††institutetext:  aPhysik-Institut, Universität Zürich, Winterthurerstrasse
190, 8057 Zürich, Switzerland
bInstitute for Particle Physics Phenomenology, Department of Physics,
University of Durham, Durham, DH1 3LE, UK

# The colourful antenna subtraction method

T. Gehrmanna, E.W.N. Gloverb, M. Marcolia,b<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present a general subtraction scheme for NNLO calculations in massless QCD:
the colourful antenna subtraction method. It is a reformulation of the antenna
subtraction approach designed to address some of the limitations of the
traditional framework, especially aiming at high-multiplicity processes. In
the context of the new formalism, structures needed to locally subtract the
infrared-divergent behaviour of real emission corrections are systematically
inferred from virtual subtraction terms, relying on the cancellation of
infrared singularities and on the correspondence between integrated and
unintegrated antenna functions. We illustrate in detail how the colourful
antenna subtraction method works up to NNLO. The algorithm is particularly
suited to be fully automated for the generation of NNLO subtraction terms for
generic processes. We employ the new formalism to assemble the subtraction
terms required for the calculation of the NNLO correction to hadronic three-
jet production and describe their validation procedure.

###### Keywords:

QCD, NNLO Computations, Antenna subtraction

††preprint: IPPP/23/63
ZU-TH 70/23

##  1 Introduction

The successful interpretation of particle collider data demands high-precision
theoretical predictions for Standard Model cross sections. Quantum
Chromodynamics (QCD) effects dominate scattering events at particle colliders
and need to be described accurately to reach the desired precision.
Computations in QCD for high-energy collisions are performed by power series
expansion in the strong coupling constant $\alpha_{s}$. According to how many
terms are computed in this expansion, one obtains increasingly more accurate
predictions, denominated Leading-Order (LO), Next-to-Leading-Order (NLO),
Next-to-Next-to-Leading-Order (NNLO), and so forth. The complexity grows with
the accuracy: the emergence of infrared divergences in theoretical
calculations poses an obstacle to the computation of higher-order terms in the
$\alpha_{s}$ expansion.

LO predictions only provide a naive estimate and are not accurate enough for
precision experiments. More precise calculations at NLO are required for any
precision analysis. For more than two decades, fully general techniques to
reach NLO accuracy have been available Catani:1996jh ; Frixione:1995ms , as
well as public multi-purpose computational frameworks for cross-section
calculations Alwall:2014hca ; Sherpa:2019gpd ; Bellm:2019zci ; Alioli:2010xd .
However, for some observables NLO predictions are still not precise enough and
NNLO calculations are needed to approach the percent-level accuracy goal.
Several methods have been proposed to compute NNLO corrections Gehrmann-
DeRidder:2005btv ; Currie:2013vh ; DelDuca:2016ily ; Catani:2007vq ;
Czakon:2010td ; Czakon:2014oma ; Gaunt:2015pea ; Cacciari:2015jma ;
Caola:2017dug ; Magnea:2018hab ; Bertolotti:2022ohq ; Devoto:2023rpv  and
NNLO-accurate predictions for essentially all $2\to 2$ processes at hadron
colliders are nowadays available. From $2020$ onwards, NNLO calculations for
$2\to 3$ processes have started to appear Chawdhry:2019bji ; Kallweit:2020gcp
; Chawdhry:2021hkp ; Czakon:2021mjy ; Chen:2022ktf ; Hartanto:2022qhh ;
Alvarez:2023fhi ; Badger:2023mgf ; Catani:2022mfv ; Buonocore:2022pqq ;
Buonocore:2023ljm , thanks to the calculation of two-loop five-point
amplitudes Chicherin:2017dob ; Gehrmann:2018yef ; Chicherin:2020oor ;
Abreu:2020cwb ; Abreu:2021oya ; Abreu:2021asb ; Badger:2021ega ;
Badger:2021imn ; Abreu:2023rco . These results represent the current state-of-
the-art for NNLO QCD corrections for LHC processes.

The currently available results at NNLO have been obtained proceeding on a
case-by-case basis, usually demanding a considerable amount of work to extend
the existing formalisms to new classes of processes. At NLO, the development
of process-independent and automatable techniques led to the industrialization
of NLO-accurate predictions. Nowadays, NLO computations are systematically
performed using public codes for any precision analysis at the LHC, thereby
raising the standards of particle phenomenology. It is foreseeable that a
similar revolution could occur at NNLO if general frameworks and tools for
NNLO calculations become available.

In this paper we formulate a process-independent approach to perform NNLO
calculations in QCD: the colourful antenna subtraction method, which was
originally presented in Chen:2022ktf  for gluon scattering. The traditional
antenna subtraction method Gehrmann-DeRidder:2005btv ; Currie:2013vh  makes
use of antenna functions Gehrmann-DeRidder:2004ttg ; Gehrmann-DeRidder:2005alt
; Gehrmann-DeRidder:2005svg  to assemble suitable counterterms to deal with
infrared divergences up to NNLO. These antenna functions are based on colour-
ordered matrix elements describing radiation from processes with two coloured
particles: $\gamma^{*}\to q\bar{q}$, $\tilde{\chi}\to\tilde{g}g$ and $H\to
gg$, covering the cases where the coloured particles are massless quarks and
gluons. It has been successfully applied to compute the NNLO correction to a
series of phenomenologically relevant processes at electron-positron,
electron-proton and hadron colliders Gehrmann-DeRidder:2007foh ;
Currie:2017tpe ; Currie:2017eqf ; Chen:2022clm ; Chen:2022ktf ; Chen:2019zmr ;
Gehrmann:2020oec ; Gehrmann-DeRidder:2015wbt ; Gehrmann-DeRidder:2016cdi ;
Gehrmann-DeRidder:2017mvr ; Gauld:2020deh ; Gauld:2023zlv ; Chen:2016zka ;
Cruz-Martinez:2018rod ; Gauld:2019yng ; Gauld:2021ule . However, it has been
gradually extended throughout the past fifteen years to deal with new
processes, demanding a substantial amount of time and manpower for each new
process. Despite being quite flexible, it presents some intrinsic limitations.
In particular, as many of the proposed NNLO subtraction schemes, it scales
poorly with the number of coloured particles involved in the process and much
of its simplicity is lost when dealing with contributions beyond the so-called
leading-colour approximation Currie:2013vh  where one cannot directly rely on
colour connections of squared amplitudes as the guiding principle for the
construction of the subtraction terms, especially at high multiplicities.

The colourful antenna subtraction method derived here is a reformulation of
the antenna subtraction approach designed to systematise the construction of
the real-emission subtraction term. The main idea behind it is to exploit the
predictability of the singularity structure of virtual amplitudes in colour
space to straightforwardly construct the virtual subtraction terms in a
completely general way. This helps overcome the aforementioned limitations of
the traditional antenna subtraction scheme in two ways. First, the complete
structure of colour correlations among external QCD particles is retained by
working in colour-space, making the construction of subtraction terms for all
colour layers straightforward. This improves on the traditional approach using
colour-decomposed antenna functions, which works best only for the leading-
colour contribution. Secondly, we exploit the correspondence between
integrated and unintegrated antenna functions and the known infrared structure
of the virtual subtraction terms to infer the subtraction terms for the real
emission corrections. This strategy goes in the opposite direction with
respect to the typical procedure implemented in infrared subtraction schemes,
in which the real emission subtraction term is constructed first, and then
analytically integrated to yield the virtual subtraction term. Nonetheless it
is firmly supported by the same principle: the cancellation of infrared
singularities between real and virtual corrections. The construction of local
subtraction terms for real radiation starting from the infrared factorization
properties of virtual amplitudes has also been investigated in Magnea:2018ebr
.

