# Fourier Calculus from Intersection Theory

Giacomo Brunello$\,{}^{a,b,c}$<EMAIL_ADDRESS>Giulio
Crisanti$\,{}^{a,b}$<EMAIL_ADDRESS>Mathieu
Giroux$\,{}^{d}$<EMAIL_ADDRESS>Pierpaolo
Mastrolia$\,{}^{a,b}$<EMAIL_ADDRESS>Sid Smith$\,{}^{a,e}$
<EMAIL_ADDRESS>aDipartimento di Fisica e Astronomia, Universita di Padova,
Via Marzolo 8, 35131 Padova, Italy  bINFN, Sezione di Padova, Via Marzolo 8,
I-35131 Padova, Italy.  cInstitut de Physique Théorique, CEA, CNRS, Université
Paris-Saclay, F–91191 Gif-sur-Yvette cedex, France  dDepartment of Physics,
McGill University, 3600 Rue University, Montréal, H3A 2T8, QC Canada  eHiggs
Centre for Theoretical Physics, University of Edinburgh, James Clerk Maxwell
Building,Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United Kingdom

###### Abstract

Building on recent advances in studying the co-homological properties of
Feynman integrals, we apply intersection theory to the computation of Fourier
integrals. We discuss applications pertinent to gravitational bremsstrahlung
and deep inelastic scattering in the saturation regime. After identifying the
bases of master integrals, the latter are evaluated by means of the
differential equation method. Finally, new results with exact dependence on
the spacetime dimension $\mathrm{D}$ are presented.

##  I Introduction

In recent years, monumental efforts have been invested in developing tools for
evaluating Feynman integrals in particle physics. Modern state-of-the-art
computations face the challenge of applying integration-by-parts (IBP)
decompositions [1, 2] in the most efficient way possible. IBP identities are
relations among Feynman integrals sharing a common set of denominators,
appearing with different propagator powers (irreducible scalar products in the
numerator can be dealt with as denominators with negative powers). These allow
for the decomposition of any Feynman integral in terms of a finite spanning
set of simpler and linearly independent integrals, often referred to as
_master integrals_. The decomposition to master integrals is fundamentally a
matter of linear algebra, and several publicly available computer programs can
efficiently execute it [3, 4, 5, 6, 7, 8, 9, 10, 8, 11].

When recognized as _twisted periods_ , Feynman integrals can alternatively be
decomposed into master bases by making use of concepts and computing tools of
_de Rham twisted cohomology theory_ [12, 13, 14, 15], as first proposed in
[16, 17, 18, 19].

Within this framework, integrals are considered as pairings of regulated
integration domains and differential $n$-forms, known as twisted cycles and
cocycles respectively, which are elements of isomorphic vector spaces,
equipped with inner products, called the _intersection numbers_. The
intersection number can be used to derive the decomposition of Feynman
integrals in terms of master integrals by projections, as an alternative to
the system-solving procedure underpinning IBP decompositions.

Although the most recent applications of intersection theory have dealt with
Feynman integrals [17, 18, 20, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31], the method is rather general, and the range of its applications can be
extended to a much wider class of cases, relevant for physical and
mathematical studies (see, e.g., [32, 33, 34, 35]).

In this letter, we propose an intersection-theory based approach to the
evaluation of dimensionally regularized Fourier transforms, herein referred to
as _Fourier integrals_.

Indeed, as is the case for any Feynman integral, expressing any Fourier
integral in _Baikov representation_ [36] enables its identification as a
twisted period. Thus, relations between Fourier integrals and, in particular,
their decompositions into a common master integral basis, can be obtained
directly from intersection numbers. Moreover, intersection theory can be used
to derive the system of differential equations satisfied by the master Fourier
integrals, which can be solved analytically (when possible) similarly to
Feynman integrals. [37, 38, 39, 40].

To illustrate our method, we identify three cases of physical interest,
requiring the evaluation of certain types of dimensionally regulated Fourier
integrals: III.1 the scalar (Feynman) propagator in position space, III.2 the
tree-level gravitational spectral waveform, and III.3 color dipole scattering
in high-energy QCD. In each of the considered cases, intersection numbers are
used to build linear relations and differential equations for the associated
master integrals in Baikov representation. Once the systems of differential
equations are obtained, the solutions are systematically derived, provided an
appropriate set of boundary conditions. Using this approach, we present _new_
, closed-form formulae for $\mathrm{D}$-dimensional Fourier integrals relevant
to cases III.2 and III.3.

From a mathematical point of view, our results offer a generalization of the
studies carried out on confluent hypergeometric integrals [14, 15], involving
intersection numbers between $1$-forms, to cases where the evaluation of
intersection numbers for $n$-forms is required.

This letter is organized as follows. In sec. II we provide the necessary
background on intersection theory and outline its application to Fourier
integrals. The method is then applied in sec. III, using the three cases of
study (III.1, III.2 and III.3) mentioned above. For III.2 and III.3, a minimal
physics background is provided for the reader’s convenience. In sec. IV, we
present our conclusions and an outlook. The four appendices contain the
derivation of the Baikov representation for Fourier integrals, and auxiliary
formulae recalled in the text.

##  II Fourier integrals and intersection theory

In this section, we provide some background material on intersection theory
and describe how it can be applied to dimensionally regularized Fourier
integrals [16, 17, 18, 19] .

#### Twisted cohomology

We consider instances of _twisted period integrals_ , which generically take
the form

$\displaystyle I=\int_{C_{R}}u(\mathbf{z})\varphi_{L}(\mathbf{z})\,,$ (1)

where the _twist_ $u(\mathbf{z})$ is a multivalued function,
$\varphi_{L}(\mathbf{z})$ is an algebraic differential $n$-form and $C_{R}$ is
a contour of integration.111It is assumed that the causality conditions have
already been incorporated into the definition of $C_{R}$, which means the
$i\varepsilon$ prescription is accounted for there, and not in the integrand.
The latter is defined such that $u(\mathbf{z})$ vanishes on its boundary:
$u(\mathbf{z})=0$ for any $\mathbf{z}\in\partial C_{R}$. This condition on $u$
together with eq. (1) give an equivalence relation between differential forms

$\displaystyle\varphi_{L}\sim\varphi_{L}+\nabla_{\omega}\xi\,,$ (2)

where $\xi$ is a differential $(n{-}1)$-form and

$\nabla_{\omega}=\mathrm{d}+\omega\wedge\quad\text{with}\quad\omega=\mathrm{d}\log{u}\,.$
(3)

The collection of all the equivalence classes forms the _twisted cohomology
group_ $H_{\omega}^{n}$.222Setting $\omega=0$ in eq. (2), one obtains the
standard (non-twisted) de Rham equivalence class. This group is always finite-
dimensional and forms a vector space [17]. We denote an element by

$\displaystyle\bra{\varphi_{L}}\in H_{\omega}^{n}\,.$ (4)

As for any other finite dimensional vector space, there exists a _dual_ vector
space $H_{-\omega}^{n}$, denoted by

$\displaystyle H_{-\omega}^{n}\ni\ket{\varphi_{R}}\,.$ (5)

Using these definitions, we write the (dual) integrals in eq. (1) as a
_pairing_ between a (dual) cycle and a (dual) cocycle

$\displaystyle I$
$\displaystyle=\int_{C_{R}}u(\mathbf{z})\varphi_{L}(\mathbf{z})=\bra{\varphi_{L}}C_{R}]\,,$
(6a)
$\displaystyle=\int_{C_{L}}u(\mathbf{z})^{-1}\varphi_{R}(\mathbf{z})=[C_{L}\ket{\varphi_{R}}\,.$
(6b)

#### Basis of master integrals

We can formally define bases for $H_{\omega}^{n}$ and its dual
$H_{-\omega}^{n}$ as

$\displaystyle\text{span}\\{\bra{e_{1}},\ldots,\bra{e_{\nu}}\\}$
$\displaystyle=H_{\omega}^{n}\,,$ (7a)
$\displaystyle\text{span}\\{\ket{\widecheck{e}_{1}},\ldots,\ket{\widecheck{e}_{\nu}}\\}$
$\displaystyle=H_{-\omega}^{n}\,,$ (7b)

respectively. Both $H_{\omega}^{n}$ and $H_{-\omega}^{n}$ have the same
dimension [41, 18], which can be computed as

$\displaystyle\nu=\left\\{\\#\leavevmode\nobreak\ \text{of solutions to
$\omega=0$}\right\\}\,.$ (8)

Any elements $\bra{\varphi_{L}}\in H_{\omega}^{n}$ and $\ket{\varphi_{R}}\in
H_{-\omega}^{n}$ can then be decomposed with respect to the choice of (dual)
bases

$\bra{\varphi_{L}}=\sum_{i=1}^{\nu}c_{i}\bra{e_{i}}\quad\text{and}\quad\ket{\varphi_{R}}=\sum_{i=1}^{\nu}\widecheck{c}_{i}\ket{\widecheck{e}_{i}}\,.$
(9)

By pairing the linear combinations in eq. (9) with the appropriate contours,
we obtain the decompositions of the corresponding twisted periods with respect
to the basis of master integrals $J_{i}=\bra{e_{i}}C_{R}]$ and
$\widecheck{J}_{i}=[C_{L}\ket{\widecheck{e}_{i}}$

$\displaystyle I$
$\displaystyle=\bra{\varphi_{L}}C_{R}]=\sum_{i=1}^{\nu}c_{i}\bra{e_{i}}C_{R}]=\sum_{i=1}^{\nu}c_{i}J_{i}\,,$
(10a)
$\displaystyle=[C_{L}\ket{\varphi_{R}}=\sum_{i=1}^{\nu}\widecheck{c}_{i}[C_{L}\ket{\widecheck{e}_{i}}=\sum_{i=1}^{\nu}\widecheck{c}_{i}\widecheck{J}_{i}\,.$
(10b)

#### Integral decompositions

Following [13, 16], we can introduce a scalar product
$\braket{\bullet,\bullet}$ between elements of $H_{\omega}^{n}$ and
$H_{-\omega}^{n}$, called the _intersection number_. With this additional
structure, the coefficients $c_{i}$ can be extracted using the _master
decomposition formula_ [16, 17]

$c_{i}=\sum_{j=1}^{\nu}\braket{\varphi_{L},\widecheck{e}_{j}}(C^{-1})_{ji}\quad\text{with}\quad
C_{ij}=\braket{e_{i},\widecheck{e}_{j}}\,.$ (11)

The computation of intersection numbers in eq. (11) has been the primary focus
of recent work in intersection theory, with significant progress made over the
past few years [42, 19, 43, 44, 45, 29].

In the univariate case ($n=1$), a compact formula for the intersection number
is known and given by

$\displaystyle\braket{\varphi_{L},\varphi_{R}}=\sum_{p\in\mathcal{P}}\text{Res}_{z=p}\left(\psi\varphi_{R}\right)\,,$
(12)

where $\mathcal{P}$ is the set of $\omega$’s poles, and $\psi$ satisfies the
differential equation

$\displaystyle\nabla_{\omega}\psi=\varphi_{L}\,.$ (13)

The calculation of multivariate intersection numbers requires more effort and
there are several strategies [13, 46, 18, 20, 25, 47, 45, 27, 29]. In this
letter, we adopt the ones introduced in [29, 25, 26, 18, 31].

As mentioned earlier, the computation of intersection numbers allows us to
build the _differential equation_ $\bm{\Omega}_{x}$ satisfied by the basis of
master integrals in any external variable $x$

$\displaystyle\partial_{x}J_{i}=[\bm{\Omega}_{x}]_{ij}J_{j}\,.$ (14)

To see this, we note that in the language of twisted cohomology, eq. (14)
translates to

$\displaystyle\partial_{x}\bra{e_{i}}$
$\displaystyle=\bra{\partial_{x}(ue_{i})/u}=[\bm{\Omega}_{x}]_{ij}\bra{e_{j}}\,,$
(15)

which implies

$\displaystyle[\bm{\Omega}_{x}]_{ij}$
$\displaystyle=\braket{\partial_{x}(ue_{i})/u,\widecheck{e}_{k}}[C^{-1}]_{kj}\,.$
(16)

We reiterate that the derivations of eqs. 11 and 16 do not involve solving
(potentially large) systems of linear equations but instead exclusively rely
on the computation of intersection numbers.

