (13)

Here, $h$ is a Heaviside function

$h:=\sum_{p\in\mathcal{P}({\omega})}\left(\theta(\epsilon-\absolutevalue{z-z_{i}})\right),\quad\mathcal{P}({\omega}):=\text{poles
of $\omega$}\,,$ (14)

and the function $\psi_{L}$ is the solution to the differential equation:

$\gradient_{\omega}\psi_{L}=\varphi_{L}\,.$ (15)

The space of integration is
$X=\mathbb{C}\mathrm{P}\setminus\mathcal{P}({\omega})$. After some
algebra111See ojm/1200788347 ; mizerascatt ; Mizera_2020_b ;
Gasparotto:2023cdl ; weinzierl2022feynman  for derivations., the univariate
intersection number localises to a straightforward residue computation:

$\innerproduct{\varphi_{L}}{\varphi_{R}}=\sum_{p\in\mathcal{P}({\omega})}\text{Res}_{z=p}(\psi_{L}\varphi_{R})\,.$
(16)

For this procedure to be well defined, it is essential that the poles in
$\varphi_{L}$ and $\varphi_{R}$ are regulated by $u$, which we define to be
the condition

$\mathcal{P}(\varphi_{L/R})\in\mathcal{P}({\omega})\,.$ (17)

Many cases of physical interest involve twisted integrals that are not
regulated. In Section (2.4) we discuss the relative cohomology approach to
overcoming this problem.

####  2.3.2 Multivariate intersection numbers

For the case $n>1$, there are several strategies one can adopt for
intersection computations Frellesvig_2021 ; PhysRevLett.123.201602 ;
Chestnov_2023 ; ojm/1200788347 ; Frellesvig_2019 ; Chestnov_2022 . For this
work, we will use the recursive or fibration based approach, which we review
now. For more details on this algorithm, see PhysRevLett.123.201602 ;
Mizera_2020_b ; Frellesvig_2021  and Gasparotto:2023cdl ; weinzierl2022feynman
for reviews.

The recursive algorithm builds $n$-form intersection numbers one variable at a
time: for concreteness, we order them as $z_{n},\dots,z_{1}$. We can consider
the $m$-th “layer” of the iterative process as treating the variables
$z_{n},\dots,z_{m+1}$ as “external”, and the variables $z_{m},\dots,z_{1}$ as
“internal” integration variables. The $m$-forms generated by these internal
variables belong to their own twisted cohomology group and thus also form a
vector space. For all $1\leq m\leq n$, we denote the dimension of this inner
space as $\nu_{m}$, and a basis of (dual) forms as
$\bra*{e_{i}^{(\mathbf{m})}}$ and $\ket*{h_{i}^{(\mathbf{m})}}$
respectively222A systematic algorithm for choosing a basis of forms and duals
is presented in Section 3.1..

Let us assume all $(m-1)$-form intersection numbers are known. The strategy is
then to calculate any $m$-form intersection number by projecting it onto
$(m-1)$-form intersection numbers, and computing the remaining $1$-form
intersections using a univariate algorithm.

Given an intersection number
$\innerproduct*{\varphi_{L}^{(\mathbf{m})}}{\varphi_{R}^{(\mathbf{m})}}$, we
start by first projecting in terms of our $(m-1)$-form basis

$\bra*{\varphi_{L}^{(\mathbf{m})}}=\bra*{\phi_{L,i}^{(m)}}\wedge\bra*{e_{i}^{(\mathbf{m-1})}},\quad\ket*{\varphi_{R}^{(\mathbf{m})}}=\ket*{\phi_{R,i}^{(m)}}\wedge\ket*{h_{i}^{(\mathbf{m-1})}}\,,$
(18)

where $\bra*{\phi_{L,i}^{(m)}}$ and $\ket*{\phi_{R,i}^{(m)}}$ are $1$-forms in
the variable $z_{m}$. These projections can be calculated using the master
decomposition formula (10)

$\bra*{\phi_{L,i}^{(m)}}=\sum_{j=1}^{\nu_{m-1}}\innerproduct*{\varphi_{L}^{(\mathbf{m})}}{h_{j}^{(\mathbf{m-1})}}(\mathbf{C}^{(m-1)})^{-1}_{ji},\quad\ket*{\phi_{R,i}^{(m)}}=\sum_{j=1}^{\nu_{m-1}}(\mathbf{C}^{(m-1)})^{-1}_{ij}\innerproduct*{e_{j}^{(\mathbf{m-1})}}{\varphi_{R}^{(\mathbf{m})}}\,,$
(19)

where
$(\mathbf{C}^{(m)})_{ij}=\innerproduct*{e_{i}^{(\mathbf{m})}}{h_{j}^{(\mathbf{m})}}$
represents the $\mathbf{C}$-matrix at the $m^{th}$ layer of the iterative
process. The $m$-form intersection number then reduces to

$\innerproduct*{\varphi_{L}^{(\mathbf{m})}}{\varphi_{R}^{(\mathbf{m})}}=\sum_{p\in\mathcal{P}_{m}}\text{Res}_{z_{m}=p}\left(\psi_{L,i}(\mathbf{C}^{(m-1)})_{ij}\phi_{R,j}^{(m)}\right)\,.$
(20)

In direct analogy with the univariate case, $\psi_{L,i}$ is the solution to
the differential equation

$\differential\psi_{L,i}+\psi_{L,j}(\mathbf{\Omega}^{(m)})_{ji}=\phi_{L,i}^{(m)}\,.$
(21)

In this expression, $\mathbf{\Omega}^{(m)}$ is known as the connection matrix,
and is given by the differential equation matrix for the basis
$\bra*{e_{i}^{(\mathbf{m-1})}}$ in the variable $z_{m}$:

$\bra*{\gradient_{\omega_{m}}e_{i}^{(\mathbf{m-1})}}:=\partial_{z_{m}}\bra*{e_{i}^{(\mathbf{m-1})}}=(\mathbf{\Omega}^{(m)})_{ij}\bra*{e_{j}^{(\mathbf{m-1})}}\,,$
(22)

Using the master decomposition formula (10), $(\mathbf{\Omega}^{(m)})$ can be
calculated as

$\displaystyle(\mathbf{\Omega}^{(m)})_{ij}=\sum_{k=1}^{\nu_{m-1}}\innerproduct*{\gradient_{\omega_{m}}e_{i}^{(\mathbf{m-1})}}{h_{k}^{(\mathbf{m-1})}}(\mathbf{C}^{(m-1)})^{-1}_{kj}\,.$
(23)

In the above expressions, in analogy to the univariate case, $\mathcal{P}_{m}$
is defined as the set of poles of the $\mathbf{\Omega}^{(m)}$ matrix. For all
computations in this work, all intersection numbers were verified using this
algorithm, as well as further techniques and ideas presented in
brunello2023intersection .

###  2.4 Relative Twisted Cohomology

In Baikov representation, Feynman integrals are represented as twisted
integrals (see Section 2.5), but they do not satisfy the condition (17),
namely that they are regulated integrals. This is because the differential
forms have the following structure

$\varphi_{L}=\frac{\mathrm{d}z_{1}\wedge\cdots\wedge\mathrm{d}z_{n}}{z_{1}^{a_{1}}\cdots
z_{n}^{a_{n}}}\,,$ (24)

but the Baikov polynomial $b$ is such that $b(\mathbf{z}=0)\neq 0$, and thus
$\mathbf{0}\notin\mathcal{P}({\omega})$.

An appropriate mathematical framework to deal with non regulated intersection
numbers is Relative Twisted Cohomology matsumoto2019relative ; Caron_Huot_2021
; Caron_Huot_2022 ; brunello2023intersection . In this formalism, the
contribution of the unregulated poles is captured through the usage of new
dual forms $\delta$. Roughly speaking, these “delta forms” act as residue
operators, “integrating out” variables from the intersection number. For
$1$-form intersection numbers, they are defined to operate as

$\innerproduct*{{\varphi}_{L}}{{\delta}_{z}}=\innerproduct*{\hat{\varphi}_{L}\mathrm{d}z}{{\delta}_{z}}:=\mathrm{Res}_{z=0}\left(\frac{u(z)}{u(0)}\hat{\varphi}_{L}\right)\,,$
(25)

where we have used the notation $f=\hat{f}\,\mathrm{d}z$. As shown in
brunello2023intersection , it is also possible to consider generalisations of
the delta forms involving their derivatives. For 1-forms we have

$\innerproduct*{\varphi_{L}}{\partial^{(k)}_{z}\delta_{z}}:=\frac{1}{k!}\,\Res_{z=0}\left(u(z)\,\varphi_{L}(z)\left(\partial_{z}^{(k)}\left.\frac{1}{u(z)}\right)\right\rvert_{z=0}\right)\,.$
(26)

For $n$-form intersection numbers, suppose

$\displaystyle\bra{\varphi_{L}}$
$\displaystyle=\hat{\varphi}\,\mathrm{d}z_{1}\wedge\cdots\wedge\mathrm{d}z_{n}\,,$
(27) $\displaystyle\ket{\varphi_{R}}$
$\displaystyle=\hat{\phi}\,\ket{\delta_{z_{1}}}\wedge\dots\wedge\ket{\delta_{z_{a}}}\wedge\,\mathrm{d}z_{a+1}\wedge\cdots\wedge\mathrm{d}z_{n}\,,$

where $\hat{\phi}$ is any rational function in the variables $\mathbf{z}$. We
then have

$\innerproduct{\varphi_{L}}{\varphi_{R}}:=\innerproduct{\mathrm{Res}_{z_{1}\cdots
z_{a}=0}\left(\frac{u\,\hat{\varphi}\,dz_{1}\wedge
dz_{n}}{u\rvert_{z_{1}\cdots z_{a}=0}}\right)}{\hat{\phi}\rvert_{z_{1}\cdots
z_{a}=0}\,\mathrm{d}z_{a+1}\wedge\cdots\wedge\mathrm{d}z_{n}}\,.$ (28)

In the above expression, the multivariate residue on the variables
$z_{1}\cdots z_{a}$ evaluates to a rational function. The remaining
$(n-a)$-form intersection number is then computed using the recursive
intersection algorithm for multivariate forms introduced above. For the rest
of this work we will interchangeably use the notations
$\ket{\delta_{z_{1}}}\wedge\dots\wedge\ket{\delta_{z_{n}}}=\ket{\delta_{z_{1}\cdots
z_{n}}}$.