The colourful antenna approach is fully compatible with other (parallel)
attempts to refine the antenna subtraction scheme by building the antenna
functions directly from a list of desired infrared limits Braun-White:2023sgd
; Braun-White:2023zwd ; Fox:2023bma . This so-called designer antenna approach
avoids the need to decompose the antenna functions into sub-antennae, it
streamlines (and reduces the size of) the subtraction terms by avoiding the
introduction of spurious limits that are inevitably present in the matrix-
element based antennae when applied to processes with multiple coloured
particles. So far, designer antenna and their integrated counterparts have
been derived for use in electron-positron annihilation, electron-hadron and
hadron-hadron collisions at NLO Braun-White:2023sgd ; Fox:2023bma , and at
NNLO for electron-positron annihilation Braun-White:2023sgd ; Braun-
White:2023zwd . In the future, one can imagine an optimal approach in which
the colourful antenna subtraction method proposed here is supplemented by the
use of designer antenna functions, and vice versa.

The outline of the paper is as follows. In Section 2 we give a brief overview
of the basic ingredients and structures of the traditional antenna subtraction
method. Section 3, after reviewing the colour space formalism and the infrared
singularity structure of QCD virtual amplitudes, introduces integrated dipoles
as operators in colour space, which play a fundamental role in the formalism
we propose. In Sections 4 and 5 we present the colourful antenna subtraction
method, respectively at NLO and NNLO, supplemented by the appendices. The
application of the new colourful formalism for the construction of NNLO
subtraction terms for three-jet production at hadron colliders is then
discussed in Section 6. Concluding remarks and an outlook are presented in
Section 7.

##  2 Basics of antenna subtraction

The antenna subtraction method has been developed in detail in Gehrmann-
DeRidder:2005btv ; Gehrmann-DeRidder:2007foh ; Currie:2013vh . We summarize
its essential features in the following.

The core ingredients of the antenna subtraction method are the antenna
functions. These objects capture the singular behaviour of matrix elements in
the presence of unresolved emissions and are directly extracted from colour-
ordered squared matrix elements for simple processes involving the radiation
of QCD particles between a pair of hard radiators Gehrmann-DeRidder:2004ttg ;
Gehrmann-DeRidder:2005alt ; Gehrmann-DeRidder:2005svg . According to the
partonic species of the hard radiators, antenna functions are classified as
quark-antiquark, quark-gluon and gluon-gluon antenna functions.

An unintegrated $\ell$-loop antenna function is a function of $n$ partonic
momenta, a subset $I$ of which are in the initial state, generically indicated
as:

$X_{n,\left\\{i\right\\}_{i\in I}}^{\ell}(j_{1},\dots,j_{n}).$ (1)

Configurations with zero, one or two partons in the initial state are
respectively denoted by final-final (FF), initial-final (IF) and initial-
initial (II) antenna functions. Different partonic arrangements are described
by different antenna functions, characterized with a specific upper-case
letter replacing the generic $X$. At one loop, $\ell=1$, antenna functions
containing a closed fermion loop are indicated by a hat
($\widehat{\phantom{X}}$), and antenna functions corresponding to the
subleading-colour virtual configurations are indicated with a tilde
($\widetilde{\phantom{X}}$). Some of the antenna functions are further split
into so-called sub-antennae. This is done for several reasons such as the
identification of two hard radiators, the separation of different reduced
matrix elements or the separation of different momentum mappings. Typically
sub-antennae have a more complicated structure than the full result. Sub-
antennae are denoted with the same notation as full ones, but with lower-case
letters or with a dedicated subscript.

Exploiting the exact factorization of the phase space Gehrmann-
DeRidder:2005alt ; Daleo:2006xa ; Daleo:2009yj ; Boughezal:2010mc ; Gehrmann-
DeRidder:2012too , antenna functions can be analytically integrated in
dimensional regularization to obtain their integrated counterparts. Integrated
antenna functions are denoted as:

$\mathcal{X}_{n,\left\\{i\right\\}_{i\in I}}^{\ell}(s_{j_{1}\dots j_{n}}),$
(2)

where $s_{j_{q}\dots j_{n}}$ is the invariant mass of the considered partonic
system:

$s_{j_{1}\dots j_{n}}=\left(\sum_{j=1}^{n}\lambda_{j}p_{j}\right)^{2},$ (3)

with $\lambda_{j}=+1$ for final-state partons and $\lambda_{j}=-1$ for
initial-state ones.

All the integrated and unintegrated antenna functions and sub-antennae used in
this paper were derived in Gehrmann-DeRidder:2005btv ; Gehrmann-
DeRidder:2007foh ; Glover:2010kwr ; Daleo:2006xa ; Daleo:2009yj ;
Boughezal:2010mc ; Gehrmann:2011wi ; Gehrmann-DeRidder:2012too .

###  2.1 Structure of antenna subtraction

In the following we review the infrastructure of the antenna subtraction
method and of the subtraction counterterms which are constructed to remove the
infrared singularities in different layers of a calculation. The detailed
construction of the subtraction counterterms is described in Section 4 and 5.

####  2.1.1 Subtraction at NLO

The NLO QCD correction to an $n$-jet partonic cross section with parton
species $a$ and $b$ in the initial state is given by:

$\text{d}{\hat{\sigma}_{ab,\mathrm{NLO}}}=\int_{n}\left(\text{d}{\hat{\sigma}^{V}_{ab,\mathrm{NLO}}}+\text{d}{\hat{\sigma}^{MF}_{ab,\mathrm{NLO}}}\right)+\int_{n+1}\text{d}{\hat{\sigma}^{R}_{ab,\mathrm{NLO}}},$
(4)

where $\text{d}{\hat{\sigma}^{V}_{ab,\mathrm{NLO}}}$ and
$\text{d}{\hat{\sigma}^{R}_{ab,\mathrm{NLO}}}$ respectively represent the
virtual and real corrections, while
$\text{d}{\hat{\sigma}^{MF}_{ab,\mathrm{NLO}}}$ is the NLO mass factorization
counterterm. The symbol $\int_{n}$ indicates an integration over a
$n$-particle phase space. The NLO cross section in (4), despite being well
defined and finite, is not suitable for numerical integration in this form,
due to the emergence of infrared singularities. These cancel in the final
result, but a proper subtraction procedure is needed to separately remove the
singularities in the real and virtual corrections and make both integrals in
(4) computable with numerical methods.

In the context of antenna subtraction, this is achieved constructing a real
subtraction term $\text{d}{\hat{\sigma}^{S}_{ab,\mathrm{NLO}}}$ Currie:2013vh
, which locally removes the singular behaviour of
$\text{d}{\hat{\sigma}^{R}_{ab,\mathrm{NLO}}}$ in the infrared (IR) limits and
can be analytically integrated over the phase space of the unresolved
radiation. For the regularization of the IR singularities, we rely on
dimensional regularization with the customary choice $d=4-2\epsilon$. After
this integration, $\text{d}{\hat{\sigma}^{S}_{ab,\mathrm{NLO}}}$ serves as
ingredient to the virtual subtraction term
$\text{d}{\hat{\sigma}^{T}_{ab,\mathrm{NLO}}}$, which cancels the explicit
$\epsilon$-poles of the virtual correction and contains the mass factorization
contribution. The NLO cross section can then be reformulated as Currie:2013vh
:

$\text{d}{\hat{\sigma}_{ab,\mathrm{NLO}}}=\int_{n}\left[\text{d}{\hat{\sigma}^{V}_{ab,\mathrm{NLO}}}-\text{d}{\hat{\sigma}^{T}_{ab,\mathrm{NLO}}}\right]+\int_{n+1}\left[\text{d}{\hat{\sigma}^{R}_{ab,\mathrm{NLO}}}-\text{d}{\hat{\sigma}^{S}_{ab,\mathrm{NLO}}}\right],$
(5)

with

$\text{d}{\hat{\sigma}^{T}_{ab,\mathrm{NLO}}}=-\int_{1}\text{d}{\hat{\sigma}^{S}_{ab,\mathrm{NLO}}}-\text{d}{\hat{\sigma}^{MF}_{ab,\mathrm{NLO}}}.$
(6)