#### Fourier integrals in Baikov representation

We consider a generic $\mathrm{D}$-dimensional _Fourier integral_ , which
takes the form

$\displaystyle\tilde{f}(\\{x_{i}\\})=\int
f(\\{q_{i}\\})\prod_{j=1}^{L}\mathrm{e}^{iq_{j}\cdot
x_{j}}{\text{d}\bar{}\hskip 1.00006pt}^{\mathrm{D}}q_{j}\,,$ (17a)
$\displaystyle\text{with measure:}\qquad{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}q_{j}=\frac{\mathrm{d}^{\mathrm{D}}q_{j}}{(2\pi)^{\mathrm{D}/2}}\,.$
(17b)

Eq. (17) is the Fourier transform of the function/distribution $f$ performed
over $L$ internal vectors $\\{q_{i}\\}$. The result is a function of $E$
external vectors $\\{x_{i}\\}$. We denote the set of
$n=\frac{L}{2}(L{+}1){+}LE$ internal scalar products as

$\displaystyle S_{i}=\\{q_{1}^{2},q_{1}\cdot q_{2},\dots,q_{L}^{2},q_{1}\cdot
x_{1},q_{1}\cdot x_{2},\dots,q_{L}\cdot x_{E}\\}\,.$ (18)

To reinterpret the Fourier transform in eq. (17) as a twisted period, we
propose to change variables to the _Baikov variables_ [36, 48]: the procedure
involves a first change of variables from the internal vectors $q_{i}$ to the
internal scalar products $S_{i}$, followed by a second change of variables,

$\displaystyle z_{i}=A_{ij}\,S_{j}+B_{j}\,,$ (19)

where $A_{ij}$ is an $n\times n$ matrix and $B_{j}$ is an $n$-dimensional
vector. Both operations only depend on the external scalar products
$\\{S_{i}\\}$. Once the dust settles, the result reads

$\displaystyle\tilde{f}=\int_{C_{R}}u(\mathbf{z})\,\varphi_{L}(\mathbf{z})\,,$
(20)

where

$C_{R}=\bigcap_{i=1}^{L}\left\\{\frac{\det
G_{\\{q_{i},\ldots,q_{L},y_{1},\ldots,y_{E}\\}}}{\det
G_{\\{q_{i+1},\ldots,q_{L},y_{1},\ldots,y_{E}\\}}}>0\right\\}\,,$ (21)

is the contour of integration. The differential form
$\varphi_{L}(\mathbf{z})=f(\mathbf{z})\,\mathrm{d}^{n}\mathbf{z}$ contains the
function/distribution $f$ we would like to Fourier transform and

$\displaystyle u(\mathbf{z})=\kappa\
\mathrm{e}^{ig(\mathbf{z})}B(\mathbf{z})^{\frac{\mathrm{D}-L-E-1}{2}}\,,$ (22)

is the _twist_. Here, $g(\mathbf{z})$ is always _linear_ in $\mathbf{z}$ and
we define

$\displaystyle G(\\{x\\})$ $\displaystyle=\det[x_{i}\cdot x_{j}]\,,$ (23a)
$\displaystyle B(\mathbf{z})$
$\displaystyle=G(q_{1},\dots,q_{L},y_{1},\dots,y_{E})\,,$ (23b)
$\displaystyle\kappa$
$\displaystyle=\frac{\pi^{\frac{L(1-L-2E)}{4}}G(y_{1},\dots,y_{E})^{\frac{E-\mathrm{D}+1}{2}}}{2^{\frac{L\mathrm{D}}{2}}\det
A\prod_{j=1}^{L}\Gamma\left(\frac{\mathrm{D}{-}L{-}E+j}{2}\right)}\,.$ (23c)

Note that $B\geq 0$ on $C_{R}$. Complementary details regarding the derivation
of eq. (20) can be found in app. A.

Representing a Fourier integral as the twisted period in eq. (20) enables the
use of intersection theory for the construction of differential equation
(c.f., eq. (14)). Thus, the master Fourier integrals $J_{i}$ can be evaluated
by solving the system of differential equations, analogously to Feynman
integrals.

##  III Applications

In this section, we apply the formalism described above to three families of
Fourier integrals arising in various corners of particle physics. An ancillary
Mathematica file (ancillary.m) containing complementary details for each
example is attached to the preprint version of this letter.

Below, $\mathcal{M}=\mathbbm{R}^{1,\mathrm{D}-1}$ denotes the Minkowski
spacetime manifold. Unless specified otherwise, we work in the mostly plus
Lorentzian signature $(-,+,+,...,+)$.

###  III.1 Fourier transform of a scalar propagator

As a first example, we consider the Fourier transform of a massive scalar
Feynman propagator,

$\displaystyle I_{n}=\int_{\mathcal{M}}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}q\frac{\mathrm{e}^{iq\cdot
x}}{(q^{2}+m^{2}-i\varepsilon)^{n}}\,.$ (24)

We work with dimensionless integrals $K_{n}$, defined by

$\displaystyle I_{n}=m^{\mathrm{D}{-}2n}K_{n}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ K_{n}=\int_{\mathcal{M}}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}k\frac{\mathrm{e}^{ik\cdot
v}}{(k^{2}{+}1{-}i\varepsilon)^{n}}\,,$ (25)

where $v=mx$ and $k=q/m$ are both dimensionless vectors. For $K_{n}$, we have
$L=1$ internal vector $\\{k\\}$ and $E=1$ external vector $\\{v\\}$. We define
the $n=2$ integration variables as $z_{1}=k^{2}+1$ and $z_{2}=k\cdot v$. Thus,
in the Baikov representation, this integral takes the form

$\displaystyle K_{n}=\int\frac{\mathrm{d}\mathbf{z}}{z_{1}^{n}}u(\mathbf{z})\
,$ (26)

where the twist is given by

$u(z_{1},z_{2})=\frac{\mathrm{e}^{iz_{2}}\tau^{\frac{2-\mathrm{D}}{2}}}{2^{\frac{\mathrm{D}}{2}}\sqrt{\pi}\Gamma((\mathrm{D}-1)/2)}((z_{1}-1)\tau-
z_{2}^{2})^{\frac{\mathrm{D}-3}{2}}\,,$ (27)

and $\tau=v^{2}$. From eq. (8), the number of master integrals is found to be
$\nu=2$. We choose them as $K_{1}$ and $K_{2}$ and form the basis vector
$\mathbf{K}=(K_{1},K_{2})^{\top}$. From intersection decompositions, we find
that $\mathbf{K}$ obeys the differential equation

$\displaystyle\partial_{\tau}\mathbf{K}$
$\displaystyle=\bm{\Omega}_{\tau}\cdot\mathbf{K}\ ,\quad{\rm
with}\quad\bm{\Omega}_{\tau}=-\begin{pmatrix}\frac{(\mathrm{D}-2)}{2\tau}&\frac{1}{\tau}\\\
\frac{1}{4}&0\end{pmatrix}\,.$ (28)

We can decouple this system of equations into second- and first-order
differential equations for $K_{1}$ and $K_{2}$ respectively, by first changing
the variable $\tau$ in favor of the intermediate variable $t=\sqrt{-\tau}$. We
find

$\displaystyle 0$
$\displaystyle=t\partial^{2}_{t}K_{1}(t)+(\mathrm{D}-1)\partial_{t}K_{1}(t)+tK_{1}(t)\,,$
(29a) $\displaystyle 0$
$\displaystyle=K_{2}(t)+\frac{1}{2}t\partial_{t}K_{1}(t)+\frac{\mathrm{D}-2}{2}K_{1}(t)\,.$
(29b)

Solving eq. (29a) first, followed by eq. (29b), results in (after changing the
variable back to $\tau$)

$\displaystyle K_{1}$
$\displaystyle=\frac{c_{1}J_{\frac{\mathrm{D}-2}{2}}\left(\sqrt{-\tau}\right)+c_{2}Y_{\frac{\mathrm{D}-2}{2}}\left(\sqrt{-\tau}\right)}{(-\tau)^{\frac{\mathrm{D}-2}{4}}}\,,$
(30a) $\displaystyle K_{2}$
$\displaystyle=-\frac{c_{1}J_{\frac{\mathrm{D}-4}{2}}\left(\sqrt{-\tau}\right)+c_{2}Y_{\frac{\mathrm{D}-4}{2}}\left(\sqrt{-\tau}\right)}{2(-\tau)^{\frac{\mathrm{D}-4}{4}}}\,,$
(30b)

where $J_{\nu}(z)$ and $Y_{\nu}(z)$ are the Bessel functions of the first and
second kind respectively.

To fix the boundary constants $c_{1}$ and $c_{2}$ for $\mathbf{K}$, we
consider its boundary values at $\tau=\infty$ and $\tau=0$.

In terms of the four-vector $v$, the boundary point at $\tau=\infty$ is
approached with constant and finite spacelike component $\mathbf{v}$ and
piecewise timelike component

$v^{0}\to\begin{cases}-i\infty&\text{for}\,k^{0}\geq 0\,,\\\
i\infty&\text{for}\,k^{0}<0\,.\end{cases}$ (31)

In this limit, the integrand in eq. (25) is exponentially suppressed

$\begin{split}\mathrm{e}^{ik\cdot
v}&=\mathrm{e}^{i(-k_{0}v^{0}+\mathbf{k}\cdot\mathbf{v})}\\\
&\sim\mathrm{e}^{-k_{0}|v^{0}|}\Theta(k_{0})+\mathrm{e}^{k_{0}|v^{0}|}\Theta(-k_{0})\quad(|v^{0}|\gg
1)\,,\end{split}$ (32)

where $\Theta(z)$ denotes the Heaviside step function. Thus, $K_{n}$ vanishes
in this limit for any $n\in\mathbbm{Z}$. At the level of eq. (30), this
condition enforces

$c_{2}=i\,c_{1}\,.$ (33)

Within dimensional regularization, the integral in eq. (25) can also be
explicitly evaluated at $\tau=0$ (approached with $v^{\mu}=0$). To do so, one
can first perform a Wick rotation on $K_{n}(v^{\mu}=0)$ and then use [49,
eq.(7.85)] to find

$\begin{split}\int_{\mathcal{M}}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}k\frac{1}{(k^{2}{+}1{-}i\varepsilon)^{n}}&=\frac{i\,\Gamma\left(n-\frac{\mathrm{D}}{2}\right)}{2^{\frac{\mathrm{D}}{2}}\Gamma(n)}\,.\end{split}$
(34)

When comparing the result of this calculation at $n=1$ or $n=2$ with eq. (30)
in the vicinity of $\tau=0$, one finds that

$c_{1}=\frac{\pi}{2}\mathrm{e}^{i\pi\mathrm{D}/2}\qquad(\mathrm{D}<2)\,.$ (35)

Putting everything together and substituting back $\tau=v^{2}=m^{2}x^{2}$, we
find

$\displaystyle I_{1}$
$\displaystyle=\frac{\pi\mathrm{e}^{\frac{i\pi\mathrm{D}}{2}}}{2}\left(-\frac{x^{2}}{m^{2}}\right)^{\frac{2-\mathrm{D}}{4}}H_{\frac{\mathrm{D}-2}{2}}^{(1)}\left(m\sqrt{-x^{2}}\right)\,,$
(36a) $\displaystyle I_{2}$
$\displaystyle=-\frac{\pi\mathrm{e}^{\frac{i\pi\mathrm{D}}{2}}}{4}\left(-\frac{x^{2}}{m^{2}}\right)^{\frac{4-\mathrm{D}}{4}}H_{\frac{\mathrm{D}-4}{2}}^{(1)}\left(m\sqrt{-x^{2}}\right)\,,$
(36b)

where $H^{(1)}_{\nu}(z)=J_{\nu}(z)+iY_{\nu}(z)$ is the Hankel function of the
first kind.333We note that since our original integral is manifestly
spherically symmetric after performing Wick rotation, it is not surprising
that its closed form involves Bessel functions, independently of spacetime
dimension.

Despite the dimensionality condition in eq. (35), we find the $\mathrm{D}\to
4$ limit of eq. (36a) to be smooth, and for timelike, spacelike and lightlike
$x$, respectively, to reduce to the known results in [50, eqs. (2.84), (2.85)
and (2.86)], once factors coming from different normalisations and metric
signatures are taken into account (see also [51]).

Moreover, taking the $m\to 0$ limit, we also verify that $I_{1}$ and $I_{2}$
agree with the result for the massless integral

$\int_{\mathcal{M}}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}q\frac{\mathrm{e}^{iq\cdot
x}}{(q^{2}{-}i\varepsilon)^{n}}=\frac{i\,\left(x^{2}\right)^{n-\frac{\mathrm{D}}{2}}\Gamma\left(\frac{\mathrm{D}}{2}{-}n\right)}{2^{\frac{4n-\mathrm{D}}{2}}\Gamma(n)}\,,$
(37)

which can be obtained directly from the result in app. B, after Wick rotating
the left-hand side of eq. (37).

###  III.2 Spectral gravitational waveform

With the prospect of future space-based gravitational wave observatories such
as LISA [52], a pressing theoretical problem is to streamline the path to
higher-precision computations as much as possible. There is a growing interest
in obtaining, from amplitude methods, the one-loop correction to the
gravitational waveform [53, 54, 55, 56, 57, 58, 59, 60], which could
potentially be measured by such observatories.

A standard dictionary used to link waveforms and scattering amplitudes is the
observable-based Kosower–Maybee–O’Connell (KMOC) formalism [57]. In this
framework, the gravitational waveform is characterized as the _expectation
value_ $\text{Exp}_{3}$ of measuring asymptotically in the future a graviton
field in the background of two black holes (modeled here as heavy scalars)
scattering off each other, from and back to the far past (see Fig. 1).

Figure 1: The KMOC momentum space waveform as the expectation value of
measuring a graviton (labelled by $3$) in the background of two Schwarzschild
black holes/heavy scalars (labelled by $1$ and $2$). The prime decorations on
$1^{\prime}$ and $2^{\prime}$ emphasize that the scattering is non forward (no
in and out states are exactly collinear). To obtain, e.g., the spectral
waveform, one needs to Fourier transform this observable to impact parameter.

To establish the connection with scattering amplitudes more precisely, it is
useful to first introduce the generators $a,a^{\dagger},b$ and $b^{\dagger}$
for the algebra of asymptotic measurements. Its existence is physically
motivated by the naive expectation that finite energy excitations in the
“bulk” should decay into a set of _stable_ and _free_ particles at asymptotic
times. This means that the asymptotic states are assumed to be free of any
external forces/fields, so that they do not radiate nor decay.444In the
context of collider physics, particles encountered near the detectors are, of
course, generally _not_ free (their motion is most likely affected by
background fields). In such cases, it is essential to also consider the
scattering of unstable particles (which can decay and radiate). Recent
literature on this subtle subject includes [61, 62, 63].

The annihilation and creation operators in the far past are denoted,
respectively, by $a$ and $a^{\dagger}$, while those in the far future are
similarly denoted by $b$ and $b^{\dagger}$. In what follows, the key property
is that $a$ and $b$ are conjugated to each other with respect to unitary time
evolution : $b=S^{\dagger}aS$ (and, similarly,
$b^{\dagger}=S^{\dagger}a^{\dagger}S$), where $SS^{\dagger}=\mathbbm{1}$. We
refer the reader to [59] for complementary details.