###  2.5 One-loop Feynman Integrals as Twisted Integrals

Any arbitrary one-loop Feynman Integral, with $N$ external legs of momentum
$p_{i}$, can be written as

$I(n_{1},\dots,n_{E+1})=\int\frac{\differential^{d}k}{(2\pi)^{d/2}}\frac{1}{D_{1}^{n_{1}}\dots
D_{N}^{n_{N}}}\,.$ (29)

We can view this integral as a twisted period by changing to Baikov
representation Baikov:1996iu ; Baikov_1997 , where it takes the form

$I\propto\int_{C_{R}}u(\mathbf{z})\,\varphi_{L}\,.$ (30)

The twist is given by

$u(\mathbf{z})=b(\mathbf{z})^{\gamma},\quad
b(\mathbf{z})=G(k,p_{1},\dots,p_{N-1})\evaluated{}_{D_{i}\rightarrow
z_{i}},\quad\gamma=\frac{d-N-1}{2}\,,$ (31)

where $G(\\{x_{i}\\})=\det[x_{i}\cdot x_{j}]$ is the Gram determinant of the
loop and external momenta $\\{x_{i}\\}$, and $\varphi_{L}$ is a differential
form given by

$\varphi_{L}=\frac{d^{N}\mathbf{z}}{z_{1}^{n_{1}}\dots z_{N}^{n_{N}}}\,.$ (32)

Due to the form of $G(\\{x_{i}\\})$, the Baikov polynomial $b(\mathbf{z})$ is
always quadratic in all $\mathbf{z}$ variables. Thus, all one-loop Feynman
integrals will be associated to quadratic twists. This property allows for the
results presented in Sections 3 and 4 to be immediately applied to one-loop
Feynman integrals.

##  3 Diagonal Basis Prescription

Our new prescription to generate diagonal $\mathbf{C}$-matrices for many one-
loop Feynman Integrals is a simple extension to the algorithm presented in
(brunello2023intersection, , Section 4.4). To this end we review this
algorithm first, before building upon it in Section 3.2. In Section 3.3 we
showcase the algorithm on three distinct examples in detail. Beyond these,
many more examples can be found in Section 5.

###  3.1 Standard Basis Generation Prescription

Our starting point is a twist, obtained from the Baikov polynomials
$b_{i}(\mathbf{z})$ of a Feynman integral family, and a set of $n$ propagators
$\mathbf{z}=\\{z_{1}\cdots z_{n}\\}$. Our goal is to find a valid basis of MIs
for this integral family. In the language of relative cohomology intersection
numbers, this condition translates to finding a basis such that the
$\mathbf{C}$-matrix is invertible.

To this end, consider a (sub)-sector $\mathcal{S}$ of our integral family,
which we represent as a set containing the propagators present in the sector

$\mathcal{S}\subseteq\\{z_{1}\cdots z_{n}\\}\ .$ (33)

To count how many master integrals are inside $\mathcal{S}$, we construct the
“regulated” twist and its respective $\omega$ form

$u(\mathbf{z})=b(\mathbf{z})^{\gamma}\,,\qquad
u_{\rho}(\mathbf{z})=\left(\prod_{z_{i}\in\mathcal{S}}z_{i}^{\rho_{i}}\right)u(\mathbf{z})\,,\qquad\omega_{\rho}(\mathbf{z})=\mathrm{d}\log(u_{\rho}(\mathbf{z}))\,.$
(34)

The number of solutions to the equation $\omega_{\rho}(\mathbf{z})=0$, denoted
as $\nu_{\mathcal{S}}$, is the number of master integrals present in the
sector $\mathcal{S}$ Lee_2013 ; Frellesvig_2021 ; brunello2023intersection .
One repeats this procedure for all sectors and subsectors to obtain a full
basis with the correct number of master integrals in each sector. The end of
this procedure is a basis $\tilde{e}$. We construct its dual basis counterpart
by replacing inverse powers of the variables $\\{z_{1}\cdots z_{n}\\}$ with
delta forms, namely

$\tilde{h}=\tilde{e}|_{(z_{i}z_{j}\cdots)^{-1}\rightarrow\
\delta_{z_{i}z_{j}\cdots}}\,.$ (35)

For an example of this algorithm in practice, see (brunello2023intersection, ,
Appendix B). It is worth noting that these choices of $\tilde{e}$ and
$\tilde{h}$ already produce a block triangular $\mathbf{C}$-matrix. This is
due to the multivariate residue operation vanishing in eq. (28) if there are
more residues of variables than poles.

###  3.2 Diagonal Basis Generation Prescription

We now turn to the new part of this work, namely a prescription for taking
$\tilde{e}$ and $\tilde{h}$ and turning them into a new basis $e$ and $h$
which produces a diagonal $\mathbf{C}$-matrix. The process is a very simple
extension of the algorithm presented above, and is valid for $b(\mathbf{z})$
quadratic.

For any element in the dual basis $\tilde{h}_{j}$, denote by $m_{j}$ the
number of delta forms appearing in the respective element. We then define new
bases $e$, $h$ as

$e_{j}=\tilde{e}_{j}\,,\qquad
h_{j}=\frac{\tilde{h}_{j}}{b(\mathbf{z})^{n-m_{j}}}\,.$ (36)

In Section 3.4, we prove this new basis choice produces diagonal
$\mathbf{C}$-matrices for $1$-form intersection numbers. Beyond $1$-forms, we
conjecture that the bases $e$ and $h$ will always provide a diagonal basis for
one-loop Feynman integrals with massive internal propagators and external legs
respectively. In practice, we have verified this statement to be true for
multiple configurations up to 6 point kinematics. For one-loop Feynman
integrals with massless propagators and legs, we find that often the above
prescription still works, but occasionally it fails to provide a diagonal
$\mathbf{C}$-matrix. In these cases, we have observed that small changes to
the bases generated by the prescription above can in many cases be implemented
to once again yield diagonal $\mathbf{C}$-matrices. An example of such a
situation is given in Section 3.3.3 and further examples are shown in Section
5. Eq. (36) is the first new result of this work, representing a simple
prescription for diagonal bases beyond $1$-form intersection numbers.

In Caron_Huot_2021 , a similar algorithm was used to derive a basis of dual
forms orthogonal to a canonical (uniform transcendental) basis of one-loop
Feynman integrals. In this approach, the physical basis of Feynman integrals
consists of pairs of integrals in varying space-time dimensions. We expect
these results to be deeply connected, and further comparisons between these
approaches are left to future works.

###  3.3 First Examples

####  3.3.1 One-loop Box for Bhabha Scattering on a Double Cut

We illustrate the prescription of Section 3.2 for creating diagonal bases
through a simple example. To this end, we return to the computation of
(brunello2023intersection, , Section 5.1), namely a one-loop two mass box on
the double cut.

Figure 1: One-loop Massive Box with two masses.

The propagators are defined as

$z_{1}=\ell^{2}-m^{2},\quad z_{2}=(\ell-p_{1})^{2},\quad z_{3}=(\ell-
p_{1}-p_{2})^{2}-m^{2},\quad z_{4}=(\ell-p_{1}-p_{2}-p_{3})^{2},$ (37)

with loop momenta $\ell$, and kinematics

$p_{i}^{2}=m^{2},\quad s=(p_{1}+p_{2})^{2},\quad t=(p_{2}+p_{3})^{2},\quad
s+t+(p_{1}+p_{3})^{2}=4m^{2}\>.$ (38)

Upon cutting the massless propagators $z_{2}$ and $z_{4}$, the twist reads up
to a constant prefactor:

$u=b(\mathbf{z})^{(d-5)/2},\quad
b(\mathbf{z})=4m^{2}\left(st+\left(z_{1}-z_{3}\right){}^{2}\right)-s^{2}t+2st\left(z_{1}+z_{3}\right)+4sz_{1}z_{3}-t\left(z_{1}-z_{3}\right){}^{2}.$
(39)

Using the algorithm outlined in Section 3.1, the internal and outer bases for
the intersection numbers are given as:

$\begin{split}\tilde{e}^{(3)}=\left\\{1\,,\;\frac{1}{z_{3}}\right\\},\qquad\tilde{e}=\left\\{1\,,\;\frac{1}{z_{1}}\,,\;\frac{1}{z_{3}}\,,\;\frac{1}{z_{1}z_{3}}\right\\}\>,\end{split}$
(40)

with dual bases

$\begin{split}\tilde{h}^{(3)}=\\{1,\delta_{3}\\},\qquad\tilde{h}=\\{1,\delta_{1},\delta_{3},\delta_{13}\\}\>.\end{split}$
(41)

To aid comparisons with the new prescription, we compute the relevant
$\mathbf{C}$-matrices
$\mathbf{\tilde{C}}^{(3)}=\innerproduct*{\tilde{e}^{(3)}}{\tilde{h}^{(3)}}$
and $\mathbf{\tilde{C}}=\innerproduct*{\tilde{e}}{\tilde{h}}$ for this basis
choice. We have:

$\mathbf{\tilde{C}}^{(3)}=\left(\begin{array}[]{cc}\frac{4(d-5)s\left(-4m^{2}+s+t\right)\left(m^{2}t+z_{1}\left(t+z_{1}\right)\right)}{(d-6)(d-4)\left(t-4m^{2}\right)^{2}}&0\\\
-\frac{z_{1}\left(-4m^{2}+2s+t\right)+st}{(d-6)\left(4m^{2}-t\right)}&1\\\
\end{array}\right)\ ,$ (42)

and

$\mathbf{\tilde{C}}=\left(\begin{array}[]{cccc}-\frac{st^{2}\left(-4m^{2}+s+t\right)}{4(d-7)(d-3)}&0&0&0\\\
\frac{st^{2}\left((44-8d)m^{2}+(d-6)t\right)\left(-4m^{2}+s+t\right)}{2(d-7)(d-6)(d-4)\left(t-4m^{2}\right)^{2}}&-\frac{4(d-5)m^{2}st\left(4m^{2}-s-t\right)}{(d-6)(d-4)\left(t-4m^{2}\right)^{2}}&0&0\\\
\frac{st^{2}\left((44-8d)m^{2}+(d-6)t\right)\left(-4m^{2}+s+t\right)}{2(d-7)(d-6)(d-4)\left(t-4m^{2}\right)^{2}}&0&-\frac{4(d-5)m^{2}st\left(4m^{2}-s-t\right)}{(d-6)(d-4)\left(t-4m^{2}\right)^{2}}&0\\\
\frac{st^{2}}{(d-7)(d-6)\left(4m^{2}-t\right)}&\frac{st}{(d-6)\left(t-4m^{2}\right)}&\frac{st}{(d-6)\left(t-4m^{2}\right)}&1\\\
\end{array}\right)\ .$ (43)

We now return to the bases and apply the procedure of Section 3.2 to obtain

$\begin{split}h^{(3)}=\left\\{\frac{1}{b},\delta_{3}\right\\},\qquad
h=\left\\{\frac{1}{b^{2}},\frac{\delta_{1}}{b},\frac{\delta_{3}}{b},\delta_{13}\right\\}\>\,,\end{split}$
(44)

and $e^{(3)}=\tilde{e}^{(3)}\,$, $e=\tilde{e}\,$. With this new choice the
$\mathbf{C}$-matrices now read

$\mathbf{C}^{(3)}=\left(\begin{array}[]{cc}\frac{1}{(d-4)t\left(4m^{2}-t\right)}&0\\\
0&1\\\ \end{array}\right)\,,$ (45)

and

$\mathbf{C}=\left(\begin{array}[]{cccc}-\frac{1}{4(d-3)^{2}st^{2}\left(-4m^{2}+s+t\right)}&0&0&0\\\
0&\frac{1}{(d-4)t\left(4m^{2}-t\right)}&0&0\\\
0&0&\frac{1}{(d-4)t\left(4m^{2}-t\right)}&0\\\ 0&0&0&1\\\
\end{array}\right)\,.$ (46)

Several comments are now in order. Firstly, with the new choices of $h^{(3)}$
and $h$ the $\mathbf{C}$-matrices $\mathbf{C}^{(3)}$ and $\mathbf{C}$ are now,
as anticipated, diagonalised. Remarkably, $\mathbf{C}^{(3)}$ is not only
diagonal, but is also independent of the external variable $z_{1}$, (cf eq.
(42)). Finally, the individual intersection numbers, both in the internal and
external $\mathbf{C}$-matrices, are greatly simplified compared to the basis
$\tilde{e}$ and $\tilde{h}$. These properties seem to be rather general and we
have found them to hold for all one-loop examples where the inner and outer
matrices have been successfully diagonalised.

####  3.3.2 One-loop Massive Triangle

As a second example, let us consider a one-loop triangle with massive
propagators and external legs shown in Figure 2.

Figure 2: One-loop Massive Triangle with external momenta $p_{1},p_{2},p_{3}$
and propagators $z_{1},z_{2},z_{3}$.

We define the propagators as

$z_{1}=l^{2}-m_{1}^{2}\,,\qquad
z_{2}=\left(l+p_{1}\right){}^{2}-m_{1}^{2}\,,\qquad
z_{3}=\left(l+p_{1}+p_{2}\right){}^{2}-m_{1}^{2}\,,$ (47)

with kinematics

$p_{i}^{2}=m_{2}^{2}\,,\qquad(p_{1}+p_{2})^{2}=m_{2}^{2}\,.$ (48)

The twist reads

$u=b(\mathbf{z})^{\frac{d-4}{2}}\,,\quad
b(\mathbf{z})=m_{2}^{2}\left(z_{1}+z_{2}+z_{3}\right)-m_{2}^{4}+3m_{1}^{2}m_{2}^{2}-z_{1}^{2}-z_{2}^{2}-z_{3}^{2}+z_{1}z_{2}+\left(z_{1}+z_{2}\right)z_{3}\,.$
(49)

We pick the variable ordering $\left\\{z_{3}\,,z_{2}\,,z_{1}\right\\}$ and,
using the algorithm described in Section 3.1 pick bases

$\displaystyle\tilde{e}^{(1)}$
$\displaystyle=\left\\{1,\frac{1}{z_{1}}\right\\}\,,\qquad\tilde{e}^{(21)}=\left\\{1,\frac{1}{z_{1}},\frac{1}{z_{2}},\frac{1}{z_{1}z_{2}}\right\\}\,,\qquad\tilde{e}=\left\\{\frac{1}{z_{1}},\frac{1}{z_{2}},\frac{1}{z_{3}},\frac{1}{z_{1}z_{2}},\frac{1}{z_{1}z_{3}},\frac{1}{z_{2}z_{3}},\frac{1}{z_{1}z_{2}z_{3}}\right\\}\,,\qquad$
(50) $\displaystyle\tilde{h}^{(1)}$
$\displaystyle=\left\\{1,\delta_{1}\right\\}\,,\qquad\,\,\,\,\tilde{h}^{(21)}=\left\\{1,\delta_{1},\delta_{2},\delta_{12}\right\\}\,,\qquad\quad\,\,\,\,\tilde{h}=\left\\{\delta_{1},\delta_{2},\delta_{3},\delta_{12},\delta_{13},\delta_{23},\delta_{123}\right\\}\,.$

Following the diagonal basis prescription we generate the new basis

$\displaystyle{e}^{(1)}$
$\displaystyle=\left\\{1,\frac{1}{z_{1}}\right\\}\,,\qquad{e}^{(21)}=\left\\{1,\frac{1}{z_{1}},\frac{1}{z_{2}},\frac{1}{z_{1}z_{2}}\right\\}\,,\qquad{e}=\left\\{\frac{1}{z_{1}},\frac{1}{z_{2}},\frac{1}{z_{3}},\frac{1}{z_{1}z_{2}},\frac{1}{z_{1}z_{3}},\frac{1}{z_{2}z_{3}},\frac{1}{z_{1}z_{2}z_{3}}\right\\}\,,\qquad$
(51) $\displaystyle{h}^{(1)}$
$\displaystyle=\left\\{\frac{1}{b},\delta_{1}\right\\}\,,\quad\,\,\,\,\,{h}^{(21)}=\left\\{\frac{1}{b^{2}},\frac{\delta_{1}}{b},\frac{\delta_{2}}{b},\delta_{12}\right\\}\,,\quad\,\,\,\,\,\,\,{h}=\left\\{\frac{\delta_{1}}{b^{2}},\frac{\delta_{2}}{b^{2}},\frac{\delta_{3}}{b^{2}},\frac{\delta_{12}}{b},\frac{\delta_{13}}{b},\frac{\delta_{23}}{b},\delta_{123}\right\\}\,.$

With this basis, the $\mathbf{C}$-matrices read

$\displaystyle\mathbf{C}^{(1)}$
$\displaystyle=\left(\begin{array}[]{cc}\frac{1}{(d-3)m_{2}^{2}}&0\\\ 0&1\\\
\end{array}\right)\,,$ (52) $\displaystyle\mathbf{C}^{(21)}$
$\displaystyle=\left(\begin{array}[]{cccc}\frac{4}{3(d-2)^{2}m_{2}^{4}}&0&0&0\\\
0&\frac{1}{(d-3)m_{2}^{2}}&0&0\\\ 0&0&\frac{1}{(d-3)m_{2}^{2}}&0\\\ 0&0&0&1\\\
\end{array}\right)\,,$ $\displaystyle\mathbf{C}$
$\displaystyle=\left(\begin{array}[]{ccccccc}\frac{4}{3(d-2)^{2}m_{2}^{4}}&0&0&0&0&0&0\\\
0&\frac{4}{3(d-2)^{2}m_{2}^{4}}&0&0&0&0&0\\\
0&0&\frac{4}{3(d-2)^{2}m_{2}^{4}}&0&0&0&0\\\
0&0&0&\frac{1}{(d-3)m_{2}^{2}}&0&0&0\\\
0&0&0&0&\frac{1}{(d-3)m_{2}^{2}}&0&0\\\
0&0&0&0&0&\frac{1}{(d-3)m_{2}^{2}}&0\\\ 0&0&0&0&0&0&1\\\
\end{array}\right)\,.$

Once again, each $\mathbf{C}$-matrix for each layer is diagonalised, and the
internal $\mathbf{C}$-matrices do not depend on the outer variables. The
intersection numbers themselves also evaluate to very simple functions.