At NLO the mass factorization counterterm is given by:

$\text{d}{\hat{\sigma}^{MF}_{ab,\mathrm{NLO}}}=-\left(\dfrac{\alpha_{s}\overline{C}(\epsilon)}{2\pi}\right)\sum_{c,d}\int{\dfrac{\text{d}x_{1}}{x_{1}}}{\dfrac{\text{d}x_{2}}{x_{2}}}\Gamma^{(1)}_{ab;cd}(x_{1},x_{2})\,\text{d}{\hat{\sigma}_{cd,\mathrm{LO}}},$
(7)

where $x_{1}$ and $x_{2}$ represent the momentum fractions transferred to the
hard process, while $\Gamma^{(1)}_{ab;cd}\left(x_{1},x_{2}\right)$ denotes the
NLO mass factorization kernel:

$\Gamma^{(1)}_{ab;cd}\left(x_{1},x_{2}\right)=\Gamma^{(1)}_{ca,\text{full}}\left(x_{1}\right)\delta_{db}\delta(1-x_{2})+\Gamma^{(1)}_{db,\text{full}}\left(x_{1}\right)\delta_{ca}\delta(1-x_{1}).$
(8)

which can be organized into different terms corresponding to different colour
factors:

$\displaystyle\Gamma^{(1)}_{qq,\text{full}}\left(x\right)$ $\displaystyle=$
$\displaystyle\left(\dfrac{N_{c}^{2}-1}{N_{c}}\right)\Gamma^{(1)}_{qq}\left(x\right),$
(9) $\displaystyle\Gamma^{(1)}_{gq,\text{full}}\left(x\right)$
$\displaystyle=$
$\displaystyle\left(\dfrac{N_{c}^{2}-1}{N_{c}}\right)\Gamma^{(1)}_{gq}\left(x\right),$
(10) $\displaystyle\Gamma^{(1)}_{qg,\text{full}}\left(x\right)$
$\displaystyle=$ $\displaystyle\Gamma^{(1)}_{qg}\left(x\right),$ (11)
$\displaystyle\Gamma^{(1)}_{gg,\text{full}}\left(x\right)$ $\displaystyle=$
$\displaystyle
N_{c}\Gamma^{(1)}_{gg}\left(x\right)+N_{f}\widehat{\Gamma}^{(1)}_{gg}\left({x}\right).$
(12)

The NLO mass factorization terms are directly related to regularized LO
Altarelli-Parisi splitting kernels Altarelli:1977zs :

$\Gamma^{(1)}_{ab,\text{full}}\left(x\right)=-\dfrac{1}{\epsilon}P^{0}_{ab}(x),$
(13)

where

$\displaystyle P^{0}_{qq}(x)$ $\displaystyle=$
$\displaystyle\left(\dfrac{N_{c}^{2}-1}{N_{c}}\right)p^{0}_{qq}(x),$ (14)
$\displaystyle P^{0}_{gq}(x)$ $\displaystyle=$
$\displaystyle\left(\dfrac{N_{c}^{2}-1}{N_{c}}\right)p^{0}_{gq}(x),$ (15)
$\displaystyle P^{0}_{qg}(x)$ $\displaystyle=$ $\displaystyle p^{0}_{qg}(x),$
(16) $\displaystyle P^{0}_{gg}(x)$ $\displaystyle=$ $\displaystyle
N_{c}p^{0}_{gg}(x)+N_{f}\hat{p}^{0}_{gg}(x).$ (17)

####  2.1.2 Subtraction at NNLO

The NNLO QCD correction to an $n$-jet cross section is given by:

$\displaystyle\text{d}{\hat{\sigma}_{ab,\mathrm{NNLO}}}$ $\displaystyle=$
$\displaystyle\int_{n}\left(\text{d}{\hat{\sigma}^{VV}_{ab,\mathrm{NNLO}}}+\text{d}{\hat{\sigma}^{MF,2}_{ab,\mathrm{NNLO}}}\right)$
(18) $\displaystyle+$
$\displaystyle\int_{n+1}\left(\text{d}{\hat{\sigma}^{RV}_{ab,\mathrm{NNLO}}}+\text{d}{\hat{\sigma}^{MF,1}_{ab,\mathrm{NNLO}}}\right)+\int_{n+2}\text{d}{\hat{\sigma}^{RR}_{ab,\mathrm{NNLO}}},$

where $\text{d}{\hat{\sigma}^{VV}_{ab,\mathrm{NNLO}}}$ represents the double-
virtual correction, $\text{d}{\hat{\sigma}^{RV}_{ab,\mathrm{NNLO}}}$ the real-
virtual correction and $\text{d}{\hat{\sigma}^{RR}_{ab,\mathrm{NNLO}}}$ the
double-real correction. The mass factorization counterterm at this
perturbative order is split into two terms associated with $n$\- and
$(n+1)$-particle final states, respectively
$\text{d}{\hat{\sigma}^{MF,2}_{ab,\mathrm{NNLO}}}$ and
$\text{d}{\hat{\sigma}^{MF,1}_{ab,\mathrm{NNLO}}}$.

As in the NLO case, the quantity in (18) cannot be computed directly with
numerical methods. The singular behaviour of both the double-real and real-
virtual corrections in infrared limits has to be subtracted and the explicit
poles in the double-virtual and real-virtual matrix elements need to be
removed properly. To achieve this, the NNLO cross section is rewritten in the
context of antenna subtraction as Currie:2013vh :

$\displaystyle\text{d}{\hat{\sigma}_{ab,\mathrm{NNLO}}}$ $\displaystyle=$
$\displaystyle\int_{n}\left[\text{d}{\hat{\sigma}^{VV}_{ab,\mathrm{NNLO}}}-\text{d}{\hat{\sigma}^{U}_{ab,\mathrm{NNLO}}}\right]$
(19) $\displaystyle+$
$\displaystyle\int_{n+1}\left[\text{d}{\hat{\sigma}^{RV}_{ab,\mathrm{NNLO}}}-\text{d}{\hat{\sigma}^{T}_{ab,\mathrm{NNLO}}}\right]$
$\displaystyle+$
$\displaystyle\int_{n+2}\left[\text{d}{\hat{\sigma}^{RR}_{ab,\mathrm{NNLO}}}-\text{d}{\hat{\sigma}^{S}_{ab,\mathrm{NNLO}}}\right],$

where the subtracted quantities are the double-virtual, the real-virtual and
the double-real subtraction term. These contributions have the following form
Currie:2013vh :

$\displaystyle\text{d}{\hat{\sigma}^{S}_{ab,\mathrm{NNLO}}}$ $\displaystyle=$
$\displaystyle\text{d}{\hat{\sigma}^{S,1}_{ab,\mathrm{NNLO}}}+\text{d}{\hat{\sigma}^{S,2}_{ab,\mathrm{NNLO}}}\,,$
$\displaystyle\text{d}{\hat{\sigma}^{T}_{ab,\mathrm{NNLO}}}$ $\displaystyle=$
$\displaystyle\text{d}{\hat{\sigma}^{VS}_{ab,\mathrm{NNLO}}}-\int_{1}\text{d}{\hat{\sigma}^{S,1}_{ab,\mathrm{NNLO}}}-\text{d}{\hat{\sigma}^{MF,1}_{ab,\mathrm{NNLO}}}\,,$
$\displaystyle\text{d}{\hat{\sigma}^{U}_{ab,\mathrm{NNLO}}}$ $\displaystyle=$
$\displaystyle-\int_{1}\text{d}{\hat{\sigma}^{VS}_{ab,\mathrm{NNLO}}}-\int_{2}\text{d}{\hat{\sigma}^{S,2}_{ab,\mathrm{NNLO}}}-\text{d}{\hat{\sigma}^{MF,2}_{ab,\mathrm{NNLO}}}\,.$
(20)