The background in which the scattering occurs is defined by perturbations of
the time-invariant vacuum $|0\rangle$ in the far past

$|12\rangle=a_{2}^{\dagger}a_{1}^{\dagger}|0\rangle\quad\text{and}\quad|1^{\prime}2^{\prime}\rangle=a_{2^{\prime}}^{\dagger}a_{1^{\prime}}^{\dagger}|0\rangle\,.$
(38)

As these two-particle states evolve over time, they can interact non-trivially
with each other (i.e., create and absorb particles). Then, $\text{Exp}_{3}$ is
defined as

$\text{Exp}_{3}={}_{\text{in}}\langle
2^{\prime}1^{\prime}|b_{3}|12\rangle_{\text{in}}\,.$ (39)

The connection between $\text{Exp}_{3}$ and amplitudes is made manifest in two
steps. First, using the relation $b=S^{\dagger}aS$ and inserting a complete
basis of states555The symbol
$\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{X}$
formally denotes an integral-sum over the on-shell phase space of the inserted
state $\ket{X}$ (see [59, eq. (3.5)]).
$\mathbbm{1}=\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{X}|X\rangle\langle
X|$ in eq. (39), we obtain

$\begin{split}\text{Exp}_{3}&={}_{\text{in}}\langle
2^{\prime}1^{\prime}|S^{\dagger}a_{3}S|12\rangle_{\text{in}}\\\
&=\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{X}{}_{\text{in}}\langle
2^{\prime}1^{\prime}|S^{\dagger}|X\rangle\langle
X3|S|12\rangle_{\text{in}}\,.\end{split}$ (40)

Next, plugging the decomposition formula $S=\mathbbm{1}+iT$ of the 4-point
S-matrix (where $T$ is the connected part) into eq. (40), we obtain

$\begin{split}\text{Exp}_{3}&={}_{\text{in}}\langle
32^{\prime}1^{\prime}|iT|12\rangle_{\text{in}}\\\
&\qquad+\operatorname*{\mathchoice{\ooalign{$\displaystyle\sum$\cr$\displaystyle\int$\cr}}{\ooalign{\raisebox{0.14pt}{\scalebox{0.7}{$\textstyle\sum$}}\cr$\textstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}{\ooalign{\raisebox{0.2pt}{\scalebox{0.6}{$\scriptstyle\sum$}}\cr$\scriptstyle\int$\cr}}}_{X}{}_{\text{in}}\langle
2^{\prime}1^{\prime}|T^{\dagger}|X\rangle\langle
X3|T|12\rangle_{\text{in}}\,.\end{split}$ (41)

The first term is a (conventional) time-ordered $3\leftarrow 2$ amplitude,
while the second term is a product of two time-ordered amplitudes glued
together by a
$s_{1^{\prime}2^{\prime}}=-(p_{1^{\prime}}{+}p_{2^{\prime}})^{2}$ channel cut.
In practice, we can therefore compute $\text{Exp}_{3}$ perturbatively,
directly from conventional time-ordered Feynman rules. (Alternatively, it was
recently explained in [64] how to obtain such observables from analytic
continuations of time-ordered scattering amplitudes.)

To eventually streamline comparison with experimental data, one may opt to
work with waveforms expressed as functions of variables other than momenta.
Such quantities can be derived from $\text{Exp}_{3}$ after performing
additional Fourier transforms. For example, obtaining the _spectral_ waveform
requires to Fourier transform $\text{Exp}_{3}$ to impact parameter space.
Similarly, to obtain the _time domain_ waveform, an additional Fourier
transform in the frequency of the outgoing graviton is needed.

It was recently demonstrated in [55] (see also [53]) that to obtain the
leading-order (tree-level) spectral waveform in pure general relativity and
$\mathcal{N}=8$ supergravity, one must perform Fourier transforms of the
form666Note that the exponential has the non-standard sign. This is due to our
use of a signature convention opposite to that in [55, 53].

$\begin{split}\mathcal{I}_{\beta_{1}\beta_{2}}^{\bm{\nu}_{2m}}&=\int_{\mathcal{M}}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}q\frac{\delta(u_{1}{\cdot}q)\delta(u_{2}{\cdot}(q{-}k))q^{\nu_{1}}{\dots}q^{\nu_{2m}}\mathrm{e}^{-iq\cdot
b}}{[q^{2}-i\varepsilon]^{\beta_{1}}[(q{-}k)^{2}-i\varepsilon]^{\beta_{2}}}\,,\end{split}$
(42)

where the $u_{i}$ s denote the (dimensionless) classical velocities of the
heavy external objects, $k$ is the (on-shell: $k^{2}=0$) graviton momentum and
$b$ the impact parameter.

In what follows, we showcase how our method can be applied to obtain new
$\mathrm{D}$-dimensional closed form formulae for eq. (42) in the case where
$\beta_{2}=0$. (Given that the ultra-soft graviton limits of the results
presented below are not smooth, a separate evaluation of eq. (42) for
$k^{\mu}=0$ is provided for completeness in app. C.)

#### Evaluation of $\mathcal{I}_{\beta_{1}0}^{\bm{\nu}_{2m}}$

The tensor reduction of eq. (42) is simple to perform. We find

$\begin{split}\mathcal{I}_{\beta_{1}0}^{\bm{\nu}_{2m}}&=\mathcal{N}\leavevmode\nobreak\
\mathcal{I}_{\beta_{1}-m}\,,\end{split}$ (43)

where we define

$\displaystyle\mathcal{I}_{\alpha}$
$\displaystyle=\int_{\mathcal{M}}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}q\frac{\delta(u_{1}{\cdot}q)\delta(u_{2}{\cdot}(q{-}k))\mathrm{e}^{-iq\cdot
b}}{[q^{2}-i\varepsilon]^{\alpha}}\,,$ (44a) $\displaystyle\mathcal{N}$
$\displaystyle=\frac{(-1)^{m}\big{[}\eta^{\otimes
m}\big{]}^{\\{\bm{\nu}_{2m}\\}}}{\prod_{j=1}^{m}(\mathrm{D}{+}2(m{-}1))}\,.$
(44b)

Above, $\eta^{\otimes m}$ denotes $m$ occurrences of the metric tensor,
$[\dots]^{\\{\bm{\nu}_{2m}\\}}$ stands for the sum over all possible shuffling
of the $2m$ Lorentz indices $\nu_{i}$ in the tensor $[\dots]$.

Defining the kinematics as

$\displaystyle u_{i}^{2}$ $\displaystyle=-1\,,$ $\displaystyle u_{1}\cdot
u_{2}$ $\displaystyle=-y\,,$ $\displaystyle u_{i}\cdot b$ $\displaystyle=0\,,$
(45) $\displaystyle k\cdot b$ $\displaystyle=0\,,$ $\displaystyle k\cdot
u_{1}$ $\displaystyle=-w_{1}\,,$ $\displaystyle k\cdot u_{2}$
$\displaystyle=-w_{2}\,,$

and denominators

$z_{1}=u_{1}\cdot q\,,\leavevmode\nobreak\ z_{2}=u_{2}\cdot
q+w_{2}\,,\leavevmode\nobreak\ z_{3}=q^{2}\,,\leavevmode\nobreak\
z_{4}=-ib\cdot q\,,$ (46)

the integral in Baikov representation reads:

$\mathcal{I}_{\alpha}=\int_{\mathcal{M}}\frac{\mathrm{d}\mathbf{z}}{z_{3}^{\alpha}}u(\mathbf{z})\delta(z_{1})\delta(z_{2})\,.$
(47)

Treating the delta functions appearing as cut propagators, we can rewrite the
integrals by taking the residues at $z_{1}=0$ and $z_{2}=0$ as

$\mathcal{I}_{\alpha}=\int_{\mathcal{M}}\frac{\mathrm{d}\mathbf{z}}{z_{3}^{\alpha}}u(\mathbf{z})\big{|}_{z_{1}=z_{2}=0}\,,$
(48)

where the twist on the cut is defined as

$\begin{split}u(\mathbf{z})\big{|}_{z_{1}=z_{2}=0}&=-i\mathrm{e}^{z_{4}}\frac{\left(-b^{2}\left(y^{2}-1\right)\right)^{2-\frac{\mathrm{D}}{2}}}{2^{\frac{\mathrm{D}}{2}}\pi^{3/2}\Gamma\left(\frac{\mathrm{D}-3}{2}\right)}\\\
&\times\left(b^{2}w_{2}^{2}-\left(y^{2}-1\right)\left(b^{2}z_{3}+z_{4}^{2}\right)\right)^{\frac{\mathrm{D}-5}{2}}\,.\end{split}$
(49)

Rescaling with respect to $|b|$ allows us to work with dimensionless integrals
$\mathcal{K}_{n}$ defined as

$\displaystyle\mathcal{I}_{\alpha}=|b|^{2+2\alpha-\mathrm{D}}\mathcal{K}_{\alpha}\,.$
(50)

We find $\nu=2$ master integrals, which we define as
$\bm{\mathcal{K}}=\\{\mathcal{K}_{1},\mathcal{K}_{2}\\}$.

Next, defining the dimensionless parameter $s=w_{2}^{2}\,b^{2}$, intersection
decompositions yield the system of differential equations

$\displaystyle\partial_{s}\bm{\mathcal{K}}$
$\displaystyle=\bm{\Omega}_{s}\cdot\bm{\mathcal{K}}\
,\quad\quad\bm{\Omega}_{s}=\begin{pmatrix}0&\frac{-1}{y^{2}-1}\\\
-\frac{1}{4s}&\frac{\mathrm{D}-6}{2s}\end{pmatrix}\,,$ (51a)
$\displaystyle\partial_{y}\bm{\mathcal{K}}$
$\displaystyle=\bm{\Omega}_{y}\cdot\bm{\mathcal{K}}\
,\quad\quad\bm{\Omega}_{y}=\begin{pmatrix}\frac{-y}{y^{2}-1}&\frac{2sy}{\left(y^{2}-1\right)^{2}}\\\
\frac{y}{2\left(y^{2}-1\right)}&\frac{(5-\mathrm{D})y}{y^{2}-1}\end{pmatrix}\,.$
(51b)

In terms of the dimensionful master integrals, the solution to eq. (51a) reads

$\displaystyle\mathcal{I}_{1}=\left(b^{2}/w_{2}^{2}\right)^{\frac{4{-}\mathrm{D}}{4}}\left(y^{2}{-}1\right)^{\frac{2{-}\mathrm{D}}{4}}$
(52)
$\displaystyle\times\left[c_{1}I_{2-\frac{\mathrm{D}}{2}}\left(\frac{\sqrt{b^{2}}w_{2}}{\sqrt{y^{2}{-}1}}\right)-c_{2}I_{\frac{\mathrm{D}}{2}-2}\left(\frac{\sqrt{b^{2}}w_{2}}{\sqrt{y^{2}{-}1}}\right)\right]\,,$

and similarly for $\mathcal{I}_{2}$. In this expression, $I_{\nu}(z)$ denotes
the modified Bessel function of the first kind.

We now fix the boundary conditions. Similarly to what happened in eq. (31), in
the neighborhood of $b^{2}=\infty$ approached from

$b^{0}\to\begin{cases}i\infty&\text{for}\,q^{0}\geq 0\,,\\\
-i\infty&\text{for}\,q^{0}<0\,,\end{cases}\quad\text{and
fixed}\quad\mathbf{b}\,,$ (53)

the integrand in eq. (44a) is exponentially suppressed (see eq. (32)) and thus
vanishes. This condition fixes

$c_{2}=c_{1}\,,$ (54)

in eq. (52).

To fix $c_{1}$, we evaluate eq. (44a) at zero impact parameter. In doing so,
it is convenient to commit to the rest frame of $u_{1}$, where

$u_{1}=(1,0,\mathbf{0}_{\perp})\quad\text{and}\quad
u_{2}=(y,\sqrt{y^{2}{-}1},\mathbf{0}_{\perp})\,,$ (55)

such that $u_{1}{\cdot}u_{2}=-y$. In this frame, the integral at $b^{\mu}=0$
becomes

$\begin{split}\mathcal{I}_{\alpha}|_{b=0}&=\int_{\mathcal{M}}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}q\frac{\delta(q^{0})\delta(yq^{0}{-}\sqrt{y^{2}{-}1}q^{1}{-}w_{2})}{[-(q^{0})^{2}+(q^{1})^{2}+(\mathbf{q}_{\perp})^{2}-i\varepsilon]^{\alpha}}\,.\end{split}$
(56)

Integrating out the two delta functions gives the $\alpha^{\text{th}}$ mass
derivative of the Euclidean $(\mathrm{D}{-}2)$-dimensional tadpole of mass
$\mathfrak{m}=w_{2}/\sqrt{y^{2}-1}$

$\displaystyle\mathcal{I}_{\alpha}|_{b=0}$
$\displaystyle=\frac{1}{2\pi\sqrt{y^{2}{-}1}}\partial^{(\alpha)}_{\mathfrak{m}^{2}}\int_{\mathbbm{R}^{\mathrm{D}-2}}\frac{{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}-2}\mathbf{q}_{\perp}}{(\mathbf{q}_{\perp})^{2}+\mathfrak{m}^{2}}$
(57)
$\displaystyle=\frac{\Gamma(\alpha{+}1{-}\mathrm{D}/2)}{2^{\mathrm{D}/2}\pi\Gamma(\alpha)\sqrt{y^{2}{-}1}}\left[\frac{y^{2}{-}1}{w_{2}^{2}}\right]^{\alpha{+}1{-}\mathrm{D}/2}\,.$

Using these results, and by computing the $b\to 0$ limit of eq. (52), $c_{1}$
is fixed to

$c_{1}=\frac{1}{4}\csc\left(\frac{\pi\mathrm{D}}{2}\right)\,.$ (58)

Putting everything together, the final expressions for the dimensionful master
integrals read

$\displaystyle\mathcal{I}_{1}$
$\displaystyle=\frac{\left(b^{2}/w_{2}^{2}\right)^{\frac{4{-}\mathrm{D}}{4}}}{2\pi\left(y^{2}{-}1\right)^{\frac{\mathrm{D}{-}2}{4}}}K_{\frac{4{-}\mathrm{D}}{2}}\left(\frac{\sqrt{b^{2}}w_{2}}{\sqrt{y^{2}{-}1}}\right)\,,$
(59a) $\displaystyle\mathcal{I}_{2}$
$\displaystyle=\frac{\left(b^{2}/w_{2}^{2}\right)^{\frac{6{-}\mathrm{D}}{4}}}{4\pi\left(y^{2}{-}1\right)^{\frac{\mathrm{D}{-}4}{4}}}K_{\frac{6{-}\mathrm{D}}{2}}\left(\frac{\sqrt{b^{2}}w_{2}}{\sqrt{y^{2}-1}}\right)\,.$
(59b)

where $K_{\nu}(z)$ stands for the modified Bessel function of the second kind.
The $\mathrm{D}\to 4$ limit of eq. (59a) is smooth and agrees with [53, eq.
(C16)], once convention differences are taken into account. The
$\mathrm{D}$-dimensional analytic expressions in eq. (59) are new and
constitute one of the main results of this work.