####  3.3.3 One-loop Box for Bhabha Scattering on a Single Cut

To illustrate a case where massless propagators require the diagonal basis
prescription algorithm to be slightly modified, we return to the example of
Section 3.3.1, but this time consider a different cut, $z_{1}=0$. By choosing
this cut, we have two massless propagators, $z_{2}$ and $z_{4}$. The twist
reads

$\begin{split}u&=b(\mathbf{z})^{\frac{d-5}{2}}\,,\\\
b(\mathbf{z})&=s^{2}z_{4}^{2}+\left(t\left(z_{3}-s\right)+sz_{2}\right){}^{2}-2sz_{4}\left(z_{2}(s+2t)+t\left(s-z_{3}\right)\right)\\\
&\quad-4m^{2}\left(s\left(-2z_{2}\left(t+z_{4}\right)+\left(t-z_{4}\right){}^{2}+z_{2}^{2}\right)+tz_{3}^{2}\right)\,.\end{split}$
(53)

We pick the variable ordering $\left\\{z_{4}\,,z_{3}\,,z_{2}\right\\}$ and
using the algorithm in Section 3.1 pick bases

$\displaystyle\tilde{e}^{(2)}$
$\displaystyle=\left\\{1,\frac{1}{z_{2}}\right\\}\,,\qquad\tilde{e}^{(32)}=\left\\{1,\frac{1}{z_{2}},\frac{1}{z_{3}},\frac{1}{z_{2}z_{3}}\right\\}\,,\qquad\tilde{e}=\left\\{1,\frac{1}{z_{3}},\frac{1}{z_{2}z_{4}},\frac{1}{z_{2}z_{3}z_{4}}\right\\}\,,$
(54) $\displaystyle\tilde{h}^{(2)}$
$\displaystyle=\left\\{1,\delta_{2}\right\\}\,,\qquad\,\,\,\,\tilde{h}^{(32)}=\left\\{1,\delta_{2},\delta_{3},\delta_{23}\right\\}\,,\qquad\quad\,\,\,\,\tilde{h}=\left\\{1,\delta_{3},\delta_{24},\delta_{234}\right\\}\,.$

Using the diagonal basis prescription, we would pick the bases $e$ and $h$ as

$\displaystyle{e}^{(2)}$
$\displaystyle=\left\\{1,\frac{1}{z_{2}}\right\\}\,,\qquad{e}^{(32)}=\left\\{1,\frac{1}{z_{2}},\frac{1}{z_{3}},\frac{1}{z_{2}z_{3}}\right\\}\,,\qquad{e}=\left\\{1,\frac{1}{z_{3}},\frac{1}{z_{2}z_{4}},\frac{1}{z_{2}z_{3}z_{4}}\right\\}\,,$
(55) $\displaystyle{h}^{(2)}$
$\displaystyle=\left\\{\frac{1}{b},\delta_{2}\right\\}\,,\quad\,\,\,{h}^{(32)}=\left\\{\frac{1}{b^{2}},\frac{\delta_{2}}{b},\frac{\delta_{3}}{b},\delta_{23}\right\\}\,,\qquad{h}=\left\\{\frac{1}{b^{3}},\frac{\delta_{3}}{b^{2}},\frac{\delta_{24}}{b},\delta_{234}\right\\}\,.$

However, this produces an almost diagonal basis. Explicitly, we have

$\displaystyle\mathbf{C}^{(2)}$
$\displaystyle=\left(\begin{array}[]{cc}\frac{1}{(d-4)s\left(4m^{2}-s\right)}&0\\\
0&1\\\ \end{array}\right)\,,$ (56) $\displaystyle\mathbf{C}^{(32)}$
$\displaystyle=\left(\begin{array}[]{cccc}-\frac{1}{4(d-3)^{2}m^{2}st\left(-4m^{2}+s+t\right)}&0&0&0\\\
0&\frac{1}{(d-4)t\left(4m^{2}-t\right)}&0&0\\\
0&0&\frac{1}{(d-4)s\left(4m^{2}-s\right)}&0\\\ 0&0&0&1\\\
\end{array}\right)\,,$ $\displaystyle\mathbf{C}$
$\displaystyle=\left(\begin{array}[]{cccc}\frac{1}{4(d-3)(d-2)(d-1)s^{2}t^{2}\left(-4m^{2}+s+t\right)^{2}}&0&0&0\\\
0&-\frac{s-4m^{2}}{4(d-5)(d-3)s}&0&0\\\
\frac{1}{4(d-3)^{2}(d-1)m^{2}s^{2}t^{3}\left(-4m^{2}+s+t\right)^{2}}&0&\frac{1}{(d-4)t\left(4m^{2}-t\right)}&0\\\
\frac{7-2d}{4(d-4)(d-3)^{2}(d-1)m^{2}s^{2}t^{3}\left(4m^{2}-s\right)\left(-4m^{2}+s+t\right)^{2}}&0&0&1\\\
\end{array}\right)\,.$

To produce a diagonal basis, we modify $e$ and $h$ to

${e}=\left\\{1,\frac{1}{z_{3}}\frac{1}{b},\frac{1}{z_{2}z_{4}},\frac{1}{z_{2}z_{3}z_{4}}\right\\}\,,\qquad{h}=\left\\{\frac{1}{b^{2}},\frac{\delta_{3}}{b},\frac{\delta_{24}}{b},\delta_{234}\right\\}\,,$
(57)

keeping the internal bases unchanged. We note that this choice, unlike the
previous examples, requires a modification of the left basis $e$. With this
new choice, we once again obtain a diagonal outer matrix

$\mathbf{C}=\left(\begin{array}[]{cccc}-\frac{m^{2}}{(d-4)(d-3)(d-2)st\left(-4m^{2}+s+t\right)}&0&0&0\\\
0&-\frac{1}{4(d-5)^{2}s^{2}t\left(-4m^{2}+s+t\right)}&0&0\\\
0&0&\frac{1}{(d-4)t\left(4m^{2}-t\right)}&0\\\ 0&0&0&1\\\
\end{array}\right)\,.$ (58)

In general, we find that some one-loop diagrams with massless propagators
require modifications of this type to the diagonal basis algorithm to obtain a
fully diagonal outer basis.

Section 5 contains many more examples of the diagonal basis algorithm in use.

###  3.4 Proof for 1-form Intersection Numbers

An algorithm for producing a diagonal basis for any univariate twist was
presented in Frellesvig_2019 . Here we present a different approach
specifically suited to the prescription of Section 3.2: As outlined in Section
2.5, all one-loop Feynman integrals produce quadratic Baikov polynomials. To
this end, we consider a generic quadratic univariate polynomial and it’s
respective twist

$b(z)=a_{0}+a_{1}z+a_{2}z^{2}=a_{2}\,(z-r_{1})(z-r_{2})\,,\quad
u(z)=b(z)^{\gamma}\,,\quad\omega(z)=\gamma\left(\frac{1}{z-r_{1}}+\frac{1}{z-r_{2}}\right)\mathrm{d}z\,,$
(59)

where $r_{1}\neq r_{2}\neq 0$ and $\gamma$ is an generic constant. As in the
Baikov representation, we interpret $z$ to be a propagator, and thus are
interested in constructing a basis for the sector $\mathcal{S}_{1}=\\{z\\}$
and its subsector $\mathcal{S}_{0}=\emptyset$. Using the algorithm of Section
3.1, we find $\mathcal{S}_{0}$ to have one master integral333For twists
originating from physical Feynman integrals, this unphysical subsector,
corresponding to a Feynman integral without propagators, contains no master
integrals. and $\mathcal{S}=\\{z\\}$ to have two. Thus, we pick bases

$\tilde{e}=\left\\{1\,,\frac{1}{z}\right\\}\,,\qquad\tilde{h}=\left\\{1\,,\delta_{z}\right\\}\,,$
(60)

and consequently

${e}=\left\\{1\,,\frac{1}{z}\right\\}\,,\qquad{h}=\left\\{\frac{1}{b(z)}\,,\delta_{z}\right\\}\,.$
(61)

The relevant $\mathbf{C}$-matrix is

$\mathbf{C}_{ij}=\innerproduct{e_{i}\,\differential z}{h_{j}\,\differential
z}=\left(\begin{array}[]{cc}c_{11}&c_{12}\\\ c_{21}&c_{22}\\\
\end{array}\right)$ (62)

From eq. (28) we have $c_{22}=1$ and $c_{12}=0$. Thus, it remains to show that
$c_{21}=\innerproduct{e_{2}}{h_{1}}=\innerproduct{1/z}{1/b(z)}=0$. To do this
we analyse the poles of the two forms appearing. We have

$\mathcal{P}(e_{2})=\\{0\,,\infty\\}\,,\qquad\mathcal{P}(h_{1})=\\{r_{1},r_{2}\\}\,.$
(63)

Furthermore, we note that all poles are simple, and as such $e_{2}$ and
$h_{1}$ can be written as $\mathrm{d}\log$ forms. We can thus leverage a known
formula for $\mathrm{d}\log$ intersection numbers in hyperplanar
arrangements444For univariate intersection numbers, the arrangement is always
hyperplanar. ojm/1200788347 .

$\innerproduct{\varphi_{L}}{\varphi_{R}}\propto\sum_{\\{r_{i},r_{j}\\}\,\in\,\mathcal{P}(\varphi_{L})\cap\mathcal{P}(\varphi_{R})}\frac{1}{\gamma_{i}\,\gamma_{j}}\,.$
(64)

Instead of explaining this result in detail, we jump straight to the point.
For $\innerproduct{e_{2}}{h_{1}}$ we have

$\mathcal{P}(e_{2})\cap\mathcal{P}(h_{1})=\emptyset\,,$ (65)

and thus there are no terms in the sum, implying $c_{21}=0$. For more details,
see ojm/1200788347 ; mizerascatt .

It is worth nothing that this argument does not imply that $c_{11}=0$, because
$1\,\differential z$ is not $\differential\log$ due to its double pole at
$\infty$. Additionally, this argument still shows
$\innerproduct{1/z}{1/b(z)}=0$ for higher order polynomials $b(z)$ beyond
quadratics. However, in these cases, the sector $\mathcal{S}_{1}$ will include
more master integrals, and thus just $c_{21}$ vanishing would not guarantee a
diagonal basis.