The double-real subtraction term has been decomposed into two contributions
which contain single and double unresolved IR limits:
$\text{d}{\hat{\sigma}^{S,1}_{ab,\mathrm{NNLO}}}$ and
$\text{d}{\hat{\sigma}^{S,2}_{ab,\mathrm{NNLO}}}$. In the real-virtual
subtraction term, $\text{d}{\hat{\sigma}^{VS}_{ab,\mathrm{NNLO}}}$ cancels the
implicit singular behaviour of the real-virtual correction in single
unresolved limits. The remaining terms in
$\text{d}{\hat{\sigma}^{T}_{ab,\mathrm{NNLO}}}$ and
$\text{d}{\hat{\sigma}^{U}_{ab,\mathrm{NNLO}}}$, which are not mass
factorization counterterms, are obtained via analytical integration of the
aforementioned contributions, over the phase space of a single or double
unresolved emission. These terms cancel explicit $\epsilon$-poles in the
virtual corrections.

Within the traditional antenna subtraction approach Gehrmann-DeRidder:2005btv
; Currie:2013dwa , the infrastructure above is constructed starting from
$\text{d}{\hat{\sigma}^{S,1}_{ab,\mathrm{NNLO}}}$,
$\text{d}{\hat{\sigma}^{S,2}_{ab,\mathrm{NNLO}}}$ and
$\text{d}{\hat{\sigma}^{VS}_{ab,\mathrm{NNLO}}}$, which are obtained by
studying the infrared behaviour of the double-real and real-virtual matrix
elements. Afterwards, analytical integrations are performed to complete the
subtraction. Several adjustments are needed during this process to ensure the
removal of spurious divergences and prevent the oversubtraction of singular
behaviour. Examples of the typical structures appearing in each layer can be
found in Gehrmann-DeRidder:2005btv ; Gehrmann-DeRidder:2007foh ;
Glover:2010kwr ; Gehrmann-DeRidder:2011jwo ; Gehrmann-DeRidder:2012dog ;
Currie:2013vh .

At NNLO, we have two different contributions to the mass factorization
counterterm: the double-virtual and the real-virtual mass factorization terms.
The real-virtual mass factorization counterterm is given by Currie:2013vh :

$\text{d}{\hat{\sigma}^{MF,1}_{ab,\mathrm{NNLO}}}=-\left(\dfrac{\alpha_{s}\overline{C}(\epsilon)}{2\pi}\right)\sum_{c,d}\int{\dfrac{\text{d}x_{1}}{x_{1}}}{\dfrac{\text{d}x_{2}}{x_{2}}}\Gamma^{(1)}_{ab;cd}\left(x_{1},x_{2}\right)\left(\text{d}{\hat{\sigma}^{R}_{cd,\mathrm{NLO}}}-\text{d}{\hat{\sigma}^{S}_{cd,\mathrm{NLO}}}\right),$
(21)

where the required mass factorization kernels are the ones used at NLO.

The double-virtual mass factorization counterterm reads Currie:2013vh :

$\displaystyle\text{d}{\hat{\sigma}^{MF,2}_{ab,\mathrm{NNLO}}}$
$\displaystyle=$
$\displaystyle-\int{\dfrac{\text{d}x_{1}}{x_{1}}}{\dfrac{\text{d}x_{2}}{x_{2}}}\sum_{c,d}\Bigg{\\{}\Bigg{.}\left(\dfrac{\alpha_{s}\overline{C}(\epsilon)}{2\pi}\right)\left[\Gamma^{(1)}_{ab;cd}\left(x_{1},x_{2}\right)\left(\text{d}{\hat{\sigma}^{V}_{cd,\mathrm{NLO}}}-\text{d}{\hat{\sigma}^{T}_{cd,\mathrm{NLO}}}\right)\right]$
(22) $\displaystyle\hskip
14.22636pt+\left(\dfrac{\alpha_{s}\overline{C}(\epsilon)}{2\pi}\right)^{2}\Big{[}\Big{.}\overline{\Gamma}^{(2)}_{ab;cd}\left(x_{x_{1},x_{2}}\right)-\dfrac{\beta_{0}}{\epsilon}\Gamma^{(1)}_{ab;cd}\left(x_{1},x_{2}\right)$
$\displaystyle\hskip
85.35826pt+\dfrac{1}{2}\sum_{\alpha,\beta}\left[\Gamma^{(1)}_{ab;\alpha\beta}\otimes\Gamma^{(1)}_{\alpha\beta;cd}\right](x_{1},x_{2})\Big{.}\Big{]}\text{d}{\hat{\sigma}_{cd,\mathrm{LO}}}\Bigg{.}\Bigg{\\}}.$

The reduced two-loop mass factorization kernel is defined as Currie:2013vh :

$\overline{\Gamma}^{(2)}_{ab;cd}\left(x_{x_{1},x_{2}}\right)=\overline{\Gamma}^{(2)}_{ca,\text{full}}\left(x_{x_{1}}\right)\delta_{db}\delta(1-x_{2})+\overline{\Gamma}^{(2)}_{db,\text{full}}\left(x_{x_{2}}\right)\delta_{ca}\delta(1-x_{1}),$
(23)

where $\overline{\Gamma}^{(2)}_{ca,\text{full}}\left(x_{x_{i}}\right)$ are
directly related to the LO and NLO Altarelli-Parisi spitting kernels
Altarelli:1977zs ; Curci:1980uw ; Furmanski:1980cm :

$\overline{\Gamma}^{(2)}_{ab,\text{full}}\left(x_{x}\right)=-\dfrac{1}{2\epsilon}\left(P^{1}_{ab}(x)+\dfrac{\beta_{0}}{\epsilon}P^{0}_{ab}(x)\right),$
(24)

and can be decomposed into colour layers as Currie:2013vh :

$\displaystyle{\overline{\Gamma}}_{qq}^{(2)}(x)$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{N_{c}^{2}-1}{N_{c}}\bigg{)}\bigg{[}N_{c}\overline{\Gamma}_{qq}^{(2)}(x)+\widetilde{\overline{\Gamma}}_{qq}^{(2)}(x)+\frac{1}{N_{c}}\widetilde{\widetilde{\overline{\Gamma}}}_{qq}^{(2)}(x)+N_{f}\widehat{\overline{\Gamma}}_{qq}^{(2)}(x)\bigg{]},$
(25) $\displaystyle{\overline{\Gamma}}_{q{\bar{q}}}^{(2)}(x)$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{N_{c}^{2}-1}{N_{c}}\bigg{)}\bigg{[}\overline{\Gamma}_{q{\bar{q}}}^{(2)}(x)+\frac{1}{N_{c}}\widetilde{\overline{\Gamma}}_{q{\bar{q}}}^{(2)}(x)\bigg{]},$
(26) $\displaystyle{\overline{\Gamma}}_{qq^{\prime}}^{(2)}(x)$
$\displaystyle=$ $\displaystyle\bigg{(}\frac{N_{c}^{2}-1}{N_{c}}\bigg{)}\
\overline{\Gamma}_{qq^{\prime}}^{(2)}(x),$ (27)
$\displaystyle{\overline{\Gamma}}_{q{\bar{q}}^{\prime}}^{(2)}(x)$
$\displaystyle=$ $\displaystyle\bigg{(}\frac{N_{c}^{2}-1}{N_{c}}\bigg{)}\
\overline{\Gamma}_{q{\bar{q}}^{\prime}}^{(2)}(x),$ (28)
$\displaystyle{\overline{\Gamma}}_{gq}^{(2)}(x)$ $\displaystyle=$
$\displaystyle\bigg{(}\frac{N_{c}^{2}-1}{N_{c}}\bigg{)}\bigg{[}N_{c}\overline{\Gamma}_{gq}^{(2)}(x)+\frac{1}{N_{c}}\widetilde{\overline{\Gamma}}_{gq}^{(2)}(x)+N_{f}\widehat{\overline{\Gamma}}_{gq}^{(2)}(x)\bigg{]},$
(29) $\displaystyle{\overline{\Gamma}}_{qg}^{(2)}(x)$ $\displaystyle=$
$\displaystyle
N_{c}\overline{\Gamma}_{qg}^{(2)}(x)+\frac{1}{N_{c}}\widetilde{\overline{\Gamma}}_{qg}^{(2)}(x)+N_{f}\widetilde{\overline{\Gamma}}_{qg}^{(2)}(x),$
(30) $\displaystyle{\overline{\Gamma}}_{gg}^{(2)}(x)$ $\displaystyle=$
$\displaystyle
N_{c}^{2}\overline{\Gamma}_{gg}^{(2)}(x)+N_{c}N_{f}\widehat{\overline{\Gamma}}_{gg}^{(2)}(x)+\frac{N_{f}}{N_{c}}\widehat{\widetilde{\overline{\Gamma}}}_{gg}^{(2)}(x)+N_{f}^{2}\widehat{\widehat{\overline{\Gamma}}}_{gg,}^{(2)}(x).$
(31)