###  III.3 QCD color dipole scattering

A central objective of future electron-ion collision experiments [65] is to
gather data on how the density of partons inside hadrons changes as a function
of energy. It is theorized that, as energy increases, this density becomes
larger and larger until it reaches the so-called _saturation regime_ of QCD,
where non-linear effects from gluon recombination ($gg\to g$) take over soft
bremsstrahlung. This prediction arises in the _color glass condensate_
interpretation of _deep inelastic scattering_ (DIS) [66, 67]. In this
framework, the incoming lepton emits a high-energy virtual photon scattering
from the color potential of the proton. This interaction is then modeled in
the frame where the virtual photon fluctuates into a color dipole (quarkonia)
that scatters eikonally from the color potential (see Fig. 2).

Figure 2: The bare color dipole cross-section discussed in the main text. The
eikonal quark and anti-quark are represented by (red) Wilson lines. The color
potential, which appears as a static two-dimensional pancake in the laboratory
rest frame, models the highly boosted target nucleus and is represented by the
blue region.

At leading order, the total cross-section for the photon polarization states
is obtained by applying the optical theorem to the color dipole forward
amplitude $\mathcal{T}$ [68]

$\begin{split}\sigma^{\gamma^{\ast}p}_{\text{LO}}=2\int\text{d}^{2}\mathbf{b}_{q}\text{d}^{2}\mathbf{b}_{\bar{q}}\text{d}z&|\psi(\Delta_{\perp},q^{2},z)|^{2}\mathcal{T}(\mathbf{b}_{q},\mathbf{b}_{\bar{q}},Y)\,.\end{split}$
(60)

Here, $\psi=\psi_{\gamma^{\ast}\uparrow q\bar{q}}$ denotes the lightcone
wavefunction of the virtual photon of momentum $q$ in the frame where it
decays into a quarkonia dipole of transverse size
$\Delta_{\perp}{=}|\mathbf{b}_{q}{-}\mathbf{b}_{\bar{q}}|$ carrying a fraction
$z$ of the photon’s longitudinal momentum. The forward amplitude $\mathcal{T}$
is related to the correlator of Wilson lines

$\tilde{U}(\mathbf{b}_{q},\mathbf{b}_{\bar{q}},Y)=\frac{1}{N_{\text{c}}}\text{tr}[U(\mathbf{b}_{q},Y)U^{\dagger}(\mathbf{b}_{\bar{q}},Y)]\,,$
(61)

via $\mathcal{T}=1-\tilde{U}$. Here, $N_{\text{c}}$ denotes the number of
colors and each Wilson line $U(\mathbf{b}_{\mathfrak{p}},Y)$ represents a
parton $\mathfrak{p}$ traversing the target at transverse position/impact
parameter $\mathbf{b}_{\mathfrak{p}}$ and rapidity $Y=Y(z)$ (see Fig. 2).

The rapidity evolution of the target color field is described by the Jalilian-
Marian–Iancu–McLerran–Weigert–Leonidov–Kovner (JIMWLK) equation [69]. An
approximate, yet more tractable, large-$N_{\text{c}}$/mean-field description
is given by the Balitsky–Kovchegov (BK) equation [70, 71, 72], which is to
leading order accuracy given by

$\begin{split}&\frac{\partial\tilde{U}(\mathbf{b}_{q},\mathbf{b}_{\bar{q}},Y)}{\partial
Y}=\int\text{d}^{2}\mathbf{b}_{g}\leavevmode\nobreak\
\mathcal{K}_{\text{BK}}^{\text{LO}}(\mathbf{b}_{q},\mathbf{b}_{\bar{q}},\mathbf{b}_{g})\\\
&\quad\times[\tilde{U}(\mathbf{b}_{q},\mathbf{b}_{g},Y)\tilde{U}(\mathbf{b}_{g},\mathbf{b}_{\bar{q}},Y)-\tilde{U}(\mathbf{b}_{q},\mathbf{b}_{\bar{q}},Y)]\,,\end{split}$
(62)

where
$\mathcal{K}_{\text{BK}}^{\text{LO}}(\mathbf{b}_{q},\mathbf{b}_{\bar{q}},\mathbf{b}_{g})=\frac{\alpha_{\text{s}}N_{\text{c}}}{2\pi^{2}}\frac{(\mathbf{b}_{\bar{q}}-\mathbf{b}_{q})^{2}}{(\mathbf{b}_{\bar{q}}-\mathbf{b}_{g})^{2}(\mathbf{b}_{q}-\mathbf{b}_{g})^{2}}$
and $\alpha_{\text{s}}$ is the strong coupling constant.777When considering
high energy QCD in situations involving dilute targets and projectiles, the
partonic Wilson lines in eq. (62) tend to stay close to unity such that
$\tilde{U}\to 1^{-}$ [73]. In such scenarios, $\mathcal{T}$ is a small
parameter and the relevant physics is governed by the linearized version eq.
(62) known as the 1-loop Balitsky–Fadin–Kuraev–Lipatov (BFKL) equation (see
[74, 75] and [68] for a recent review).

The solution to the BK equation predicts an interesting feature of the DIS
total cross-section known as _geometrical scaling_ [76]. This scaling is
indicative of gluon saturation within the hadron in the Regge limit.

Over the past decade, significant efforts have been made to refine the BK
equation by including next-to-leading order corrections and beyond (see, e.g.,
[77, 78, 79, 73]). These refinements involve calculating higher-order
corrections in the strong coupling constant, which can be quite cumbersome. In
particular, as intermediate steps, it is often necessary to trade the
transverse-momentum dependence in expressions in favor of transverse position.
This step necessarily leads to complicated Fourier integrals.

As illustrative examples, we consider two $\mathrm{D}$-dimensional families of
integrals relevant to deep inelastic scattering in the saturation regime

$\displaystyle I^{ij}$
$\displaystyle=\int_{\mathbbm{R}^{2\mathrm{D}}}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}q_{1}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}q_{2}\frac{N_{I}^{ij}(q_{1},q_{2})\mathrm{e}^{i(q_{1}\cdot
x_{1}{+}q_{2}\cdot x_{2})}}{q_{1}^{2}(q_{1}^{2}\tau{+}q_{2}^{2})}\,,$ (63a)
$\displaystyle G^{ij}$
$\displaystyle=\int_{\mathbbm{R}^{2\mathrm{D}}}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}q_{1}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}q_{2}\frac{N_{G}^{ij}(q_{1},q_{2})\mathrm{e}^{i(q_{1}\cdot
x_{1}+q_{2}\cdot x_{2})}}{(q_{1}+q_{2})^{2}(q_{1}^{2}\tau+q_{2}^{2})}\,,$
(63b)

where the $q_{i}\equiv\mathbf{q}^{\perp}_{i}$s are Euclidean, $1\geq\tau>0$
and

$\begin{split}N_{I}^{ij}&=q_{1}^{i}q_{2}^{j}\,,\\\
N_{G}^{ij}&=\delta^{ij}(q_{1}^{2}{-}q_{2}^{2}){-}\frac{2q_{1}^{i}(q_{1}{+}q_{2})^{j}}{u}{+}\frac{2(q_{1}{+}q_{2})^{i}q_{2}^{j}}{u\tau}\,.\end{split}$
(64)

In particular, in $\mathrm{D}=2$, eqs. 63a and 63b appear in the derivation of
the NLO BK equation [77, eq. (42)]. A small subset of diagrams leading to
their appearance is shown in Fig. 3.

In the following, we present new closed-form formulae for eqs. 63a and 63b in
$\mathrm{D}$ dimensions. We anticipate these results to be useful considering
that the $\mathcal{O}(\epsilon)$ correction to the NLO BK equation yields non-
trivial contributions to the NNLO BK equation in the critical
dimension.888More precisely, as prescribed by the “spacelike-timelike
correspondence” [80, 81, 79], at any fixed order in $\alpha_{\text{s}}$, the
non-global log Hamiltonian is independent of $\epsilon$ in dimensional
regularization and equals the BK Hamiltonian in the critical dimension (recall
that non-global observables (e.g., jet shapes) involve incomplete/“non-global”
integrals over final states phase space. These phase-space cuts lead soft
radiation to not be integrated over _all_ angles, resulting in “non-global”
large logarithms that need to be resummed). Concretely,
$\text{\emph{if}}\leavevmode\nobreak\ \leavevmode\nobreak\
H_{\text{BK}}^{(2)}=H_{\text{BK}}^{(2,0)}{+}\epsilon
H_{\text{BK}}^{(2,1)}{+}...\leavevmode\nobreak\ \leavevmode\nobreak\
\text{\emph{then}}\leavevmode\nobreak\ \leavevmode\nobreak\
H_{\text{BK}}^{(3)}{-}H_{\text{NGL}}^{(3)}=H_{\text{BK}}^{(2,1)}\,.$ This
situation bears similarity to the relation between the soft anomalous
dimension $\gamma_{\text{s}}$, which is independent of $\epsilon$, and the
rapidity anomalous dimension, as mentioned in [82, eq. (6.21)].

Figure 3: A small sample of NLO diagrams relevant to the rapidity evolution of
a color dipole in lightcone coordinates (the transverse direction is left
implicit). The first diagram exemplifies a cut self-energy correction, while
the subsequent ones illustrate cut vertices. The Fourier integrals discussed
in the main text emerge as intermediate steps in the computation of NLO BK
observables in $\mathrm{D}$-dimensions.

#### Tensor decomposition

We first perform the tensor decomposition of $I^{ij}$ and $G^{ij}$, namely

$\displaystyle I^{ij}=\sum_{a=1}^{5}I_{a}t_{a}^{ij}\quad\text{and}\quad
G^{ij}=\sum_{a=1}^{5}G_{a}t_{a}^{ij}\,,$ (65)

with basis

$\displaystyle t_{1}^{ij}=x_{1}^{i}x_{1}^{j},\quad t_{2}^{ij}$
$\displaystyle=x_{1}^{i}x_{2}^{j},\quad t_{3}^{ij}=x_{2}^{i}x_{1}^{j},$ (66)
$\displaystyle t_{4}^{ij}=x_{2}^{i}x_{2}^{j},$ $\displaystyle\quad
t_{5}^{ij}=\delta^{ij}\,.$

Here, $I_{a}$ and $G_{a}$ are respectively given by

$\displaystyle\\{I_{a},G_{a}\\}=\sum_{b=1}^{5}(T^{-1})_{ab}\\{K_{a}^{(I)},K^{(G)}_{a}\\}\,,$
(67)

where we have defined

$K_{a}^{(I)}=t_{a}^{ij}I^{ij}\,,\quad K_{a}^{(G)}=t_{a}^{ij}I^{ij}\,,\quad
T_{ab}=t_{a}^{ij}t_{b}^{ij}\,.$ (68)

Therefore, in order to find the tensor decompsoition of these integrals, our
first task is to compute the scalar integrals $K_{a}^{(I)}$ and $K_{a}^{(G)}$
for $a=1,\dots,5$.