##  4 Closed Formula for Intersection Numbers

In Section 3 a prescription was introduced for diagonalising intersection
numbers related to quadratic twists $u(\mathbf{z})=b(\mathbf{z})^{\gamma}$. In
this section, we focus on the properties of intersection numbers generated by
this prescription, specifically of the form
$\innerproduct{b(\mathbf{z})^{p}}{b(\mathbf{z})^{q}}$555For the case $p=q=1$ a
similar result was derived in (Caron_Huot_2021, , Section 4 & Appendix B). We
find that such intersection numbers follow very regular patterns, and in
Section 4.1 propose a closed form formula for their evaluation, for any
$p\,,\,q$ integer. To the author’s knowledge, this is the first such
proposition beyond $\mathrm{d}\log$ forms and as such constitutes the second
main result of this work.

In Section 4.2 we show that all the intersection numbers appearing in the
diagonal basis prescription are of this form. Thus, the full
$\mathbf{C}$-matrix can be computed with just this closed formula, without the
need of any algorithmic evaluation strategies.

###  4.1 Closed Formula For Intersection Numbers

We consider intersection numbers of the kind

$u(\mathbf{z})=b(\mathbf{z})^{\gamma}\,,\qquad\innerproduct{b(\mathbf{z})^{p}}{b(\mathbf{z})^{q}}\,,$
(66)

where we stress again $b(\mathbf{z})$ is a quadratic polynomial in $n$
variables. The closed formula for this intersection number reads:

$\innerproduct{b(\mathbf{z})^{p}}{b(\mathbf{z})^{q}}=f_{n}(p,q;\gamma)\times\frac{\det\left(\mathbf{H}(b_{h})\right)^{n+p+q}}{\det\left(\mathbf{H}(b)\right)^{n+p+q+1}}\,,$
(67)

In this expression, $b_{h}$ is the polynomial obtained by homogenising $b$
with an extra variable $z_{n+1}$, and $\mathbf{H}$ is the Hessian matrix
computed with respect to the variables $\mathbf{z}$ for $b$ and
$\\{\mathbf{z},z_{n+1}\\}$ for $b_{h}$. Since $b$ and $b_{h}$ are quadratic,
their Hessian matrices do not depend on the variables $\mathbf{z}$. In
Appendix A, we show that $\det\left(\mathbf{H}(b_{h})\right)$, up to a
numerical prefactor, is equivalent to the multivariate discriminant of $b$.
Finally, $f_{n}(p,q;\gamma)$ is a $\gamma$ dependent prefactor given by

$f_{n}(p,q;\gamma)=\frac{(-1)^{n}}{(\gamma+p+n/2)}\left(\prod_{i=1}^{n-1}\frac{1}{(\gamma+p+i)}\right)\left(\prod_{i=1}^{n+p+q}\frac{\gamma+p-i+n}{2\gamma+2p-2i+n}\right)\left(\prod_{i=0}^{-n-p-q-1}\frac{2\gamma+2p+2i+n}{\gamma+p+i+n}\right)\,.$
(68)

###  4.2 Application to the $\mathbf{C}$-matrix

The formula presented above can be immediately put to use in computing
$\mathbf{C}$-matrices. In general, using the construction of Sections 3.1 and
3.2, the on diagonal elements of the $\mathbf{C}$-matrix are $n$-form
intersection numbers of the type

$\mathbf{C}_{ii}=\innerproduct{e_{i}}{h_{i}}=\innerproduct{\left(\prod_{z_{j}\in\mathcal{S}_{i}}\frac{1}{z_{j}}\right)b(\mathbf{z})^{p}}{\left(\prod_{z_{j}\in\mathcal{S}_{i}}\delta_{j}\right)b(\mathbf{z})^{q}}\,,$
(69)

where each $\mathcal{S}_{i}$ is a (sub)sector of the integral corresponding to
the variables appearing in the denominator of $e_{i}$. The quantities $p$ and
$q$ are kept as generic integer coefficients, to allow for cases such as eq.
(57). From eq. (28) the delta functions are immediately integrated out: If the
sector $\mathcal{S}_{i}$ has $k$ propagators, the dual basis will contain $k$
delta forms, and thus upon integrating them out we obtain an $(n-k)$-form
intersection number of the form

$\mathbf{C}_{ii}=\innerproduct{b(\mathbf{z})^{p}\rvert_{z_{j}\in\mathcal{S}_{i}\to
0}}{b(\mathbf{z})^{q}\rvert_{z_{j}\in\mathcal{S}_{i}\to 0}}\,.$ (70)

The polynomial $b(\mathbf{z})\rvert_{z_{j}\in\mathcal{S}_{i}\to 0}$ is the
polynomial $b(\mathbf{z})$ with all $k$ variables appearing in the sector
$\mathcal{S}_{i}$ set to zero. After this operation one is left still with a
quadratic polynomial but in fewer variables. Thus eq. (67) can be used
immediately for their computation.

By combining these results with the prescription presented in Section 3, one
can form an orthonormal basis, similarly to Caron_Huot_2021 , by normalising
the dual forms $h_{j}$ by the result of eq. (67). This procedure results in
$\mathbf{C}=\mathbb{I}$ and eq. (10) simplifies to

$c_{i}=\innerproduct{\varphi_{L}}{h_{i}}\,.$ (71)

###  4.3 Example Applications

To illustrate eq. (67), let us recompute the intersection number

$\innerproduct{e_{1}}{h_{1}}=\innerproduct{1}{\frac{1}{b^{2}}}\,.$ (72)

From eq. (44). We have $n=2$, $p=0$, $q=-2$ and $\gamma=(d-5)/2$. The various
pieces in the formula evaluate to

$\displaystyle f_{2}\left(0,-2;\frac{d-5}{2}\right)$
$\displaystyle=\frac{4}{(d-3)^{2}}\,,$ (73)
$\displaystyle\det\left(\mathbf{H}(b)\right)$
$\displaystyle=-16st^{2}(-4m^{2}+s+t)\,,$
$\displaystyle\det\left(\mathbf{H}(b_{h})\right)$
$\displaystyle=-32s^{2}t^{4}(-4m^{2}+s+t)^{2}\,,$

and thus

$\innerproduct*{e_{1}}{h_{1}}=f_{2}\left(0,-2;\frac{d-5}{2}\right)\times\frac{1}{\det\left(\mathbf{H}(b)\right)}=-\frac{1}{4(d-3)^{2}st^{2}\left(-4m^{2}+s+t\right)}\,,$
(74)

in agreement with eq. (46). It is worth noting that for $n$-form intersection
numbers of the form $\innerproduct{1}{1/b^{n}}$ the more complicated
polynomial $\det\left(\mathbf{H}(b_{h})\right)$ never appears in the final
result, as it is always raised to the 0th power. In cases where the diagonal
basis prescription requires no modification, the intersection numbers will
always be of this form.

To showcase the true power of this result, the evaluation of any other
intersection numbers with large powers of $p$ and $q$ require very little
extra computational effort. For example, consider
$\innerproduct*{b^{10}}{b^{-16}}$, which immediately evaluates to

$\innerproduct*{b^{10}}{b^{-16}}=f_{2}\left(10,-16;\frac{d-5}{2}\right)\times\frac{\det\left(\mathbf{H}(b_{h})\right)^{-4}}{\det\left(\mathbf{H}(b)\right)^{-3}}=\frac{64}{(d+17)(d+25)}\times\frac{-1}{256s^{5}t^{10}\left(-4m^{2}+s+t\right)^{5}}\,,$
(75)

which we independently verify to be the correct result.

##  5 Examples

In Sections 3 and 4 we introduced a prescription for generating an orthogonal
basis of differential forms related to one-loop Feynman Integrals, and we
proposed a closed form formula for the direct computation of the diagonal
$\mathbf{C}$-matrix given by this basis. In this section we will present a
series of one-loop examples for which we have found an orthogonal basis, and
have verified that the closed formula gives all intersection numbers in the
$\mathbf{C}$-matrices correctly. For the fully massive diagrams the diagonal
basis comes from the prescription described in Section 3. As anticipated
previously, for the massless diagrams, we find that we must modify this
prescription slightly in order to obtain an orthogonal basis. When drawing
diagrams in this section, dotted lines represent massless propagators, thin
lines represent masses $m_{1}$, and thick lines represent masses $m_{2}$. For
the simpler examples we show explicitly the Baikov polynomial, for the more
complicated examples we omit the Baikov polynomial due to its length, and
consider a spanning set of cuts to simplify the computation. Finally, we
always consider the external momenta to be $d$-dimensional.

###  5.1 Massive Bubble

The one-loop massive bubble is shown in Fig. 3

Figure 3: One-loop Massive Bubble with external momenta $p_{1},p_{2}$ and
propagators $z_{1},z_{2}$.

The propagators are

$\displaystyle z_{1}=\ell^{2}-m_{2}^{2}\,,\quad z_{2}$ $\displaystyle=(\ell-
p_{1})^{2}-m_{2}^{2}\,.$ (76)

The kinematics for this problem are $p_{i}^{2}=m_{1}^{2}$ and the Baikov
polynomial is

$\displaystyle
b(\mathbf{z})=4m_{2}^{2}m_{1}^{2}-m_{1}^{4}+2m_{1}^{2}(z_{1}+z_{2})-(z_{1}-z_{2})^{2}\,.$
(77)

The diagonal basis is given by

$\displaystyle
e=\left\\{\frac{1}{z_{1}},\frac{1}{z_{2}},\frac{1}{z_{1}z_{2}}\right\\}\,,\quad
h=\left\\{\frac{\delta_{1}}{b},\frac{\delta_{2}}{b},\delta_{12},\right\\}\,.$
(78)

###  5.2 Massive Triangle

The massive triangle is discussed in detail in Section 3.3.2.

###  5.3 Massless Box

The massless box is shown in Fig. 4

Figure 4: One-loop Massless Box with external momenta
$p_{1},p_{2},p_{3},p_{4}$ and propagators $z_{1},z_{2},z_{3},z_{4}$.