The expressions for the two-loop mass factorization kernels are rather lengthy
and therefore we do not report them here. Their expressions as well as
detailed explanations about how they are extracted from the associated LO and
NLO splitting kernels Furmanski:1980cm  can be found in Appendix A.2 of
Currie:2013vh .

##  3 Integrated dipoles in colour space

In this section we introduce key ingredients for the colourful antenna
subtraction approach: integrated dipoles Currie:2013vh ; Currie:2013dwa ;
Chen:2022ktf ; Chen:2022tpk . Such objects conveniently collect integrated
antenna functions and mass factorization kernels and are cast as insertion
operators in colour space. Integrated dipoles allow to systematically describe
the infrared singularity structure of one- and two-loop matrix elements in
terms of integrated antenna functions. This is a pivotal prerequisite for the
formulation of the colourful antenna subtraction method.

Before illustrating the integrated dipoles, we provide a brief overview of the
colour space formalism and of the infrared singularities of virtual
amplitudes.

###  3.1 Colour space formalism

The treatment of QCD amplitudes in colour space provides a universal
description of the infrared singularity structure, in the presence of both
virtual and real corrections Bassetto:1983mvz ; Catani:1996jh ; Catani:1998bh
; Catani:1999ss ; Becher:2009qa ; Gardi:2009qi ; Gardi:2009zv .

In colour space, an $\ell$-loop amplitude with $n$ external partons is
represented by an abstract vector
$\ket{\mathcal{A}_{n}^{\ell}(\left\\{p\right\\}_{n})}$, whose components carry
the colour indices of the external partons in a suitable representation of
SU$(N_{c})$. If a set of generating vectors
$\left\\{\ket{\mathcal{C}_{n,c}^{\ell}}\right\\}$ is defined, which span the
$n$-parton colour space, any amplitude can be decomposed as:

$\ket{\mathcal{A}_{n}^{\ell}(\left\\{p\right\\}_{n})}=\sum_{c\in
I^{\ell}}A^{\ell}_{n,c}(\left\\{p\right\\}_{n})\,\ket{\mathcal{C}_{n,c}^{\ell}},$
(32)

where $I^{\ell}$ indicates a suitable subset of generating vectors. The scalar
quantities $A^{\ell}_{n,c}(\left\\{p\right\\}_{n})$ are colour-ordered partial
amplitudes. In equation (32), the dependence on the helicities of the external
partons is implicit and in the following a sum over helicity configurations is
always assumed when squared quantities are considered. For $\ell\geq 1$, the
dependence on the renormalization scale $\mu_{r}$ is understood.

An arbitrary number of colourless particles, such as photons, weak bosons or
Higgs bosons, could be involved in the considered scattering process, however
this does not modify the structure of the amplitude in colour space or the
construction of the subtraction terms for infrared singularities in QCD. The
presence of colourless emissions only affects the colour-ordered partial
amplitudes, which clearly change for different processes. For this reason, in
the following we restrict the discussion and the notation to $n$-parton
amplitudes, modulo additional colourless particles.

Our convention is to strip the partial amplitudes of overall coefficients such
as couplings and incoming particles average factors, which are inserted later
at the cross section level. In particular, we strip an $\ell$-loop amplitude
of an overall factor
$\left(\tfrac{\alpha_{s}\bar{C}(\epsilon)}{2\pi}\right)^{\ell}$ with respect
to the corresponding tree-level amplitude, where
$\overline{C}(\epsilon)=(4\pi)^{\epsilon}e^{-\epsilon\gamma_{E}}$.

The description of infrared singularities of QCD amplitudes can be achieved
through the evaluation of infrared insertion operators in colour space
Catani:1996jh ; Catani:1998bh ; Becher:2009qa ; DelDuca:2016ily . The action
of an operator $\bm{\mathcal{O}}$ in colour space can be generically computed
as

$\braket{\mathcal{A}_{n}^{\ell_{1}}(\left\\{p\right\\}_{n})}{\bm{\mathcal{O}}}{\mathcal{A}_{n}^{\ell_{2}}(\left\\{p\right\\}_{n})}.$
(33)

In particular, the coherent emission of a gluon between a dipole formed by
parton $i$ and parton $j$ is described in colour space by the colour charge
dipole operator:

$\bm{T}_{i}\cdot\bm{T}_{j},$ (34)

where we use bold symbols to emphasize that $\bm{T}_{i}=(T_{i})^{a}_{bc}$ is a
vector of $a$ SU$(N_{c})$ generators in the appropriate representation, with
$a$ indicating the colour index of the emitted gluon. In general, in the
remainder of the paper, bold symbols are used to denote operators in colour
space, while Roman symbols indicate scalars in colour space. For different
partonic species we have:

$(T_{i})^{a}_{bc}=\begin{cases}if_{bac}\quad&\text{for }i=\text{ gluon}\\\
t^{a}_{bc}\quad&\text{for }i=\text{ final-state quark or initial-state
antiquark}\\\ (t^{a}_{bc})^{\dagger}=-t^{a}_{cb}\quad&\text{for }i=\text{
final-state antiquark or initial state quark}\\\ \end{cases}$ (35)

where $f_{abc}$ and $t^{a}_{bc}$ are the colour-charge matrices in the adjoint
and fundamental representations of SU$(N_{c})$. The following properties hold:

$\displaystyle\bm{T}_{i}\cdot\bm{T}_{j}$ $\displaystyle=$
$\displaystyle\bm{T}_{j}\cdot\bm{T}_{i},$ (36)
$\displaystyle\bm{T}_{i}\cdot\bm{T}_{i}$ $\displaystyle=$
$\displaystyle\bm{T}_{i}^{2}=C_{i}\,\mathbf{Id},$ (37)

where $\mathbf{Id}$ represents the identity operator in colour space and
$C_{i}$ indicates the Casimir coefficient for the SU$(N_{c})$ representation
associated to parton $i$: $C_{g}=C_{A}=N_{c}$ and
$C_{q}=C_{\bar{q}}=C_{F}=(N_{c}^{2}-1)/(2N_{c})$.