#### Change of variables

For $K_{a}^{(I)}$, we make the change of variables

$\displaystyle
q_{1}=\frac{\kappa_{1}}{\sqrt{\tau}\absolutevalue{x_{2}}}\quad\text{and}\quad
q_{2}=\frac{\kappa_{2}}{\absolutevalue{x_{2}}}\,,$ (69)

while for $K_{a}^{(G)}$, we instead consider

$\displaystyle
q_{1}=\frac{\kappa_{1}-\sqrt{\tau}\kappa_{2}}{\sqrt{\tau}\absolutevalue{x_{2}-x_{1}}}\quad\text{and}\quad
q_{2}=\frac{\sqrt{\tau}\kappa_{1}+\kappa_{2}}{\absolutevalue{x_{2}-x_{1}}}\,.$
(70)

From there, we define the dimensionless vectors

$\displaystyle\xi_{1}=\frac{x_{1}}{\sqrt{\tau}\absolutevalue{x_{2}}},$
$\displaystyle\quad\xi_{2}=\frac{x_{2}}{\absolutevalue{x_{2}}}\,,$ (71a)
$\displaystyle\zeta_{1}=\frac{x_{1}+\tau
x_{2}}{\sqrt{\tau}\absolutevalue{x_{2}-x_{1}}},$
$\displaystyle\quad\zeta_{2}=\frac{x_{2}-x_{1}}{\absolutevalue{x_{2}-x_{1}}}\,,$
(71b)

such that both integrals take the universal form

$\displaystyle K_{a}^{(I)}$
$\displaystyle=(\sqrt{\tau}x_{2}^{2})^{2-\mathrm{D}}\mathcal{I}\left(\xi_{1},\xi_{2};\mathcal{N}_{a}^{(I)}\right)\,,$
(72a) $\displaystyle K_{a}^{(G)}$
$\displaystyle=\frac{(1+\tau)^{\mathrm{D}-3}}{(\sqrt{\tau}(x_{2}-x_{1})^{2})^{\mathrm{D}-2}}\mathcal{I}\left(\zeta_{1},\zeta_{2};\mathcal{N}_{a}^{(G)}\right)\,,$
(72b)

where

$\displaystyle\mathcal{I}\left(\eta_{1},\eta_{2};\mathcal{N}\right)=\int_{\mathbbm{R}^{2\mathrm{D}}}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}\kappa_{1}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}\kappa_{2}\
\frac{\mathcal{N}\mathrm{e}^{i(\kappa_{1}\cdot\eta_{1}+\kappa_{2}\cdot\eta_{2})}}{\kappa_{1}^{2}(\kappa_{1}^{2}+\kappa_{2}^{2})}\,,$
(73)

and $\eta_{2}^{2}=1$. The numerators are respectively given by

$\displaystyle\mathcal{N}_{a}^{(I)}$
$\displaystyle=t_{a}^{ij}N_{I}^{ij}\left(\frac{\kappa_{1}}{\sqrt{\tau}\absolutevalue{x_{2}}},\frac{\kappa_{2}}{\absolutevalue{x_{2}}}\right)\,,$
(74a) $\displaystyle\mathcal{N}_{a}^{(G)}$
$\displaystyle=t_{a}^{ij}N_{G}^{ij}\left(\frac{\kappa_{1}-\sqrt{\tau}\kappa_{2}}{\sqrt{\tau}\absolutevalue{x_{2}-x_{1}}},\frac{\sqrt{\tau}\kappa_{1}+\kappa_{2}}{\absolutevalue{x_{2}-x_{1}}}\right)\,.$
(74b)

#### Master integrals of $\mathcal{I}$

In Baikov representation, the family of integrals defined in eq. (73) reads

$\displaystyle\mathcal{I}(\eta_{1},\eta_{2};\mathcal{N})=\int_{C_{R}}\mathrm{d}^{7}\mathbf{z}\frac{u(\mathbf{z})f(\mathbf{z})}{z_{1}z_{2}}\,,$
(75)

where $z_{1}=\kappa_{1}^{2}$, $z_{2}=\kappa_{1}^{2}+\kappa_{2}^{2}$,
$z_{3}=\kappa_{1}\cdot\kappa_{2}$, $z_{4}=\kappa_{1}\cdot\eta_{1}$,
$z_{5}=\kappa_{1}\cdot\eta_{2}$, $z_{6}=\kappa_{2}\cdot\eta_{1}$,
$z_{7}=\kappa_{2}\cdot\eta_{2}$, and only $z_{1}$ and $z_{2}$ can appear as
denominators. The twist is given by

$\displaystyle
u(\mathbf{z})=-\frac{\mathrm{e}^{i(z_{4}+z_{7})}(\eta_{1}^{2}-(\eta_{1}\cdot\eta_{2})^{2})^{\frac{3-\mathrm{D}}{2}}}{2^{\mathrm{D}}\pi^{5/2}\Gamma((\mathrm{D}{-}3)/2)\Gamma(\mathrm{D}/2{-}1)}B(\mathbf{z})^{\frac{\mathrm{D}-5}{2}}\,,$
(76)

where $B(\mathbf{z})$ is given in the ancillary file and
$f(\mathbf{z}(\kappa_{1},\kappa_{2},\eta_{1},\eta_{2}))=\mathcal{N}$. In this
case, there are two master integrals

$\displaystyle J_{1}(\eta_{1},\eta_{2})$
$\displaystyle=\mathcal{I}(\eta_{1},\eta_{2};\kappa_{1}^{2})=\int_{C_{R}}\mathrm{d}^{7}\mathbf{z}\frac{u(\mathbf{z})}{z_{2}}\,,$
(77a) $\displaystyle\frac{J_{2}(\eta_{1},\eta_{2})}{(\mathrm{D}-2)^{2}}$
$\displaystyle=\mathcal{I}(\eta_{1},\eta_{2};1)=\int_{C_{R}}\mathrm{d}^{7}\mathbf{z}\frac{u(\mathbf{z})}{z_{1}z_{2}}\,,$
(77b)

where the factor $(\mathrm{D}-2)^{2}$ is introduced for later convenience.

Using intersection decompositions, the differential equations for the basis
$\mathbf{J}=(J_{1},J_{2})^{T}$ read

$\displaystyle\partial_{\eta_{1}^{2}}\mathbf{J}$
$\displaystyle=\bm{\Omega}_{\eta_{1}^{2}}\cdot\mathbf{J},\quad\bm{\Omega}_{\eta_{1}^{2}}=\begin{pmatrix}\frac{1-\mathrm{D}}{1+\eta_{1}^{2}}&0\\\
\frac{(\mathrm{D}-2)(1+\eta_{1}^{2})}{4\eta_{1}^{2}}&\frac{2-\mathrm{D}}{2\eta_{1}^{2}}\end{pmatrix}\,,$
(78) $\displaystyle\partial_{\eta_{1}\cdot\eta_{2}}\mathbf{J}$
$\displaystyle=\mathbf{0}\,.$

The solutions to this system of differential equations is

$\displaystyle J_{1}(\eta_{1})$
$\displaystyle=J_{10}(1+\eta_{1}^{2})^{1-\mathrm{D}}\,,$ (79a) $\displaystyle
J_{2}(\eta_{1})$
$\displaystyle=J_{20}(\eta_{1}^{2})^{\frac{2{-}\mathrm{D}}{2}}{+}\frac{J_{10}}{2}\mathcal{F}^{(I)}\,,$
(79b)

with
$\mathcal{F}^{(I)}={}_{2}F_{1}\left(\frac{\mathrm{D}{-}2}{2},\mathrm{D}{-}2,\frac{\mathrm{D}}{2},-\eta_{1}^{2}\right)$.

To fix the constants $J_{10}$ and $J_{20}$, we examine the boundary conditions
at $\eta_{1}=0$. For $\mathbf{J}$, the limits evaluate to

$\displaystyle J_{1}(0)$
$\displaystyle=2^{\mathrm{D}-2}\Gamma(\mathrm{D}-1)\,,$ (80a) $\displaystyle
J_{2}(0)$ $\displaystyle=2^{\mathrm{D}-3}\Gamma(\mathrm{D}-1)\,,$ (80b)

as shown in app. D. Thus, since eq. (79) yields $J_{1}(0)=J_{10}$, we
immediately see that

$\displaystyle J_{10}$ $\displaystyle=2^{\mathrm{D}-2}\Gamma(\mathrm{D}-1)\,.$
(81)

To fix $J_{20}$ we use the fact that $J_{2}(0)$ is finite. This necessarily
requires $J_{20}=0$.

#### Computing $I^{ij}$

For $I^{ij}$, we have that $\eta_{1}=\xi_{1}$ and $\eta_{2}=\xi_{2}$. Thus,
the master integrals are

$\displaystyle J_{1}^{(I)}$
$\displaystyle=2^{\mathrm{D}-2}\Gamma(\mathrm{D}-1)\left(1+\frac{x_{1}^{2}}{\tau
x_{2}^{2}}\right)^{1-\mathrm{D}}\,,$ (82a) $\displaystyle J_{2}^{(I)}$
$\displaystyle=2^{\mathrm{D}-3}\Gamma(\mathrm{D}-1)\mathcal{F}^{(I)}\,,$ (82b)

with
$\mathcal{F}^{(I)}={}_{2}F_{1}\left(\frac{\mathrm{D}-2}{2},\mathrm{D}-2,\frac{\mathrm{D}}{2},-\frac{x_{1}^{2}}{\tau
x_{2}^{2}}\right)$.

We can then decompose the $\mathcal{I}$ integrals in eq. (73) in terms of
these master integrals to find

$\displaystyle K^{(I)}_{1}$
$\displaystyle=\frac{\sqrt{\tau}x_{12}\mathcal{J}}{(\sqrt{\tau}x_{2}^{2})^{\mathrm{D}-1}},\quad
K^{(I)}_{2}=\frac{\mathcal{J}}{(\sqrt{\tau}x_{2}^{2})^{\mathrm{D}-2}}\,,$
(83a) $\displaystyle K^{(I)}_{3}$
$\displaystyle=\frac{\sqrt{\tau}x_{12}^{2}\mathcal{J}}{x_{1}^{2}(\sqrt{\tau}x_{2}^{2})^{\mathrm{D}-1}},\quad
K^{(I)}_{4}=\frac{x_{12}\mathcal{J}}{x_{1}^{2}(\sqrt{\tau}x_{2}^{2})^{\mathrm{D}-2}}\,,$
(83b) $\displaystyle K^{(I)}_{5}$
$\displaystyle=\frac{\sqrt{\tau}x_{12}\mathcal{J}}{x_{1}^{2}(\sqrt{\tau}x_{2}^{2})^{\mathrm{D}-1}}\,,$
(83c)

where $x_{12}=x_{1}\cdot x_{2}$ and $\mathcal{J}=\left(\frac{\tau
x_{2}^{2}-x_{1}^{2}}{2\tau x_{2}^{2}}J_{1}^{(I)}-J_{2}^{(I)}\right)$.

Applying the transformation described earlier in eq. (67), we find

$I_{1}=I_{3}=I_{4}=I_{5}=0\quad\text{and}\quad
I_{2}=\frac{\sqrt{\tau}\mathcal{J}}{x_{1}^{2}(\sqrt{\tau}x_{2}^{2})^{\mathrm{D}-1}}\,.$
(84)

The final answer is then

$\displaystyle
I^{ij}=\frac{\sqrt{\tau}\mathcal{J}x_{1}^{i}x_{2}^{j}}{x_{1}^{2}(\sqrt{\tau}x_{2}^{2})^{\mathrm{D}-1}}\,.$
(85)

In $\mathrm{D}=2$, we find

$\displaystyle I^{ij}=-\frac{x_{1}^{i}x_{2}^{j}}{x_{2}^{2}(x_{1}^{2}+\tau
x_{2}^{2})}\,,$ (86)

in agreement with the known result given in [77, eq. (42)]. (Subleading terms
in the $\mathrm{D}=2-2\epsilon$ expansion are provided in app. E for ‘QCD
practitioners’ convenience.)

#### Computing $G^{ij}$

For $G^{ij}$, we have instead that $\eta_{1}=\zeta_{1}$ and
$\eta_{2}=\zeta_{2}$. Thus, the master integrals are

$\displaystyle J_{1}^{(G)}$
$\displaystyle=2^{\mathrm{D}-2}\Gamma(\mathrm{D}-1)\left(\frac{(1+\tau)(x_{1}^{2}+\tau
x_{2}^{2})}{\tau(x_{2}-x_{1})^{2}}\right)^{1-\mathrm{D}}\,,$ (87a)
$\displaystyle J_{2}^{(G)}$
$\displaystyle=2^{\mathrm{D}-3}\Gamma(\mathrm{D}-1)\mathcal{F}^{(G)}\,,$ (87b)

with
$\mathcal{F}^{(G)}={}_{2}F_{1}\left(\frac{\mathrm{D}-2}{2},\mathrm{D}-2,\frac{\mathrm{D}}{2},-\frac{(x_{1}+\tau
x_{2})^{2}}{\tau(x_{2}-x_{1})^{2}}\right)$.

We again decompose our integrals and transform back to our tensor
decomposition in eq. (67) to find

$\displaystyle G_{1}$
$\displaystyle=\frac{(1+\tau)^{\mathrm{D}-1}\left(Y_{1}J_{1}^{(G)}+2\tau(x_{2}-x_{1})^{2}J_{2}^{(G)}\right)}{u\tau^{1+\frac{\mathrm{D}}{2}}(x_{1}+\tau
x_{2})^{2}((x_{2}-x_{1})^{2})^{\mathrm{D}}}\,,$ (88) $\displaystyle G_{2}$
$\displaystyle=(\tau-1)G_{1}\,,\quad G_{3}=0\,,\quad G_{4}=-\tau G_{1}\,,$
$\displaystyle G_{5}$
$\displaystyle=-\frac{(1+\tau)^{\mathrm{D}-2}\left(Y_{2}J_{1}^{(G)}+Y_{3}J_{2}^{(G)}\right)}{\tau^{\frac{\mathrm{D}}{2}}(x_{1}+\tau
x_{2})^{2}((x_{2}-x_{1})^{2})^{\mathrm{D}}}\,,$

where

$\displaystyle Y_{1}$ $\displaystyle=4\tau x_{12}+(\tau-1)(\tau
x_{2}^{2}-x_{1}^{2})\,,$ (89a) $\displaystyle Y_{2}$
$\displaystyle=(1+\tau)^{2}((\tau-1)x_{1}^{2}x_{2}^{2}+(x_{1}^{2}-\tau
x_{2}^{2})x_{12})\,,$ (89b) $\displaystyle Y_{3}$
$\displaystyle=2\tau(x_{2}-x_{1})^{2}((\tau-1)x_{12}+x_{1}^{2}-\tau
x_{2}^{2}))\,.$ (89c)

The final result is then given by eq. (65) and reads

$\displaystyle G^{ij}=\sum_{a=1}^{5}G_{a}t_{a}^{ij}\,.$ (90)

In $\mathrm{D}=2$, we find that

$\displaystyle
G^{ij}=\frac{2\tau(x_{1}{-}x_{2})^{i}x_{2}^{j}{+}2x_{1}^{i}(x_{1}{-}x_{2})^{j}{+}u\tau(x_{2}^{2}{-}x_{1}^{2})\delta^{ij}}{u\tau(x_{2}{-}x_{1})^{2}(x_{1}^{2}{+}\tau
x_{2}^{2})}\,,$ (91)

which is also in agreement with [77]. (Once again, subleading terms in the
$\epsilon$-expansion are provided in app. E.)