The propagators are

$\displaystyle z_{1}=\ell^{2},\quad z_{2}$ $\displaystyle=(\ell-
p_{1})^{2},\quad z_{3}=(\ell-p_{1}-p_{2})^{2}\,,$ (79) $\displaystyle z_{4}$
$\displaystyle=(\ell-p_{1}-p_{2}-p_{3})^{2}\,.$

The kinematics are

$\displaystyle p_{i}^{2}=0\,,\quad s=(p_{1}+p_{2})^{2}\,,\quad
t=(p_{2}+p_{3})^{2}\,,\quad s+t+(p_{1}+p_{3})^{2}=0\,.$ (80)

The Baikov polynomial is

$\displaystyle b(\mathbf{z})$
$\displaystyle=s^{2}\left(t^{2}-2t(z_{2}+z_{4})+(z_{2}-z_{4})^{2}\right)+2st(-t(z_{1}+z_{3})$
(81)
$\displaystyle\quad+z_{4}(z_{1}-2z_{2}+z_{3})+z_{1}z_{2}-2z_{1}z_{3}+z_{2}z_{3})+t^{2}(z_{1}-z_{3})^{2}\,.$

The diagonal basis is given by

$\displaystyle
e=\left\\{\frac{1}{z_{1}z_{3}},\frac{1}{z_{2}z_{4}},\frac{1}{z_{1}z_{2}z_{3}z_{4}}\right\\}\,,\quad
h=\left\\{\frac{\delta_{13}}{b},\frac{\delta_{24}}{b},\delta_{1234}\right\\}\,.$
(82)

###  5.4 Massive Box

The massive box is shown in Fig. 5

Figure 5: One-loop Massive Box with external momenta $p_{1},p_{2},p_{3},p_{4}$
and propagators $z_{1},z_{2},z_{3},z_{4}$.

The propagators are

$\displaystyle z_{1}=\ell^{2}-m_{2}^{2},\quad z_{2}$ $\displaystyle=(\ell-
p_{1})^{2}-m_{2}^{2},\quad z_{3}=(\ell-p_{1}-p_{2})^{2}-m_{2}^{2}\,,$ (83)
$\displaystyle z_{4}$ $\displaystyle=(\ell-
p_{1}-p_{2}-p_{3})^{2}-m_{2}^{2}\,.$

The kinematics are

$\displaystyle p_{i}^{2}=m_{1}^{2}\,,\quad s=(p_{1}+p_{2})^{2}\,,\quad
t=(p_{2}+p_{3})^{2}\,,\quad s+t+(p_{1}+p_{3})^{2}=4m_{1}^{2}\,.$ (84)

The Baikov polynomial is

$\displaystyle b(\mathbf{z})$
$\displaystyle=-4m_{1}^{2}st\left(-4m_{2}^{2}+s+t\right)-4m_{2}^{4}st-4stz_{1}z_{3}+2stz_{2}z_{3}+t^{2}z_{1}^{2}-2t^{2}z_{1}z_{3}+t^{2}z_{3}^{2}$
(85)
$\displaystyle\quad+4m_{2}^{2}\left(st(z_{1}+z_{2}+z_{3}+z_{4})-s(z_{2}-z_{4})^{2}-t(z_{1}-z_{3})^{2}\right)+s^{2}t^{2}-2s^{2}tz_{2}$
$\displaystyle\quad+s^{2}z_{2}^{2}+s^{2}z_{4}^{2}-2st^{2}z_{1}-2st^{2}z_{3}-2sz_{4}(s(t+z_{2})-t(z_{1}-2z_{2}+z_{3}))+2stz_{1}z_{2}\,.$

The diagonal basis is given by

$\displaystyle e$
$\displaystyle=\bigg{\\{}\frac{1}{z_{1}},\frac{1}{z_{2}},\frac{1}{z_{3}},\frac{1}{z_{4}},\frac{1}{z_{1}z_{2}},\frac{1}{z_{1}z_{3}},\frac{1}{z_{1}z_{4}},\frac{1}{z_{2}z_{3}},\frac{1}{z_{2}z_{4}},\frac{1}{z_{3}z_{4}},$
(86)
$\displaystyle\qquad\frac{1}{z_{1}z_{2}z_{3}},\frac{1}{z_{1}z_{2}z_{4}},\frac{1}{z_{1}z_{3}z_{4}},\frac{1}{z_{2}z_{3}z_{4}},\frac{1}{z_{1}z_{2}z_{3}z_{4}}\bigg{\\}}\,,$
$\displaystyle h$
$\displaystyle=\bigg{\\{}\frac{\delta_{1}}{b^{3}},\frac{\delta_{2}}{b^{3}},\frac{\delta_{3}}{b^{3}},\frac{\delta_{4}}{b^{3}},\frac{\delta_{12}}{b^{2}},\frac{\delta_{13}}{b^{2}},\frac{\delta_{14}}{b^{2}},\frac{\delta_{23}}{b^{2}},\frac{\delta_{24}}{b^{2}},\frac{\delta_{34}}{b^{2}},\frac{\delta_{123}}{b},\frac{\delta_{124}}{b},\frac{\delta_{134}}{b},\frac{\delta_{234}}{b},\delta_{1234}\bigg{\\}}\,.$

###  5.5 Massless Pentagon

The massless pentagon is shown in Fig. 6

Figure 6: One-loop Massless Pentagon with external momenta
$p_{1},p_{2},p_{3},p_{4},p_{5}$ and propagators
$z_{1},z_{2},z_{3},z_{4},z_{5}$.

The propagators are

$\displaystyle z_{1}$ $\displaystyle=\ell^{2}\,,\quad z_{2}=(\ell-
p_{1})^{2}\,,\quad z_{3}=(\ell-p_{1}-p_{2})^{2}\,,$ (87) $\displaystyle z_{4}$
$\displaystyle=(\ell-p_{1}-p_{2}-p_{3})^{2}\,,\quad z_{5}=(\ell-
p_{1}-p_{2}-p_{3}-p_{4})^{2}\,.$

The kinematics are

$\displaystyle p_{i}^{2}=0\,,\quad
s_{ij}=(p_{i}+p_{j})^{2}\,,\quad\sum_{0<i<j<5}s_{ij}=0\,.$ (88)

The spanning set of cuts up to symmetry relations is

$\displaystyle\\{z_{4},z_{5}\\}\,,\quad\\{z_{3},z_{5}\\}\,.$ (89)

####  5.5.1 Cut $\\{z_{4},z_{5}\\}$

On this cut, the diagonal basis is given by

$\displaystyle
e=\left\\{\frac{1}{z_{1}z_{2}},\frac{1}{z_{1}z_{3}},\frac{1}{z_{2}z_{3}},\frac{1}{z_{1}z_{2}z_{3}}\right\\}\,,\quad
h=\left\\{\frac{\delta_{12}}{b},\frac{\delta_{13}}{b},\frac{\delta_{23}}{b},\delta_{123}\right\\}\,.$
(90)

####  5.5.2 Cut $\\{z_{3},z_{5}\\}$

On this cut, the diagonal basis is given by

$\displaystyle
e=\left\\{1,\frac{1}{z_{1}z_{2}},\frac{1}{z_{1}z_{4}},\frac{1}{z_{2}z_{4}},\frac{1}{z_{1}z_{2}z_{4}}\right\\}\,,\quad
h=\left\\{\frac{1}{b^{2}},\frac{\delta_{12}}{b},\frac{\delta_{14}}{b},\frac{\delta_{24}}{b},\delta_{124}\right\\}\,.$
(91)

###  5.6 Massive Pentagon

The massive pentagon is shown in Fig. 7

Figure 7: One-loop Massive Pentagon with external momenta
$p_{1},p_{2},p_{3},p_{4},p_{5}$ and propagators
$z_{1},z_{2},z_{3},z_{4},z_{5}$.

The propagators are

$\displaystyle z_{1}$ $\displaystyle=\ell^{2}-m_{2}^{2},\quad z_{2}=(\ell-
p_{1})^{2}-m_{2}^{2},\quad z_{3}=(\ell-p_{1}-p_{2})^{2}-m_{2}^{2}$ (92)
$\displaystyle z_{4}$ $\displaystyle=(\ell-
p_{1}-p_{2}-p_{3})^{2}-m_{2}^{2},\quad z_{5}=(\ell-
p_{1}-p_{2}-p_{3}-p_{4})^{2}-m_{2}^{2}.$

The kinematics are

$\displaystyle p_{i}^{2}=m_{1}^{2},\quad
s_{ij}=(p_{i}+p_{j})^{2},\quad\sum_{0<i<j<5}s_{ij}=8m_{1}^{2}$ (93)

The spanning set of cuts up to symmetry relations is

$\displaystyle\\{z_{5}\\}.$ (94)

####  5.6.1 Cut $\\{z_{5}\\}$

On this cut, the diagonal basis is given by

$\displaystyle e$
$\displaystyle=\bigg{\\{}1,\frac{1}{z_{1}},\frac{1}{z_{2}},\frac{1}{z_{3}},\frac{1}{z_{4}},\frac{1}{z_{1}z_{2}},\frac{1}{z_{1}z_{3}},\frac{1}{z_{1}z_{4}},\frac{1}{z_{2}z_{3}},\frac{1}{z_{2}z_{4}},\frac{1}{z_{3}z_{4}},$
(95)
$\displaystyle\qquad\frac{1}{z_{1}z_{2}z_{3}},\frac{1}{z_{1}z_{2}z_{4}},\frac{1}{z_{1}z_{3}z_{4}},\frac{1}{z_{2}z_{3}z_{4}},\frac{1}{z_{1}z_{2}z_{3}z_{4}}\bigg{\\}}$
$\displaystyle h$
$\displaystyle=\bigg{\\{}\frac{1}{b^{4}},\frac{\delta_{1}}{b^{3}},\frac{\delta_{2}}{b^{3}},\frac{\delta_{3}}{b^{3}},\frac{\delta_{4}}{b^{3}},\frac{\delta_{12}}{b^{2}},\frac{\delta_{13}}{b^{2}},\frac{\delta_{14}}{b^{2}},\frac{\delta_{23}}{b^{2}},\frac{\delta_{24}}{b^{2}},\frac{\delta_{34}}{b^{2}},\frac{\delta_{123}}{b},\frac{\delta_{124}}{b},\frac{\delta_{134}}{b},\frac{\delta_{234}}{b},\delta_{1234}\bigg{\\}}.$

###  5.7 Massless Hexagon

The massless hexagon is shown in Fig. 8

Figure 8: One-loop Massless Hexagon with external momenta
$p_{1},p_{2},p_{3},p_{4},p_{5},p_{6}$ and propagators
$z_{1},z_{2},z_{3},z_{4},z_{5},z_{6}$.