Each state $\ket{\mathcal{A}_{n}^{\ell}(\left\\{p\right\\}_{n})}$ is a colour
singlet and colour conservation implies:

$\sum_{i=1}^{n}\bm{T}_{i}\ket{\mathcal{A}_{n}^{\ell}(\left\\{p\right\\}_{n})}=0.$
(38)

Since in what follows we always consider colour singlet states, we can employ
the previous identity as $\sum_{j\neq i}\bm{T}_{j}=-\bm{T}_{i}$.

The explicit evaluation of squared matrix elements or colour operator
insertions depends on the form of the generating vectors
$\left\\{\ket{\mathcal{C}_{n,i}^{\ell}}\right\\}$, and, in general, produces
numerous terms and non-trivial structures, especially for high-multiplicity
processes. Nevertheless one can observe that the result of such calculations
is given by real functions of the external momenta and possibly the
renormalization scale. At tree-level, the general form of such a function is:

$f_{0}\left(\set{p}_{n}\right)=\sum_{c,c^{\prime}\in
I^{0}}C^{0}_{n}(c,c^{\prime})\,a_{n}^{0}(c,c^{\prime};\left\\{p\right\\}_{n}),$
(39)

where we generically assume that the indices $c$ and $c^{\prime}$ here run
over the set of possible colour orderings, along with the possible colour
structures. The coefficients $C^{0}_{n}(c,c^{\prime})$ are colour factors
which depend on $N_{c}$ and the number of partons (possibly depending on
flavour) and

$a_{n}^{0}(c,c^{\prime};\left\\{p\right\\}_{n})=\begin{cases}\left|A_{n,c}^{0}(\left\\{p\right\\}_{n})\right|^{2}\quad&\text{if
}c=c^{\prime},\\\
A_{n,c}^{0}(\left\\{p\right\\}_{n})^{\dagger}A_{n,c^{\prime}}^{0}(\left\\{p\right\\}_{n})\quad&\text{if
}c\neq c^{\prime}.\end{cases}$ (40)

This quantity represents the squared interference of two colour-ordered
partial amplitudes, with generic colour structures dictated by the indices $c$
and $c^{\prime}$. When $c=c^{\prime}$, we have squared coherent partial
amplitudes, while we refer to the case $c\neq c^{\prime}$ as incoherent
interference. We notice that even in the case $c\neq c^{\prime}$,
$a_{n}^{0}(c,c^{\prime};\left\\{p\right\\}_{n})$ is a real quantity, since a
sum over helicities is always assumed Chen:2022ktf . Analogously, at one loop
we have:

$f_{1}\left(\set{p}_{n}\right)=\sum_{c\in I^{0},c^{\prime}\in
I^{1}}C^{1}_{n}(c,c^{\prime})\,a_{n}^{1}(c,c^{\prime};\left\\{p\right\\}_{n}),$
(41)

where

$a_{n}^{1}(c,c^{\prime};\left\\{p\right\\}_{n})=2\text{Re}\left[A_{n,c}^{0}(\left\\{p\right\\}_{n})^{\dagger}A_{n,c^{\prime}}^{1}(\left\\{p\right\\}_{n})\right],$
(42)

which is manifestly a real quantity.

###  3.2 Infrared singularity structure of QCD loop amplitudes

We review here how the infrared singularities of virtual corrections
amplitudes in QCD can be predicted in a general way by means of infrared
operators in colour space Catani:1998bh ; Becher:2009qa ; Gardi:2009qi ;
Gardi:2009zv .

####  3.2.1 Infrared singularity structure at one loop

The singularity structure of renormalized $(n+2)$-parton one-loop amplitudes
in QCD can be described in colour space with Catani:1998bh :

$\ket{\mathcal{A}_{n+2}^{1}}=\bm{I}^{(1)}\left(\epsilon,\mu_{r}^{2}\right)\ket{\mathcal{A}_{n+2}^{0}}+\ket{\mathcal{A}_{n+2}^{1,\text{fin}}(\mu_{r}^{2})},$
(43)

where $\mu_{r}$ is the renormalization scale,
$\ket{\mathcal{A}_{n+2}^{1,\text{fin}}(\mu_{r}^{2})}$ is a finite remainder
and $\bm{I}^{(1)}\left(\epsilon,\mu_{r}^{2}\right)$ is Catani’s infrared
insertion operator given by Catani:1998bh :

$\bm{I}^{(1)}\left(\epsilon,\mu_{r}^{2}\right)=\dfrac{1}{2}\dfrac{e^{\epsilon\gamma_{E}}}{\Gamma(1-\epsilon)}\sum_{i=1}^{n+2}\dfrac{1}{\bm{T}_{i}^{2}}\mathcal{V}_{i}(\epsilon)\sum_{j\neq
i}(\bm{T}_{i}\cdot\bm{T}_{j})\left(\dfrac{-s_{ij}}{\mu_{r}^{2}}\right)^{-\epsilon}\,.$
(44)

The singular functions $\mathcal{V}_{i}(\epsilon)$ contain double and single
$\epsilon$-poles:

$\mathcal{V}_{i}(\epsilon)=\bm{T}^{2}_{i}\dfrac{1}{\epsilon^{2}}+\gamma_{i}\dfrac{1}{\epsilon},$
(45)

with

$\displaystyle\bm{T}^{2}_{q}=\bm{T}^{2}_{\bar{q}}=C_{F},\quad\bm{T}^{2}_{g}=C_{A},\quad\gamma_{q}=\gamma_{\bar{q}}=\dfrac{3}{2}C_{F},\quad\gamma_{g}=\dfrac{11}{6}C_{A}-\dfrac{1}{3}N_{f}.$
(46)

Equation (44) can be rewritten as

$\displaystyle\bm{I}^{(1)}\left(\epsilon,\mu_{r}^{2}\right)$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\sum_{i=1}^{n+2}\sum_{j\neq
i}\left(\bm{T}_{i}\cdot\bm{T}_{j}\right)\mathcal{I}^{(1)}_{ij}\left(\epsilon,\mu_{r}^{2}\right)$
(47) $\displaystyle=$
$\displaystyle\sum_{(i,j)}\left(\bm{T}_{i}\cdot\bm{T}_{j}\right)\mathcal{I}^{(1)}_{ij}\left(\epsilon,\mu_{r}^{2}\right),$

where in the last line the sum runs over pairs of partons and:

$\displaystyle\mathcal{I}^{(1)}_{i_{g}j_{g}}\left(\epsilon,\mu_{r}^{2}\right)$
$\displaystyle=$
$\displaystyle\dfrac{e^{\epsilon\gamma_{E}}}{\Gamma(1-\epsilon)}\left[\dfrac{1}{\epsilon^{2}}+\dfrac{1}{\epsilon}\left(\dfrac{11}{6}-\dfrac{1}{3}\dfrac{N_{f}}{N_{c}}\right)\right]\left(\dfrac{-s_{ij}}{\mu_{r}^{2}}\right)^{-\epsilon},$
(48)
$\displaystyle\mathcal{I}^{(1)}_{i_{g}j_{q}}\left(\epsilon,\mu_{r}^{2}\right)=\mathcal{I}^{(1)}_{i_{q}j_{g}}\left(\epsilon,\mu_{r}^{2}\right)$
$\displaystyle=$
$\displaystyle\dfrac{e^{\epsilon\gamma_{E}}}{\Gamma(1-\epsilon)}\left[\dfrac{1}{\epsilon^{2}}+\dfrac{1}{\epsilon}\left(\dfrac{5}{3}-\dfrac{1}{6}\dfrac{N_{f}}{N_{c}}\right)\right]\left(\dfrac{-s_{ij}}{\mu_{r}^{2}}\right)^{-\epsilon},$
(49)
$\displaystyle\mathcal{I}^{(1)}_{i_{q}j_{q^{\prime}}}\left(\epsilon,\mu_{r}^{2}\right)$
$\displaystyle=$
$\displaystyle\dfrac{e^{\epsilon\gamma_{E}}}{\Gamma(1-\epsilon)}\left[\dfrac{1}{\epsilon^{2}}+\dfrac{3}{2\epsilon}\right]\left(\dfrac{-s_{ij}}{\mu_{r}^{2}}\right)^{-\epsilon},$
(50)

independently from the flavour of $q$ and $q^{\prime}$. $N_{f}$ indicates the
number of light quark flavours.