Let us close this application by noting that for both eqs. 85 and 90, the
$\epsilon$-expansions (see app. E) involve only polylogarithms with rational
coefficients in the kinematics. This observation resonates with the fact that
the known results for NNLO BK in $\mathcal{N}=4$ super Yang-Mills involve only
(weight three) polylogarithms with similar coefficients [73]. Therefore, one
might naively expect the full QCD result to be constructed from similar
functions.

The differential equation matrices evaluated in applications III.1, III.2 and
III.3 by means of intersection numbers, have also been verified by computing
integration-by-parts identities in momentum space.

##  IV Conclusions

In this letter, we applied intersection theory to dimensionally regularized
Fourier integrals, extending the range of applicability of this technique in
particle physics beyond Feynman integrals.

We showed how to express Fourier integrals as twisted periods by deriving
their Baikov representation. From there, we explained how to derive relations
between Fourier integrals within a given family and how to obtain the system
of differential equations they satisfy using intersection numbers. This
offered a fresh perspective on Fourier calculus, inspired by recent advances
in Feynman calculus.

We showcased our method by computing explicitly relevant instances of Fourier
integrals appearing in the tree-level spectral waveform within pure general
relativity and $\mathcal{N}=8$ supergravity, as well as in color dipole
scattering at next-to-leading order (NLO) in QCD. Each time, we were able to
verify our new (dimensionally regularized) results with existing data from the
literature in specific limits, highlighting the method’s accuracy and
potential. We hope that these developments open up a number of new directions,
some of which are outlined below.

It would be interesting to see to what extent the method described in this
letter proves efficient in tackling the remaining integrals relevant to the
tree-level spectral waveform (i.e., those with $\beta_{2}\neq 0$ in eq. (42)),
and more ambitiously, those pertinent to the one-loop spectral waveform (these
come with an additional loop integration on top of the impact parameter
Fourier transform). To the authors’ knowledge, none of these has been
analytically computed in $\mathrm{D}$ dimensions.

Similarly, it would also be interesting to investigate the extent to which our
approach can be adapted to address the remaining steps in calculating the full
NLO evolution of color dipoles within dimensional regularization (with or
without the incorporation of running coupling corrections [83, 77]; the latter
requiring the computation of Fourier transforms involving additional
logarithms [83]). As stressed in the main text, such NLO results would contain
valuable non-trivial information about higher-order corrections to the
rapidity evolution equation in critical dimension, which is relevant for the
phenomenology of high-energy hadronic and nuclear content.

From the mathematical point of view, the study of the cohomology of Fourier
integrals, poses interesting questions on the type of systems of differential
equations they obey, as well as on the type of functions that are expected to
appear in their solutions, possibly involving new types of transcendental
leading singularities/maximal cuts. Finally, it would be interesting to
explore the extent to which tropical geometry [84, 85, 86, 87, 88] could be
used in the context of Fourier integrals, offering new directions for their
analytic and numerical evaluation.

###### Acknowledgements.

We thank Simon Caron-Huot, Vsevolod Chestnov, Hjalte Frellesvig, Aidan
Herderschee, Manoj K. Mandal, Matteo Pegorin and Henrik Munch for useful
discussions. We thank Ian Balitsky, Sergio Cacciatori, Simon Caron-Huot,
Hjalte Frellesvig, Federico Gasparotto, Sebastian Mizera, and Franziska
Porkert for comments on the manuscript. In particular, we thank Sebastian
Mizera for pointing out to us that in cases where $u$ has essential
singularities (as is manifestly the case at infinity in eq. (20) due to the
exponential term), the theory of twisted cohomology changes [14, 15]. However,
all the examples considered in this letter seem to indicate that the
simplified algorithm for the intersection number in [31] remains valid. We
thank the organizers and participants of the RADCOR23 conference, which
triggered the development of this project. M.G. is grateful to the Institute
for Advanced Study for its hospitality while part of this work was being
completed. M.G.’s work is supported by the National Science and Engineering
Council of Canada (NSERC) and the Canada Research Chair program. The work of
S.S. received support from Gini’s Foundation, in Padova.

##  Appendix A Derivation of the Baikov representation for Fourier integrals

Let us consider the generic Fourier integral given by eq. (17). To derive the
Baikov representation, we first split each momentum $q_{j}$ into components
perpendicular and parallel to the subspace spanned by
$\\{q_{j+1},\dots,q_{L},y_{1},\dots,y_{E}\\}$. The effect on the measure is

$\displaystyle\mathrm{d}^{\mathrm{D}}q_{j}=\mathrm{d}^{E-L-j}q_{j\parallel}\wedge\mathrm{d}^{\mathrm{D}-E+L+j}q_{j\bot}\,.$
(92)

The parallel component gives

$\displaystyle\mathrm{d}^{E-L-j}q_{j\parallel}=\frac{\bigwedge_{k=j+1}^{L}\mathrm{d}(q_{j}\cdot
q_{k})\wedge\mathrm{d}(q_{j}\cdot
y_{E})}{G^{1/2}(q_{j+1},\dots,q_{L},y_{1},\dots,y_{E})}\,,$ (93)

while the perpendicular component gives (in spherical coordinates)

$\displaystyle\mathrm{d}^{N}q_{j\bot}$
$\displaystyle=q_{j\bot}^{N-1}\mathrm{d}\Omega_{N-2}\wedge\mathrm{d}q_{j\bot}$
$\displaystyle=\frac{1}{2}q_{j\bot}^{N-2}\mathrm{d}\Omega_{N-2}\wedge\mathrm{d}(q_{j\bot}^{2})$
(94)
$\displaystyle=\frac{1}{2}\left(\frac{G(q_{j},\dots,y_{E})}{G(q_{j+1},\dots,y_{E})}\right)^{\frac{N-2}{2}}\mathrm{d}\Omega_{N-2}\wedge\mathrm{d}(q_{j\bot}^{2})\,,$

where $N=\mathrm{D}-E+L+j$ and $\mathrm{d}\Omega_{N-2}$ is the measure
accounting for the $N-2$ angular degrees of freedom. Substituting eqs. 93 and
94 into eq. (92), we observe that the Gram determinants simplify, leading to

$\displaystyle\int_{\mathbbm{R}^{L\mathrm{D}}}\prod_{j=1}^{L}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}q_{j}$
$\displaystyle=\frac{\pi^{-\frac{L(L-1)}{4}-\frac{LE}{2}}G(y_{1},\dots,y_{E})^{\frac{-\mathrm{D}+E+1}{2}}}{2^{\frac{L\mathrm{D}}{2}}\prod_{j=1}^{L}\Gamma\left(\frac{\mathrm{D}-L-E+j}{2}\right)}$
$\displaystyle\times\int_{\mathbbm{R}^{n}}G(q_{1},\dots,y_{E})^{\frac{\mathrm{D}-L-E-1}{2}}\prod_{j=1}^{n}\mathrm{d}S_{j}\,.$
(95)

We then conveniently redefine our propagators such that

$\displaystyle z_{i}=A_{ij}S_{j}+B_{j}\,,$ (96)

which introduces a Jacobian for the change of variables, given by $\det A$.
Thus, we find

$\displaystyle\int_{\mathbbm{R}^{L\mathrm{D}}}\prod_{j=1}^{L}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}q_{j}=\kappa\int_{\mathbbm{R}^{n}}\mathrm{d}^{n}\mathbf{z}B(\mathbf{z})^{(\mathrm{D}-L-E-1)/2}\,,$
(97)

where $\kappa$ and $B(\mathbf{z})$ are defined in eqs. 23c and 23b,
respectively.

##  Appendix B Fourier transform of a power-law

From the generalized Schwinger trick, we have

$\begin{split}\int_{\mathbbm{R}^{\mathrm{D}}}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}q\frac{\mathrm{e}^{iq\cdot
x}}{A(q)^{n}}&=\frac{2\pi^{n/2}}{\Gamma(n/2)}\int_{\mathbbm{R}^{\mathrm{D}}}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}q\leavevmode\nobreak\ \mathrm{e}^{iq\cdot x}\\\
&\quad\times\int_{0}^{\infty}\mathrm{d}\xi\leavevmode\nobreak\
\xi^{n-1}\mathrm{e}^{-\pi\xi^{2}A(q)^{2}}\,.\end{split}$ (98)

For $A(q)=q$, and after swapping the order of integration on the right-hand
side, the inner integral becomes much easier to perform. This is because the
Fourier transform of a Gaußian is itself a Gaußian

$\int_{\mathbbm{R}^{\mathrm{D}}}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}q\leavevmode\nobreak\
\mathrm{e}^{-\pi\xi^{2}q^{2}}\mathrm{e}^{iq\cdot
x}=\frac{\xi^{-\mathrm{D}}\mathrm{e}^{-\frac{x^{2}}{4\pi\xi^{2}}}}{(2\pi)^{\mathrm{D}/2}}\,.$
(99)

Plugging this into eq. (98), we find (for $0<n<\mathrm{D}$)

$\begin{split}\eqref{eq:todo0}|_{A=q}&=N\int_{0}^{\infty}\mathrm{d}\xi\xi^{n{-}\mathrm{D}{-}1}\mathrm{e}^{-\frac{x^{2}}{4\pi\xi^{2}}}\\\
&=N\int_{0}^{\infty}\mathrm{d}\zeta\zeta^{(\mathrm{D}-n)-1}\mathrm{e}^{-\frac{x^{2}\zeta^{2}}{4\pi}}\\\
&=\frac{\Gamma((\mathrm{D}-n)/2)}{2^{n-\mathrm{D}/2}\Gamma(n/2)}(x^{2})^{\frac{n-\mathrm{D}}{2}}\,,\end{split}$
(100)

where $N=\frac{2\pi^{n/2}}{(2\pi)^{\mathrm{D}/2}\Gamma(n/2)}$ and
$\zeta=1/\xi$. After a trivial change of variables, eq. (100) is seen to agree
with [89, eq. (A.1)].

##  Appendix C Ultra-soft graviton spectral waveform

As complementary material to the results presented in sec. III.1, we examine
the integral in eq. (42) in the ultra-soft graviton ($k^{\mu}=0$) regime

$\displaystyle I_{\alpha}^{\mu}=\int_{\mathcal{M}}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}q\delta(u_{1}\cdot q)$ $\displaystyle\delta(u_{2}\cdot
q)\frac{\mathrm{e}^{-iq\cdot b}q^{\mu}}{(q^{2})^{\alpha}}\,,$ (101)

using intersection theory and differential equations. This integral, while
solvable through more conventional methods such as Schwinger parameters,
remains a simple enough example to showcase the application of the techniques
used to address the more challenging calculations discussed in the main text.

#### Evaluation of $I_{\alpha}^{\mu}$

We have $L=1$ internal vector $\\{q\\}$ and $E=3$ external vectors
$\\{b,u_{1},u_{2}\\}$, with kinematics $u_{1}^{2}=u_{2}^{2}=-1$, $u_{1}\cdot
u_{2}=-y$, $u_{1}\cdot b=u_{2}\cdot b=0$. We define the $n=4$ integration
variables such that $z_{1}=q^{2}$, $z_{2}=-iq\cdot b$, $z_{3}=u_{1}\cdot q$
and $z_{4}=u_{2}\cdot q$.

By performing a tensor decomposition, the integral takes the form

$\displaystyle
I_{\alpha}^{\mu}=I_{\alpha}^{(1)}b^{\mu}+I_{\alpha}^{(2)}u_{1}^{\mu}+I_{\alpha}^{(3)}u_{2}^{\mu}\,.$
(102)

It is easy to see from eq. (101) that contractions with $u_{1}$ or $u_{2}$
vanish due to the delta functions. Moreover, $u_{1}$ and $u_{2}$ are, by
definition, transverse to the impact parameter $b$. Put together, these
conditions give

$\displaystyle
0=I_{\alpha}^{(2)}+yI_{\alpha}^{(3)}=yI_{\alpha}^{(2)}+I_{\alpha}^{(3)}\,.$
(103)

and thus $I_{\alpha}^{(2)}=I_{\alpha}^{(3)}=0$. This implies

$\displaystyle
I_{\alpha}^{\mu}=I_{\alpha}^{(1)}b^{\mu}=\frac{b_{\nu}I_{\alpha}^{\nu}}{b^{2}}b^{\mu}\,,$
(104)

such that only $I_{\alpha}^{(1)}$ needs to be computed. Its Baikov
representation takes the form

$I_{\alpha}^{(1)}=\frac{1}{b^{2}}\int\mathrm{d}^{4}\mathbf{z}\,\frac{i\,z_{2}}{z_{1}^{\alpha}}u(\mathbf{z})\delta(z_{3})\delta(z_{4})\,,$
(105)

where the twist reads

$\displaystyle
u(\mathbf{z})=-i\frac{\left(-b^{2}\left(y^{2}-1\right)\right)^{2-\frac{\mathrm{D}}{2}}}{2^{\mathrm{D}/2}\pi^{3/2}\Gamma\left(\frac{\mathrm{D}-3}{2}\right)}\times$
(106)
$\displaystyle\left(b^{2}\left(-\left(y^{2}-1\right)z_{1}-2yz_{4}z_{3}+z_{3}^{2}+z_{4}^{2}\right)-\left(y^{2}-1\right)z_{2}^{2}\right){}^{\frac{\mathrm{D}-5}{2}}\,.$

We can then integrate out $z_{3}$ and $z_{4}$ using the delta functions to
find

$\displaystyle
I_{\alpha}^{(1)}=\frac{1}{b^{2}}\int\mathrm{d}z_{1}\mathrm{d}z_{2}\frac{i\,z_{2}}{z_{1}^{\alpha}}u(z_{1},z_{2},0,0)\,.$
(107)