The propagators are

$\displaystyle z_{1}$ $\displaystyle=\ell^{2}\,,\quad z_{2}=(\ell-
p_{1})^{2}\,,\quad z_{3}=(\ell-p_{1}-p_{2})^{2}\,,\quad z_{4}=(\ell-
p_{1}-p_{2}-p_{3})^{2}\,,$ (96) $\displaystyle z_{5}$ $\displaystyle=(\ell-
p_{1}-p_{2}-p_{3}-p_{4})^{2}\,,\quad z_{6}=(\ell-
p_{1}-p_{2}-p_{3}-p_{4}-p_{5})^{2}\,.$

The kinematics are

$\displaystyle p_{i}^{2}=0\,,\quad
s_{ij}=(p_{i}+p_{j})^{2}\,,\quad\sum_{0<i<j<6}s_{ij}=0\,.$ (97)

The spanning set of cuts up to symmetry relations is

$\displaystyle\\{z_{5},z_{6}\\}\,,\quad\\{z_{4},z_{6}\\}\,,\quad\\{z_{3},z_{6}\\}\,.$
(98)

It is important to reiterate here that we are considering the external momenta
to be $d$-dimensional, and therefore the hexagon appears in the basis, unlike
the $4$-dimensional case.

####  5.7.1 Cut $\\{z_{5},z_{6}\\}$

On this cut, the diagonal basis is given by

$\displaystyle e$
$\displaystyle=\left\\{\frac{1}{z_{1}z_{2}},\frac{1}{z_{1}z_{3}},\frac{1}{z_{1}z_{4}},\frac{1}{z_{2}z_{3}},\frac{1}{z_{2}z_{4}},\frac{1}{z_{3}z_{4}},\frac{1}{z_{1}z_{2}z_{3}},\frac{1}{z_{1}z_{2}z_{4}},\frac{1}{z_{1}z_{3}z_{4}},\frac{1}{z_{2}z_{3}z_{4}},\frac{1}{z_{1}z_{2}z_{3}z_{4}}\right\\}\,,$
(99) $\displaystyle h$
$\displaystyle=\left\\{\frac{\delta_{12}}{b^{2}},\frac{\delta_{13}}{b^{2}},\frac{\delta_{14}}{b^{2}},\frac{\delta_{23}}{b^{2}},\frac{\delta_{24}}{b^{2}},\frac{\delta_{34}}{b^{2}},\frac{\delta_{123}}{b},\frac{\delta_{124}}{b},\frac{\delta_{134}}{b},\frac{\delta_{234}}{b},\delta_{1234}\right\\}\,.$

####  5.7.2 Cut $\\{z_{4},z_{6}\\}$

On this cut, the diagonal basis is given by

$\displaystyle e$
$\displaystyle=\left\\{1,\frac{1}{bz_{2}},\frac{1}{z_{1}z_{2}},\frac{1}{z_{1}z_{3}},\frac{1}{z_{1}z_{5}},\frac{1}{z_{2}z_{3}},\frac{1}{z_{2}z_{5}},\frac{1}{z_{3}z_{5}},\frac{1}{z_{1}z_{2}z_{3}},\frac{1}{z_{1}z_{2}z_{5}},\frac{1}{z_{1}z_{3}z_{5}},\frac{1}{z_{2}z_{3}z_{5}},\frac{1}{z_{1}z_{2}z_{3}z_{5}}\right\\}\,,$
(100) $\displaystyle h$
$\displaystyle=\left\\{\frac{1}{b^{3}},\frac{\delta_{2}}{b^{3}},\frac{\delta_{12}}{b^{2}},\frac{\delta_{13}}{b^{2}},\frac{\delta_{15}}{b^{2}},\frac{\delta_{23}}{b^{2}},\frac{\delta_{25}}{b^{2}},\frac{\delta_{35}}{b^{2}},\frac{\delta_{123}}{b},\frac{\delta_{125}}{b},\frac{\delta_{135}}{b},\frac{\delta_{235}}{b},\delta_{1235}\right\\}\,.$

####  5.7.3 Cut $\\{z_{3},z_{6}\\}$

On this cut, the diagonal basis is given by

$\displaystyle e$
$\displaystyle=\left\\{1,\frac{1}{z_{1}z_{2}},\frac{1}{z_{1}z_{4}},\frac{1}{z_{1}z_{5}},\frac{1}{z_{2}z_{4}},\frac{1}{z_{2}z_{5}},\frac{1}{z_{4}z_{5}},\frac{1}{z_{1}z_{2}z_{4}},\frac{1}{z_{1}z_{2}z_{5}},\frac{1}{z_{1}z_{4}z_{5}},\frac{1}{z_{2}z_{4}z_{5}},\frac{1}{z_{1}z_{2}z_{4}z_{5}}\right\\}\,,$
(101) $\displaystyle h$
$\displaystyle=\left\\{\frac{1}{b^{3}},\frac{\delta_{12}}{b^{2}},\frac{\delta_{14}}{b^{2}},\frac{\delta_{15}}{b^{2}},\frac{\delta_{24}}{b^{2}},\frac{\delta_{25}}{b^{2}},\frac{\delta_{45}}{b^{2}},\frac{\delta_{124}}{b},\frac{\delta_{125}}{b},\frac{\delta_{145}}{b},\frac{\delta_{245}}{b},\delta_{1245}\right\\}\,.$

###  5.8 Massive Hexagon

The massive hexagon is shown in Fig. 9

Figure 9: One-loop Massive Hexagon with external momenta
$p_{1},p_{2},p_{3},p_{4},p_{5},p_{6}$ and propagators
$z_{1},z_{2},z_{3},z_{4},z_{5},z_{6}$.

The propagators are

$\displaystyle z_{1}$ $\displaystyle=\ell^{2}-m_{2}^{2}\,,\quad z_{2}=(\ell-
p_{1})^{2}-m_{2}^{2}\,,\quad z_{3}=(\ell-p_{1}-p_{2})^{2}-m_{2}^{2}\,,$ (102)
$\displaystyle z_{4}$ $\displaystyle=(\ell-
p_{1}-p_{2}-p_{3})^{2}-m_{2}^{2}\,,\quad z_{5}=(\ell-
p_{1}-p_{2}-p_{3}-p_{4})^{2}-m_{2}^{2}\,,$ $\displaystyle z_{6}$
$\displaystyle=(\ell-p_{1}-p_{2}-p_{3}-p_{4}-p_{5})^{2}-m_{2}^{2}\,.$

The kinematics are

$\displaystyle p_{i}^{2}=m_{1}^{2}\,,\quad
s_{ij}=(p_{i}+p_{j})^{2}\,,\quad\sum_{0<i<j<6}s_{ij}=15m_{1}^{2}\,.$ (103)

The spanning set of cuts up to symmetry relations is

$\displaystyle\\{z_{6}\\}\,.$ (104)

####  5.8.1 Cut $\\{z_{6}\\}$

On this cut, the diagonal basis is given by

$\displaystyle e$
$\displaystyle=\bigg{\\{}1,\frac{1}{z_{1}},\frac{1}{z_{2}},\frac{1}{z_{3}},\frac{1}{z_{4}},\frac{1}{z_{5}},\frac{1}{z_{1}z_{2}},\frac{1}{z_{1}z_{3}},\frac{1}{z_{1}z_{4}},\frac{1}{z_{1}z_{5}},\frac{1}{z_{2}z_{3}},\frac{1}{z_{2}z_{4}},\frac{1}{z_{2}z_{5}},\frac{1}{z_{3}z_{4}},\frac{1}{z_{3}z_{5}},\frac{1}{z_{4}z_{5}},$
(105)
$\displaystyle\qquad\frac{1}{z_{1}z_{2}z_{3}},\frac{1}{z_{1}z_{2}z_{4}},\frac{1}{z_{1}z_{2}z_{5}},\frac{1}{z_{1}z_{3}z_{4}},\frac{1}{z_{1}z_{3}z_{5}},\frac{1}{z_{1}z_{4}z_{5}},\frac{1}{z_{2}z_{3}z_{4}},\frac{1}{z_{2}z_{3}z_{5}},\frac{1}{z_{2}z_{4}z_{5}},\frac{1}{z_{3}z_{4}z_{5}},$
$\displaystyle\qquad\frac{1}{z_{1}z_{2}z_{3}z_{4}},\frac{1}{z_{1}z_{2}z_{3}z_{5}},\frac{1}{z_{1}z_{2}z_{4}z_{5}},\frac{1}{z_{1}z_{3}z_{4}z_{5}},\frac{1}{z_{2}z_{3}z_{4}z_{5}},\frac{1}{z_{1}z_{2}z_{3}z_{4}z_{5}}\bigg{\\}}\,.$
$\displaystyle h$
$\displaystyle=\bigg{\\{}\frac{1}{b^{5}},\frac{\delta_{1}}{b^{4}},\frac{\delta_{2}}{b^{4}},\frac{\delta_{3}}{b^{4}},\frac{\delta_{4}}{b^{4}},\frac{\delta_{5}}{b^{4}},\frac{\delta_{12}}{b^{3}},\frac{\delta_{13}}{b^{3}},\frac{\delta_{14}}{b^{3}},\frac{\delta_{15}}{b^{3}},\frac{\delta_{23}}{b^{3}},\frac{\delta_{24}}{b^{3}},\frac{\delta_{25}}{b^{3}},\frac{\delta_{34}}{b^{3}},\frac{\delta_{35}}{b^{3}},\frac{\delta_{45}}{b^{3}},$
$\displaystyle\qquad\frac{\delta_{123}}{b^{2}},\frac{\delta_{124}}{b^{2}},\frac{\delta_{125}}{b^{2}},\frac{\delta_{134}}{b^{2}},\frac{\delta_{135}}{b^{2}},\frac{\delta_{145}}{b^{2}},\frac{\delta_{234}}{b^{2}},\frac{\delta_{235}}{b^{2}},\frac{\delta_{245}}{b^{2}},\frac{\delta_{345}}{b^{2}},$
$\displaystyle\qquad\frac{\delta_{1234}}{b},\frac{\delta_{1235}}{b},\frac{\delta_{1245}}{b},\frac{\delta_{1345}}{b},\frac{\delta_{2345}}{b},\delta_{12345}\bigg{\\}}\,.$