There is only one master integral ($\nu{=}1$), defined as

$\displaystyle J=\int{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}q\delta(u_{1}\cdot q)\delta(u_{2}\cdot
q)\frac{\mathrm{e}^{-iq\cdot b}}{q^{2}}\,.$ (108)

$I_{\alpha}^{(1)}$ can then be decomposed onto this basis as

$I_{\alpha}^{(1)}=\frac{i(4-\mathrm{D})}{b^{2}}J\,.$ (109)

The master integral $J$ obeys the differential equations

$\displaystyle\partial_{b^{2}}J=\frac{4{-}\mathrm{D}}{2b^{2}}J\quad\text{and}\quad\partial_{y}J=\frac{-y}{y^{2}{-}1}J\,.$
(110)

The solution to these equations is easily seen to be

$\displaystyle J=J_{0}\frac{(b^{2})^{(4-\mathrm{D})/2}}{\sqrt{y^{2}-1}}\,.$
(111)

To fix the boundary constant $J_{0}$ a direct evaluation of $J$ in eq. (108)
is performed at $b^{2}=1$: first, the two delta functions in the rest frame of
$u_{2}$ are integrated over. Up to a normalization factor, this gives an
Euclidean $(\mathrm{D}{-}2)$-dimensional tadpole integral with numerator
$\mathrm{e}^{-iq\cdot b}$. After mapping this result back to Schwinger
parameter space, the integral can be easily evaluated for $b^{2}=1$, similarly
to the computation outlined in app. B. This allows us to fix

$J_{0}=-\frac{i2^{\frac{\mathrm{D}}{2}}}{32}\Gamma(\mathrm{D}/2-2)\,.$ (112)

Putting everything together, we obtain

$I_{\alpha}^{\mu}=i(4-\mathrm{D})J_{0}\frac{(b^{2})^{(2-\mathrm{D})/2}}{\sqrt{y^{2}-1}}q^{\mu}\,.$
(113)

##  Appendix D Boundary conditions for application III.3

We examine the $\eta_{1}=0$ limit of the master integrals in III.3

$\displaystyle
J_{1}(\eta_{1})=\mathcal{I}(\eta_{1},\eta_{2};\kappa_{1}^{2}),\quad
J_{2}(\eta_{1})=\mathcal{I}(\eta_{1},\eta_{2};1)\,,$ (114)

with $\mathcal{I}(\eta_{1},\eta_{2},\mathcal{N})$ being the family of
integrals defined in eq. (73). If we set $\eta_{1}=0$, we have for the first
master integral

$\displaystyle J_{1}(0)$
$\displaystyle=\int_{\mathbb{R}^{\mathrm{D}}}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}\kappa_{2}\mathrm{e}^{i\kappa_{2}\cdot\eta_{2}}\int_{\mathbb{R}^{\mathrm{D}}}\frac{{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}\kappa_{1}}{\kappa_{1}^{2}+\kappa_{2}^{2}}$
$\displaystyle=\frac{\Gamma\left(1-\frac{\mathrm{D}}{2}\right)}{2^{\frac{\mathrm{D}}{2}}}\int_{\mathbb{R}^{\mathrm{D}}}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}\kappa_{2}\frac{\mathrm{e}^{i\kappa_{2}\cdot\eta_{2}}}{(\kappa_{2}^{2})^{1-\frac{\mathrm{D}}{2}}}$
$\displaystyle=2^{\mathrm{D}-2}\Gamma(\mathrm{D}-1)\,.$ (115)

For the second master integral we have

$\displaystyle\frac{J_{2}(0)}{(\mathrm{D}-2)^{2}}=\int_{\mathbb{R}^{\mathrm{D}}}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}\kappa_{2}\mathrm{e}^{i\kappa_{2}\cdot\eta_{2}}\int_{\mathbb{R}^{\mathrm{D}}}\frac{{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}\kappa_{1}}{\kappa_{1}^{2}(\kappa_{1}^{2}+\kappa_{2}^{2})}$
$\displaystyle=\int_{\mathbb{R}^{\mathrm{D}}}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}\kappa_{2}\mathrm{e}^{i\kappa_{2}\cdot\eta_{2}}\int_{0}^{1}\mathrm{d}x\int_{\mathbb{R}^{\mathrm{D}}}\frac{{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}\kappa_{1}}{(\kappa_{1}^{2}+\kappa_{2}^{2}x)^{2}}$
$\displaystyle=\frac{\Gamma\left(2-\frac{\mathrm{D}}{2}\right)}{2^{\frac{\mathrm{D}}{2}}}\int_{\mathbb{R}^{\mathrm{D}}}{\text{d}\bar{}\hskip
1.00006pt}^{\mathrm{D}}\kappa_{2}\frac{\mathrm{e}^{i\kappa_{2}\cdot\eta_{2}}}{(\kappa_{2}^{2})^{2-\frac{\mathrm{D}}{2}}}\times\left(\int_{0}^{1}\mathrm{d}xx^{\frac{\mathrm{D}}{2}-2}\right)$
$\displaystyle=\frac{2^{\mathrm{D}-3}\Gamma(\mathrm{D}-1)}{(\mathrm{D}-2)^{2}}\,.$
(116)

In order to find these boundary conditions, we have used the result from [49,
eq.(7.85)], the result from app. B, Feynman parameters.

##  Appendix E $\epsilon$-expansions for application III.3

In this appendix, we provide explicit formulae for the $\epsilon$-expansions
of eqs. 85 and 90 up to $\mathcal{O}(\epsilon^{2})$, namely

$\\{I^{ij},G^{ij}\\}=\sum_{n=0}^{2}\epsilon^{n}\big{\\{}I^{ij}_{(n)},G^{ij}_{(n)}\big{\\}}+\mathcal{O}(\epsilon^{3})\,.$
(117)

The coefficients $I^{ij}_{(n)}$ are recorded explicitly in the text below, as
well as in a computer-friendly form in the ancillary file ancillary.m. The
coefficients $G^{ij}_{(n)}$ are much larger and, therefore, are only recorded
in ancillary.m.

For the former, setting $I^{ij}_{(n)}=x_{1}^{i}x_{2}^{j}I_{(n)}$, we have

$\displaystyle I_{(0)}$ $\displaystyle=\eqref{eq:ILO}\,,$ (118a)
$\displaystyle I_{(1)}$ $\displaystyle=\frac{\Delta_{-}\log\left[\tfrac{\tau
x_{2}^{2}}{\Delta_{+}}\right]{-}2x_{1}^{2}\left(\log\left[\tfrac{\sqrt{\tau}x_{2}^{2}}{2}\right]{+}\gamma_{\text{E}}\right)}{\Delta_{+}x_{1}^{2}x_{2}^{2}}\,,$
(118b) $\displaystyle I_{(2)}$
$\displaystyle=\Big{[}3\Delta_{-}\log\left[\tfrac{\Delta_{+}}{\tau
x_{2}^{2}}\right]\left(\log\left[\tfrac{\Delta_{+}}{\tau
x_{2}^{2}}\right]{+}2\left(\log\left[\tfrac{\sqrt{\tau}x_{2}^{2}}{2}\right]{+}\gamma_{\text{E}}\right)\right)$
$\displaystyle{+}3\Delta_{+}\text{Li}_{2}\left[\tfrac{-x_{1}^{2}}{\tau
x_{2}^{2}}\right]{+}6x_{1}^{2}\left(\log\left[\tfrac{\sqrt{\tau}x_{2}^{2}}{4}\right]{+}2\gamma_{\text{E}}\right)\log\left[\sqrt{\tau}x_{2}^{2}\right]$
$\displaystyle+x_{1}^{2}\left(\pi^{2}+6(\gamma_{\text{E}}{-}\log
2)^{2}\right)\Big{]}/(3\Delta_{+}x_{1}^{2}x_{2}^{2})\,,$ (118c)

where $\Delta_{\pm}=x_{1}^{2}\pm\tau x_{2}^{2}$.

## References

  * Chetyrkin and Tkachov [1981] K. G. Chetyrkin and F. V. Tkachov, Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops, Nucl. Phys. B192, 159 (1981).
  * Laporta [2000] S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15, 5087 (2000), arXiv:hep-ph/0102033 .
  * Anastasiou and Lazopoulos [2004] C. Anastasiou and A. Lazopoulos, AIR: Automatic integral reduction for higher order perturbative calculations, JHEP 07 (1), 046, arXiv:hep-ph/0404258 .
  * von Manteuffel and Studerus [2012] A. von Manteuffel and C. Studerus, Reduze2: Distributed feynman integral reduction (2012), arXiv:1201.4330 [hep-ph] .
  * Lee [2014] R. N. Lee, LiteRed1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser. 523, 012059 (2014), arXiv:1310.1145 [hep-ph] .
  * Georgoudis  _et al._ [2017] A. Georgoudis, K. J. Larsen, and Y. Zhang, Azurite: An algebraic geometry based package for finding bases of loop integrals, Comput. Phys. Commun. 221, 203 (2017), arXiv:1612.04252 [hep-th] .
  * Smirnov and Chuharev [2020] A. V. Smirnov and F. S. Chuharev, FIRE6: Feynman Integral REduction with Modular Arithmetic, Comput. Phys. Commun. 247, 106877 (2020), arXiv:1901.07808 [hep-ph] .
  * Klappert  _et al._ [2021] J. Klappert, F. Lange, P. Maierhöfer, and J. Usovitsch, Integral reduction with Kira2.0 and finite field methods, Comput. Phys. Commun. 266, 108024 (2021), arXiv:2008.06494 [hep-ph] .
  * Klappert and Lange [2020] J. Klappert and F. Lange, Reconstructing rational functions with FireFly, Comput. Phys. Commun. 247, 106951 (2020), arXiv:1904.00009 [cs.SC] .
  * Peraro [2019] T. Peraro, FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs, JHEP 07 (1), 031, arXiv:1905.08019 [hep-ph] .
  * Wu  _et al._ [2024] Z. Wu, J. Boehm, R. Ma, H. Xu, and Y. Zhang, NeatIBP1.0: a package generating small-size integration-by-parts relations for Feynman integrals, Comput. Phys. Commun. 295, 108999 (2024), arXiv:2305.08783 [hep-ph] .
  * Cho and Matsumoto [1995] K. Cho and K. Matsumoto, Intersection theory for twisted cohomologies and twisted Riemann’s period relations I, Nagoya Math. J. 139, 67 (1995).
  * Matsumoto [1998a] K. Matsumoto, Intersection numbers for logarithmic $k$-forms, Osaka Journal of Mathematics 35, 873 (1998a).
  * Matsumoto [1998b] K. Matsumoto, Intersection numbers for $1$-forms associated with confluent hypergeometric functions, Funkcial. Ekvac. 41, 291 (1998b).
  * Majima  _et al._ [2000] H. Majima, K. Matsumoto, and N. Takayama, Quadratic relations for confluent hypergeometric functions, Tohoku Math. J. (2) 52, 489 (2000).
  * Mastrolia and Mizera [2019] P. Mastrolia and S. Mizera, Feynman Integrals and Intersection Theory, JHEP 02 (1), 139, arXiv:1810.03818 [hep-th] .
  * Frellesvig  _et al._ [2019a] H. Frellesvig, F. Gasparotto, S. Laporta, M. K. Mandal, P. Mastrolia, L. Mattiazzi, and S. Mizera, Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers, JHEP 05 (1), 153, arXiv:1901.11510 [hep-ph] .
  * Frellesvig  _et al._ [2019b] H. Frellesvig, F. Gasparotto, M. K. Mandal, P. Mastrolia, L. Mattiazzi, and S. Mizera, Vector Space of Feynman Integrals and Multivariate Intersection Numbers, Phys. Rev. Lett. 123, 201602 (2019b), arXiv:1907.02000 [hep-th] .
  * Frellesvig  _et al._ [2021] H. Frellesvig, F. Gasparotto, S. Laporta, M. K. Mandal, P. Mastrolia, L. Mattiazzi, and S. Mizera, Decomposition of Feynman Integrals by Multivariate Intersection Numbers, JHEP 03 (1), 027, arXiv:2008.04823 [hep-th] .
  * Mizera and Pokraka [2020] S. Mizera and A. Pokraka, From Infinity to Four Dimensions: Higher Residue Pairings and Feynman Integrals, JHEP 02 (1), 159, arXiv:1910.11852 [hep-th] .
  * Weinzierl [2022] S. Weinzierl, Applications of intersection numbers in physics, PoS MA2019, 021 (2022), arXiv:2011.02865 [hep-th] .
  * Chen  _et al._ [2021] J. Chen, X. Jiang, X. Xu, and L. L. Yang, Constructing canonical Feynman integrals with intersection theory, Phys. Lett. B 814, 136085 (2021), arXiv:2008.03045 [hep-th] .
  * Chen  _et al._ [2022] J. Chen, X. Jiang, C. Ma, X. Xu, and L. L. Yang, Baikov representations, intersection theory, and canonical Feynman integrals, JHEP 07 (1), 066, arXiv:2202.08127 [hep-th] .
  * Cacciatori  _et al._ [2021] S. L. Cacciatori, M. Conti, and S. Trevisan, Co-Homology of Differential Forms and Feynman Diagrams, Universe 7, 328 (2021), arXiv:2107.14721 [hep-th] .
  * Caron-Huot and Pokraka [2022] S. Caron-Huot and A. Pokraka, Duals of Feynman Integrals. Part II. Generalized unitarity, JHEP 04 (1), 078, arXiv:2112.00055 [hep-th] .
  * Caron-Huot and Pokraka [2021] S. Caron-Huot and A. Pokraka, Duals of Feynman integrals. Part I. Differential equations, JHEP 12 (1), 045, arXiv:2104.06898 [hep-th] .
  * Giroux and Pokraka [2023] M. Giroux and A. Pokraka, Loop-by-loop differential equations for dual (elliptic) Feynman integrals, JHEP 03 (1), 155, arXiv:2210.09898 [hep-th] .
  * Chen  _et al._ [2023] J. Chen, B. Feng, and L. L. Yang, Intersection theory rules symbology (2023), arXiv:2305.01283 [hep-th] .
  * Fontana and Peraro [2023] G. Fontana and T. Peraro, Reduction to master integrals via intersection numbers and polynomial expansions, JHEP 08 (1), 175, arXiv:2304.14336 [hep-ph] .
  * Duhr and Porkert [2023] C. Duhr and F. Porkert, Feynman integrals in two dimensions and single-valued hypergeometric functions (2023), arXiv:2309.12772 [hep-th] .
  * [31] G. Brunello, V. Chestnov, G. Crisanti, H. Frellesvig, M. K. Mandal, and P. Mastrolia, Intersection Numbers, Polynomial Division and Relative Cohomology, to appear.
  * Cacciatori and Mastrolia [2022] S. L. Cacciatori and P. Mastrolia, Intersection numbers in quantum mechanics and field theory (2022), arXiv:2211.03729 [hep-th] .
  * Gasparotto  _et al._ [2023a] F. Gasparotto, A. Rapakoulias, and S. Weinzierl, Nonperturbative computation of lattice correlation functions by differential equations, Phys. Rev. D 107, 014502 (2023a), arXiv:2210.16052 [hep-th] .
  * Gasparotto  _et al._ [2023b] F. Gasparotto, S. Weinzierl, and X. Xu, Real time lattice correlation functions from differential equations, JHEP 06 (1), 128, arXiv:2305.05447 [hep-th] .
  * De and Pokraka [2023] S. De and A. Pokraka, Cosmology meets cohomology (2023), arXiv:2308.03753 [hep-th] .
  * Baikov [1997] P. A. Baikov, Explicit solutions of the multiloop integral recurrence relations and its application, Nucl. Instrum. Meth. A 389, 347 (1997), arXiv:hep-ph/9611449 .
  * Kotikov [1991] A. V. Kotikov, Differential equation method: The Calculation of $n$-point Feynman diagrams, Phys. Lett. B 267, 123 (1991), [Erratum: Phys.Lett.B 295, 409–409 (1992)].
  * Remiddi [1997] E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110, 1435 (1997), arXiv:hep-th/9711188 .
  * Henn [2013] J. M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110, 251601 (2013), arXiv:1304.1806 [hep-th] .
  * Frellesvig and Papadopoulos [2017] H. Frellesvig and C. G. Papadopoulos, Cuts of Feynman Integrals in Baikov representation, JHEP 04 (1), 083, arXiv:1701.07356 [hep-ph] .
  * Lee and Pomeransky [2013] R. N. Lee and A. A. Pomeransky, Critical points and number of master integrals, JHEP 11 (1), 165, arXiv:1308.6676 [hep-ph] .
  * Weinzierl [2021] S. Weinzierl, On the computation of intersection numbers for twisted cocycles, J. Math. Phys. 62, 072301 (2021), arXiv:2002.01930 [math-ph] .
  * Frellesvig and Mattiazzi [2022] H. A. Frellesvig and L. Mattiazzi, On the Application of Intersection Theory to Feynman Integrals: the univariate case, PoS MA2019, 017 (2022), arXiv:2102.01576 [hep-ph] .
  * Mandal and Gasparotto [2022] M. Mandal and F. Gasparotto, On the Application of Intersection Theory to Feynman Integrals: the multivariate case, PoS 383, 10.22323/1.383.0019 (2022).
  * Chestnov  _et al._ [2023] V. Chestnov, H. Frellesvig, F. Gasparotto, M. K. Mandal, and P. Mastrolia, Intersection numbers from higher-order partial differential equations, JHEP 06 (1), 131, arXiv:2209.01997 [hep-th] .
  * Mizera [2020] S. Mizera,  _Aspects of Scattering Amplitudes and Moduli Space Localization_ , Ph.D. thesis, Princeton, Inst. Advanced Study (2020), arXiv:1906.02099 [hep-th] .
  * Chestnov  _et al._ [2022] V. Chestnov, F. Gasparotto, M. K. Mandal, P. Mastrolia, S. J. Matsubara-Heo, H. J. Munch, and N. Takayama, Macaulay matrix for Feynman integrals: linear relations and intersection numbers, JHEP 09 (1), 187, arXiv:2204.12983 [hep-th] .
  * Grozin [2011] A. G. Grozin, Integration by parts: An Introduction, Int. J. Mod. Phys. A 26, 2807 (2011), arXiv:1104.3993 [hep-ph] .
  * Peskin and Schroeder [1995] M. E. Peskin and D. V. Schroeder,  _An Introduction to quantum field theory_ (Addison-Wesley, Reading, USA, 1995).
  * Huang [1998] K. Huang, Scalar fields, in  _Quantum Field Theory: From Operators to Path Integrals_ (John Wiley & Sons, Ltd, 1998) Chap. 2, pp. 17–39.
  * Cacciatori  _et al._ [2023] S. L. Cacciatori, H. Epstein, and U. Moschella, Banana integrals in configuration space, Nucl. Phys. B 995, 116343 (2023), arXiv:2304.00624 [hep-th] .
  * Seoane  _et al._ [2023] P. A. Seoane _et al._ (LISA), Astrophysics with the Laser Interferometer Space Antenna, Living Rev. Rel. 26, 2 (2023), arXiv:2203.06016 [gr-qc] .
  * Cristofoli  _et al._ [2022] A. Cristofoli, R. Gonzo, D. A. Kosower, and D. O’Connell, Waveforms from amplitudes, Phys. Rev. D 106, 056007 (2022), arXiv:2107.10193 [hep-th] .
  * Damgaard  _et al._ [2023] P. H. Damgaard, E. R. Hansen, L. Planté, and P. Vanhove, The relation between KMOC and worldline formalisms for classical gravity, JHEP 09 (1), 059, arXiv:2306.11454 [hep-th] .
  * Herderschee  _et al._ [2023] A. Herderschee, R. Roiban, and F. Teng, The sub-leading scattering waveform from amplitudes, JHEP 06 (1), 004, arXiv:2303.06112 [hep-th] .
  * Brandhuber  _et al._ [2023] A. Brandhuber, G. R. Brown, G. Chen, S. De Angelis, J. Gowdy, and G. Travaglini, One-loop gravitational bremsstrahlung and waveforms from a heavy-mass effective field theory, JHEP 06 (1), 048, arXiv:2303.06111 [hep-th] .
  * Kosower  _et al._ [2019] D. A. Kosower, B. Maybee, and D. O’Connell, Amplitudes, Observables, and Classical Scattering, JHEP 02 (1), 137, arXiv:1811.10950 [hep-th] .
  * Georgoudis  _et al._ [2023] A. Georgoudis, C. Heissenberg, and I. Vazquez-Holm, Inelastic exponentiation and classical gravitational scattering at one loop, JHEP 06 (1), 126, arXiv:2303.07006 [hep-th] .
  * Caron-Huot  _et al._ [2023a] S. Caron-Huot, M. Giroux, H. S. Hannesdottir, and S. Mizera, What can be measured asymptotically? (2023a), arXiv:2308.02125 [hep-th] .
  * Bini  _et al._ [2023] D. Bini, T. Damour, and A. Geralico, Comparing one-loop gravitational bremsstrahlung amplitudes to the multipolar-post-minkowskian waveform (2023), arXiv:2309.14925 [gr-qc] .
  * Hannesdottir and Schwartz [2023] H. Hannesdottir and M. D. Schwartz, Finite $S$ matrix, Phys. Rev. D 107, L021701 (2023), arXiv:1906.03271 [hep-th] .
  * Hannesdottir and Schwartz [2020] H. Hannesdottir and M. D. Schwartz, $S$ -Matrix for massless particles, Phys. Rev. D 101, 105001 (2020), arXiv:1911.06821 [hep-th] .
  * Hannesdottir and Mizera [2023] H. S. Hannesdottir and S. Mizera,  _What is the i $\varepsilon$ for the S-matrix?_, SpringerBriefs in Physics (Springer, 2023) arXiv:2204.02988 [hep-th] .
  * Caron-Huot  _et al._ [2023b] S. Caron-Huot, M. Giroux, H. S. Hannesdottir, and S. Mizera, Crossing beyond scattering amplitudes (2023b), arXiv:2310.12199 [hep-th] .
  * Accardi  _et al._ [2016] A. Accardi _et al._ , Electron Ion Collider: The Next QCD Frontier: Understanding the glue that binds us all, Eur. Phys. J. A 52, 268 (2016), arXiv:1212.1701 [nucl-ex] .
  * Iancu  _et al._ [2001] E. Iancu, A. Leonidov, and L. D. McLerran, Nonlinear gluon evolution in the color glass condensate. 1., Nucl. Phys. A 692, 583 (2001), arXiv:hep-ph/0011241 .
  * Ferreiro  _et al._ [2002] E. Ferreiro, E. Iancu, A. Leonidov, and L. McLerran, Nonlinear gluon evolution in the color glass condensate. 2., Nucl. Phys. A 703, 489 (2002), arXiv:hep-ph/0109115 .
  * Gelis [2013] F. Gelis, Color Glass Condensate and Glasma, Int. J. Mod. Phys. A 28, 1330001 (2013), arXiv:1211.3327 [hep-ph] .
  * Mueller [2001] A. H. Mueller, A Simple derivation of the JIMWLK equation, Phys. Lett. B 523, 243 (2001), arXiv:hep-ph/0110169 .
  * Balitsky [1996] I. Balitsky, Operator expansion for high-energy scattering, Nucl. Phys. B 463, 99 (1996), arXiv:hep-ph/9509348 .
  * Kovchegov [1999] Y. V. Kovchegov, Small-x $F_{2}$ structure function of a nucleus including multiple pomeron exchanges, Phys. Rev. D 60, 034008 (1999), arXiv:hep-ph/9901281 .
  * Balitsky [2001] I. Balitsky, High eneergy QCD and wilson lines, in  _At The Frontier of Particle Physics_ (Wolrd Scientific, 2001) pp. 1237–1342.
  * Caron-Huot and Herranen [2018] S. Caron-Huot and M. Herranen, High-energy evolution to three loops, JHEP 02 (1), 058, arXiv:1604.07417 [hep-ph] .
  * Fadin  _et al._ [1975] V. S. Fadin, E. A. Kuraev, and L. N. Lipatov, On the Pomeranchuk Singularity in Asymptotically Free Theories, Phys. Lett. B 60, 50 (1975).
  * Balitsky and Lipatov [1978] I. I. Balitsky and L. N. Lipatov, The Pomeranchuk Singularity in Quantum Chromodynamics, Sov. J. Nucl. Phys. 28, 822 (1978).
  * Iancu  _et al._ [2002] E. Iancu, K. Itakura, and L. McLerran, Geometric scaling above the saturation scale, Nucl. Phys. A 708, 327 (2002), arXiv:hep-ph/0203137 .
  * Balitsky and Chirilli [2008] I. Balitsky and G. A. Chirilli, Next-to-leading order evolution of color dipoles, Phys. Rev. D 77, 014019 (2008), arXiv:0710.4330 [hep-ph] .
  * Balitsky and Chirilli [2009] I. Balitsky and G. A. Chirilli, NLO evolution of color dipoles in $\mathcal{N}=4$ SYM, Nucl. Phys. B 822, 45 (2009), arXiv:0903.5326 [hep-ph] .
  * Caron-Huot [2018] S. Caron-Huot, Resummation of non-global logarithms and the BFKL equation, JHEP 03 (1), 036, arXiv:1501.03754 [hep-ph] .
  * Weigert [2004] H. Weigert, Nonglobal jet evolution at finite $N_{\text{c}}$, Nucl. Phys. B 685, 321 (2004), arXiv:hep-ph/0312050 .
  * Hatta [2008] Y. Hatta, Relating $\mathrm{e}^{+}/\mathrm{e}^{-}$ annihilation to high energy scattering at weak and strong coupling, JHEP 11 (1), 057, arXiv:0810.0889 [hep-ph] .
  * Vladimirov [2018] A. Vladimirov, Structure of rapidity divergences in multi-parton scattering soft factors, JHEP 04 (1), 045, arXiv:1707.07606 [hep-ph] .
  * Kovchegov and Weigert [2007] Y. V. Kovchegov and H. Weigert, Triumvirate of Running Couplings in Small-x Evolution, Nucl. Phys. A 784, 188 (2007), arXiv:hep-ph/0609090 .
  * Panzer [2022] E. Panzer, Hepp’s bound for feynman graphs and matroids, Annales de l’Institut Henri Poincaré D 10, 31 (2022).
  * Arkani-Hamed  _et al._ [2022] N. Arkani-Hamed, A. Hillman, and S. Mizera, Feynman polytopes and the tropical geometry of UV and IR divergences, Phys. Rev. D 105, 125013 (2022), arXiv:2202.12296 [hep-th] .
  * Borinsky  _et al._ [2023] M. Borinsky, H. J. Munch, and F. Tellander, Tropical Feynman integration in the Minkowski regime, Comput. Phys. Commun. 292, 108874 (2023), arXiv:2302.08955 [hep-ph] .
  * Hillman [2023] A. Hillman, A subtraction scheme for feynman integrals (2023), arXiv:2311.03439 [hep-th] .
  * Arkani-Hamed  _et al._ [2023] N. Arkani-Hamed, H. Frost, G. Salvatori, P.-G. Plamondon, and H. Thomas, All loop scattering for all multiplicity (2023), arXiv:2311.09284 [hep-th] .
  * Caron-Huot [2023] S. Caron-Huot, Holographic cameras: an eye for the bulk, JHEP 03 (1), 047, arXiv:2211.11791 [hep-th] .