###  5.9 Two-loop Example: Elliptic Sunrise

The elliptic sunrise is shown in Fig. 10

Figure 10: Two-loop Massive Sunrise with external momenta $p_{1},p_{2}$ and
propagators $z_{1},z_{2},z_{3}$.

The propagators are

$\displaystyle z_{1}=\ell_{2}^{2}-m_{2}^{2},\quad
z_{2}=(\ell_{1}-\ell_{2})^{2}-m_{2}^{2},\quad
z_{3}=(\ell_{1}+p_{1})^{2}-m_{2}^{2},\quad z_{4}=\ell_{1}^{2},\quad
z_{5}=(\ell_{2}+p_{1})^{2}\,.$ (106)

The kinematics are $p_{i}^{2}=m_{1}^{2}$. Our basis choice is

$\displaystyle e$
$\displaystyle=\left\\{\frac{1}{z_{1}z_{2}},\frac{1}{z_{1}z_{3}},\frac{1}{z_{2}z_{3}},\frac{1}{z_{1}z_{2}^{2}z_{3}},\frac{1}{z_{1}z_{2}z_{3}^{2}},\frac{1}{z_{1}z_{2}z_{3}}\right\\}$
(107) $\displaystyle h$
$\displaystyle=\left\\{\frac{\delta_{12}}{b^{2}},\frac{\delta_{13}}{b^{2}},\frac{\delta_{23}}{b^{2}},\frac{\partial_{z_{2}}\delta_{123}}{b},\frac{\partial_{z_{3}}\delta_{123}}{b},\frac{\delta_{123}}{b}\right\\}\,.$

where we use the “derivative” of the delta form as defined in eq. (26). The
$\mathbf{C}$-matrix is block diagonal and is given by

$\mathbf{C}=\left(\begin{array}[]{cccccc}\frac{64}{(d-2)^{3}}&0&0&0&0&0\\\
0&\frac{64}{(d-2)^{3}}&0&0&0&0\\\ 0&0&\frac{64}{(d-2)^{3}}&0&0&0\\\
0&0&0&-\frac{2}{m^{2}}&0&\frac{8}{3d-10}\\\
0&0&0&0&-\frac{4}{m^{2}}&\frac{8}{3d-10}\\\
0&0&0&-\frac{8}{3d-8}&-\frac{8}{3d-8}&\frac{16\left(3m^{2}+s\right)}{(3d-10)(3d-8)}\\\
\end{array}\right)\,.$ (108)

##  6 Conclusions

In this paper we presented two advancements for the computation of
intersection numbers for twists with quadratic polynomials. The first of these
was a new prescription that produces duals orthogonal to Laporta bases for
relative twisted cohomology groups, resulting in diagonal
$\mathbf{C}$-matrices. We tested the procedure with a number of one-loop
Feynman integrals and found that it works consistently in the fully massive
case. If the diagrams were massless our prescription also worked but with some
minor modifications. Beyond one-loop, we also tested our result for the
elliptic sunrise, and produced a block diagonal $\mathbf{C}$-matrix, with each
block representing a (sub)-sector, by making use of “derivatives” of the delta
forms.

The second result was the introduction of a new, closed formula for
intersection numbers between quadratic polynomials raised to arbitrary powers.
We believe this result to be significant as it is the first such proposition
beyond $\differential\log$ forms. We verified this formula to hold with many
arbitrarily chosen quadratic polynomials, as well as multiple $1$-loop Feynman
Integrals presented in the work. The closed formula, combined with the
diagonal basis prescription, allowed for a very efficient computation of
$\mathbf{C}$-matrices, by requiring no explicit algorithmic computations of
intersection numbers.

Our results presented in this work open many possibilities for further
exploration. An important first question would be why the diagonal basis
prescription requires modifications when fewer masses are present in the
diagrams. In a different direction, one could attempt to generalise the
algorithm to two-loop diagrams, beyond the elliptic sunrise. For these cases,
where irreducible scalar products play a role and there exist multiple master
integrals in each sector, one might hope for a similar prescription to produce
block-diagonal $\mathbf{C}$-matrices for more general classes of integrals.

Our new closed formula also presents new interesting questions. Firstly, it
would be of great importance to prove this result. Beyond quadratics, one
could also ask whether intersection numbers between cubic polynomials or
beyond follow similar patterns, as this would represent Feynman integrals
beyond one-loop. It would not be unreasonable to expect the multivariate
discriminant to appear in such cases also, although in practice from a
computational standpoint no closed formula is expected for multivariate
discriminants of higher order polynomials.

Finally, our closed formula is specific to twists with a single polynomial
factor. Extending this to even just two factors would allow for the
computation of a much larger class of intersection numbers, including the
projections necessary to complete IBP reductions once the $\mathbf{C}$-matrix
has been computed. This could pave the way for a closed formulae for complete
IBP reductions at one-loop and perhaps beyond.

Indeed, in this work we have uncovered mathematical structures in intersection
numbers beyond $1$\- and $\differential\log$ forms, venturing beyond where
many of their properties are well known and studied. Whilst just a small
start, a deeper understanding of the characteristics of more general types of
intersection numbers could prove instrumental in our future understanding of
the Feynman integral vector space and beyond.

### Acknowledgements

We wish to thank Pierpaolo Mastrolia, Wojciech Flieger, Manoj Mandal, Vsevolod
Chestnov, Federico Gasparotto, Mathieu Giroux, Giacomo Brunello and Sebastian
Mizera for ideas, comments and feedback. We would additionally like to thank
Andrzej Pokraka and Hjalte Frellesvig for invaluable discussions relating to
all parts of the work. G.C. thanks Saiei-Jaeyeong Matsubara-Heo, Julian
Miczajka and Francesco Calisto for stimulating conversations related to
multivariate discriminants, as well as Johannes Henn and the Max Planck
institute for Physics for hosting him during the development of this project.

##  Appendix A Relation Between the Closed Formula and Multivariate
Discriminant

The multivariate discriminant Gelfand1994DiscriminantsRA ; citeulike:14125041
; DANDREA200159  for a homogeneous polynomial $p(\mathbf{z})$ in $n>1$
variables $\mathbf{z}$ is a polynomial in the coefficients of $p$.
Specifically, $\mathrm{Disc}_{\mathbf{z}}(p)=0$ if and only if there exists a
non trivial solution to the set of equations:

$\displaystyle\frac{\partial p}{\partial z_{1}}$ $\displaystyle=0$ (109)
$\displaystyle\vdots$ $\displaystyle\frac{\partial p}{\partial z_{n}}$
$\displaystyle=0\ $

for the variables $\mathbf{z}$. This fixes the multivariate discriminant
uniquely up to a numerical prefactor. The “non trivial” is necessary because
for homogeneous polynomials there will always be the trivial solution
$\mathbf{z}=\mathbf{0}$ to eq. (109), which we exclude. For an inhomogeneous
polynomial $p$ in $n$ variables, the polynomial is first homogenised to obtain
an $n+1$ variable polynomial $p_{h}$. The above definition is then applied to
$p_{h}$. Namely, we have

$\mathrm{Disc}_{\mathbf{z}}(p):=\mathrm{Disc}_{\mathbf{y}}(p_{h})\,,\qquad\mathbf{y}=\\{\mathbf{z},z_{n+1}\\}\,.$
(110)

Let us now specify to $n$ variable quadratic polynomials, such as
$b(\mathbf{z})$ in Section (4.1). To show that
$\det\left(\mathbf{H}(b_{h})\right)=\mathrm{Disc}_{\mathbf{z}}(b)$ we note
that if $b$ is quadratic, then $b_{h}$ is purely quadratic (only contains
quadratic terms). Thus, $b_{h}$ can be written as

$b_{h}=\mathbf{y}^{T}\,A\,\mathbf{y}\,.$ (111)

where $A$ is a matrix of coefficients. It is easy show that

$\frac{\partial
b_{h}}{\partial\mathbf{y}}=(A^{T}+A)\,\mathbf{y}\,,\qquad\mathbf{H}(b_{h})=A^{T}+A\,.$
(112)

Thus, eq. (109) reduces to

$\mathbf{H}(b_{h})\,\mathbf{y}=0\,.$ (113)

For any solutions other than $\mathbf{y}=\mathbf{0}$ to exist, we must
necessarily have $\det\mathbf{H}(b_{h})=0$. Thus, we conclude that
$\det\mathbf{H}(b_{h})$ (up to a numerical prefactor) must be the multivariate
discriminant of $b_{h}$ and equivalently of $b$.

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