Using (43) it is possible to extract the poles of one-loop matrix elements in
the following way:

$\displaystyle\mathcal{P}oles\left(\mathcal{M}_{n+2}^{1}\right)$
$\displaystyle=$
$\displaystyle\mathcal{P}oles\left(\braket{\mathcal{A}_{n+2}^{0}}{\mathcal{A}_{n+2}^{1}}+\braket{\mathcal{A}_{n+2}^{1}}{\mathcal{A}_{n+2}^{0}}\right)$
(51) $\displaystyle=$
$\displaystyle\mathcal{P}oles(\braket{\mathcal{A}_{n+2}^{0}}{\bm{I}^{(1)}\left(\epsilon\right)+\bm{I}^{(1),\dagger}\left(\epsilon\right)}{\mathcal{A}_{n+2}^{0}})\,.$

The appearance of the sum
$\bm{I}^{(1)}\left(\epsilon\right)+\bm{I}^{(1),\dagger}\left(\epsilon\right)$
indicates that only the real part of the insertion operator affects the
description of the poles at the matrix element level, as expected. We can then
write

$\mathcal{P}oles\left(\mathcal{M}_{n+2}^{1}\right)=\mathcal{P}oles\left[\sum_{(i,j)}\braket{\mathcal{A}_{n+2}^{0}}{\bm{T}_{i}\cdot\bm{T}_{j}}{\mathcal{A}_{n+2}^{0}}\,2\text{Re}\left(\mathcal{I}^{(1)}_{ij}\left(\epsilon,\mu_{r}^{2}\right)\right)\right]\,,$
(52)

where we considered that the colour charge evaluation on the tree-level
amplitude gives a real quantity, as argued in Section 3. At the cross section
level we have

$\displaystyle\mathcal{P}oles\left({\hat{\sigma}^{V}_{NLO}}\right)$
$\displaystyle=$
$\displaystyle\mathcal{N}^{V}_{NLO}\int\text{d}\Phi_{n}(p_{3},\dots,p_{n+2};p_{1},p_{2})\,J_{n}^{(n)}(\left\\{p\right\\}_{n})$
(53)
$\displaystyle\times\mathcal{P}oles\left[\sum_{(i,j)}\braket{\mathcal{A}_{n+2}^{0}}{\bm{T}_{i}\cdot\bm{T}_{j}}{\mathcal{A}_{n+2}^{0}}2\,\text{Re}\left(\mathcal{I}^{(1)}_{ij}\left(\epsilon,\mu_{r}^{2}\right)\right)\right].$

The factor $\mathcal{N}^{V}_{NLO}$ is given by

$\mathcal{N}^{V}_{NLO}=\left(\dfrac{\alpha_{s}\overline{C}(\epsilon)}{2\pi}\right)\mathcal{N}_{LO},$
(54)

where $\mathcal{N}_{LO}$ contains the overall factors appropriate for the LO
process, such as the strong coupling, symmetry factors and the spin- and
colour-average over the initial-state partons.

####  3.2.2 Infrared singularity structure at two loops

The singularity structure of renormalized two-loop amplitudes in QCD can be
described in colour space by Catani:1998bh ; Becher:2009qa ; Gardi:2009qi ;
Gardi:2009zv :

$\ket{\mathcal{A}_{n+2}^{2}}=\bm{I}^{(1)}\left(\epsilon,\mu_{r}^{2}\right)\ket{\mathcal{A}_{n+2}^{1}}+\bm{I}^{(2)}\left(\epsilon,\mu_{r}^{2}\right)\ket{\mathcal{A}_{n+2}^{0}}+\ket{\mathcal{A}_{n+2}^{2,\text{fin}}(\mu_{r}^{2})},$
(55)

where, as before, $\ket{\mathcal{A}_{n+2}^{2,\text{fin}}(\mu_{r}^{2})}$ is a
finite remainder. The two-loop infrared insertion operator has the following
expression Catani:1998bh :

$\displaystyle\bm{I}^{(2)}\left(\epsilon,\mu_{r}^{2}\right)$ $\displaystyle=$
$\displaystyle-\dfrac{\beta_{0}}{\epsilon}\bm{I}^{(1)}\left(\epsilon,\mu_{r}^{2};\left\\{p\right\\}_{n}\right)-\dfrac{1}{2}\bm{I}^{(1)}\left(\epsilon,\mu_{r}^{2}\right)\bm{I}^{(1)}\left(\epsilon,\mu_{r}^{2}\right)$
(56)
$\displaystyle+e^{-\epsilon\gamma_{E}}\dfrac{\Gamma(1-2\epsilon)}{\Gamma(1-\epsilon)}\left(\dfrac{\beta_{0}}{\epsilon}+K\right)\bm{I}^{(1)}\left(2\epsilon,\mu_{r}^{2}\right)$
$\displaystyle+\bm{H}^{(2)}\left(\epsilon,\mu_{r}^{2}\right)\,.$

where

$\displaystyle\beta_{0}$ $\displaystyle=$
$\displaystyle\dfrac{11}{6}N_{c}-\dfrac{1}{3}N_{f},$ (57) $\displaystyle K$
$\displaystyle=$
$\displaystyle\left(\dfrac{67}{18}-\dfrac{\pi^{2}}{6}\right)N_{c}-\dfrac{5}{9}N_{f}.$
(58)

The colour structure of (56) is more involved than the colour charge dipole
structure of (44), with products of two colour charge dipoles appearing. The
last line of (56) contains the hard radiation function
$\bm{H}^{(2)}\left(\epsilon,\mu_{r}^{2}\right)$ Catani:1998bh ; Bern:2003ck ;
Becher:2009qa , which can be decomposed in the following manner:

$\bm{H}^{(2)}\left(\epsilon,\mu_{r}^{2}\right)=\sum_{i}C_{i}\mathcal{H}^{(2)}_{i}\left(\epsilon\right)\mathbf{Id}+\check{\bm{H}}^{(2)}(\epsilon,\mu_{r}^{2}),$
(59)

where the sum runs over the $n+2$ external partons and $C_{i}$ are Casimir
coefficients. The first term in (59) is proportional to the identity in colour
space, while the second term has a non-trivial colour structure, which cannot
be in general expressed in terms of colour charge dipoles. However, the second
term vanishes when evaluated between tree-level states Bern:2003ck ;
Becher:2009qa :

$\bra{\mathcal{A}_{n+2}^{0}}\check{\bm{H}}^{(2)}(\epsilon,\mu_{r}^{2})\ket{\mathcal{A}_{n+2}^{0}}=0.$
(60)

For the purpose of describing the infrared singularity structure of two-loop
squared matrix elements, the hard radiation function
$\bm{H}^{(2)}\left(\epsilon,\mu_{r}^{2}\right)$ needs to be evaluated on tree-
level states and therefore it is possible to neglect
$\check{\bm{H}}^{(2)}(\epsilon,\mu_{r}^{2})$ in its decomposition. We can
express a colour operator proportional to the identity as a sum of colour
charge dipoles using colour conservation (38). We can then rewrite:

$\displaystyle\sum_{i}C_{i}\mathcal{H}^{(2)}_{i}\left(\epsilon\right)\mathbf{Id}$
$\displaystyle=$
$\displaystyle-\sum_{i}\mathcal{H}^{(2)}_{i}\left(\epsilon\right)\sum_{j\neq
i}\bm{T}_{i}\cdot\bm{T}_{j}$ (61) $\displaystyle=$
$\displaystyle-\sum_{(i,j)}\mathcal{H}^{(2)}_{ij}\left(\epsilon\right)\bm{T}_{i}\cdot\bm{T}_{j}\,,$

where, as usual, the sum runs over pairs of partons and
$\mathcal{H}^{(2)}_{ij}=\mathcal{H}^{(2)}_{i}+\mathcal{H}^{(2)}_{j}$. The hard
radiation functions are given by Catani:1998bh ; Becher:2009qa :

$\displaystyle\mathcal{H}^{(2)}_{g}\left(\epsilon\right)=\dfrac{e^{\epsilon\gamma_{E}}}{4\Gamma(1-\epsilon)\epsilon}N_{c}\Biggl{\\{}$
$\displaystyle\left[\dfrac{5}{12}+\dfrac{11}{144}\pi^{2}+\dfrac{\zeta_{3}}{2}\right]+\dfrac{N_{f}}{N_{c}}\left[-\dfrac{89}{108}-\dfrac{\pi^{2}}{72}\right]$
(62)
$\displaystyle+\dfrac{N_{f}}{N_{c}^{3}}\left[\dfrac{-1}{4}\right]+\dfrac{N_{f}^{2}}{N_{c}^{2}}\left[\dfrac{5}{27}\right]\Biggr{\\}}\,,$
$\displaystyle\mathcal{H}^{(2)}_{q}\left(\epsilon\right)=\dfrac{e^{\epsilon\gamma_{E}}}{4\Gamma(1-\epsilon)\epsilon}N_{c}\Biggl{\\{}$
$\displaystyle\left[\dfrac{409}{432}-\dfrac{11}{48}\pi^{2}+\dfrac{7}{2}\zeta_{3}\right]+\dfrac{N_{f}}{N_{c}}\left[-\dfrac{25}{108}+\dfrac{\pi^{2}}{24}\right]$
(63)
$\displaystyle+\dfrac{1}{N_{c}^{2}}\left[\dfrac{3}{16}-\dfrac{\pi^{2}}{4}+3\zeta_{3}\right]\Biggr{\\}}\,.$

Using (47) and (61) and neglecting
$\check{\bm{H}}^{(2)}(\epsilon,\mu_{r}^{2})$, we can rearrange equation (56)
as:

$\displaystyle\bm{I}^{(2)}\left(\epsilon,\mu_{r}^{2}\right)$ $\displaystyle=$
$\displaystyle-\dfrac{\beta_{0}}{\epsilon}\sum_{(i,j)}\mathcal{I}^{(1)}_{ij}\left(\epsilon,\mu_{r}^{2}\right)\bm{T}_{i}\cdot\bm{T}_{j}$
(64)
$\displaystyle-\dfrac{1}{2}\sum_{(i,j)}\sum_{(k,l)}\mathcal{I}^{(1)}_{ij}\left(\epsilon,\mu_{r}^{2}\right)\mathcal{I}^{(1)}_{kl}\left(\epsilon,\mu_{r}^{2}\right)(\bm{T}_{i}\cdot\bm{T}_{j})(\bm{T}_{k}\cdot\bm{T}_{l})$
$\displaystyle+\sum_{(i,j)}\mathcal{I}^{(2)}_{ij}\left(\epsilon,\mu_{r}^{2}\right)\bm{T}_{i}\cdot\bm{T}_{j},$

where

$\mathcal{I}^{(2)}_{ij}\left(\epsilon,\mu_{r}^{2}\right)=e^{-\epsilon\gamma_{E}}\dfrac{\Gamma(1-2\epsilon)}{\Gamma(1-\epsilon)}\left(\dfrac{\beta_{0}}{\epsilon}+K\right)\mathcal{I}^{(1)}_{ij}\left(2\epsilon,\mu_{r}^{2}\right)-\mathcal{H}^{(2)}_{ij}\left(\epsilon\right)\,.$
(65)

We can now use (43), (55) and (56) to express the singularity structure of a
two-loop matrix element:

$\displaystyle\mathcal{P}oles\left(\mathcal{M}_{n+2}^{2}\right)$
$\displaystyle=$
$\displaystyle\mathcal{P}oles\left(\braket{\mathcal{A}_{n+2}^{2}}{\mathcal{A}_{n+2}^{0}}+\braket{\mathcal{A}_{n+2}^{0}}{\mathcal{A}_{n+2}^{2}}+\braket{\mathcal{A}_{n+2}^{1}}{\mathcal{A}_{n+2}^{1}}\right)$
(66) $\displaystyle=$
$\displaystyle\mathcal{P}oles\Big{\\{}\Big{.}\braket{\mathcal{A}_{n+2}^{1}}{\bm{I}^{(1)}\left(\epsilon\right)+\bm{I}^{(1),\dagger}\left(\epsilon\right)}{\mathcal{A}_{n+2}^{0}}+\braket{\mathcal{A}_{n+2}^{0}}{\bm{I}^{(1)}\left(\epsilon\right)+\bm{I}^{(1),\dagger}\left(\epsilon\right)}{\mathcal{A}_{n+2}^{1}}$
$\displaystyle-\dfrac{1}{2}\braket{\mathcal{A}_{n+2}^{0}}{\left(\bm{I}^{(1)}\left(\epsilon\right)+\bm{I}^{(1),\dagger}\left(\epsilon\right)\right)\left(\bm{I}^{(1)}\left(\epsilon\right)+\bm{I}^{(1),\dagger}\left(\epsilon\right)\right)}{\mathcal{A}_{n+2}^{0}}$
$\displaystyle-\dfrac{\beta_{0}}{\epsilon}\braket{\mathcal{A}_{n+2}^{0}}{\bm{I}^{(1)}\left(\epsilon\right)+\bm{I}^{(1),\dagger}\left(\epsilon\right)}{\mathcal{A}_{n+2}^{0}}$
$\displaystyle+\text{e}^{-\epsilon\gamma_{E}}\dfrac{\Gamma(1-2\epsilon)}{\Gamma(1-\epsilon)}\left(\dfrac{\beta_{0}}{\epsilon}+K\right)\braket{\mathcal{A}_{n+2}^{0}}{\bm{I}^{(1)}\left(2\epsilon\right)+\bm{I}^{(1),\dagger}\left(2\epsilon\right)}{\mathcal{A}_{n+2}^{0}}$
$\displaystyle+\braket{\mathcal{A}_{n+2}^{0}}{\bm{H}^{(2)}\left(\epsilon\right)+\bm{H}^{(2),\dagger}\left(\epsilon\right)}{\mathcal{A}_{n+2}^{0}}\Big{.}\Big{\\}}\,.$

We see again that only the real parts of the insertion operators are needed to
describe the singularity structure. Using (47) and (64) it is possible to
recast equation (66) as:

$\displaystyle\mathcal{P}oles\left(\mathcal{M}_{n+2}^{2}\right)$
$\displaystyle=$
$\displaystyle\mathcal{P}oles\Big{\\{}\Big{.}\sum_{(i,j)}2\text{Re}\left[\mathcal{I}^{(1)}_{ij}\left(\epsilon,\mu_{r}^{2}\right)\right]\left[\braket{\mathcal{A}_{n+2}^{1}}{\bm{T}_{i}\cdot\bm{T}_{j}}{\mathcal{A}_{n+2}^{0}}+\braket{\mathcal{A}_{n+2}^{0}}{\bm{T}_{i}\cdot\bm{T}_{j}}{\mathcal{A}_{n+2}^{1}}\right]$
(67)
$\displaystyle-\dfrac{1}{2}\sum_{(i,j)}\sum_{(k,l)}2\text{Re}\left[\mathcal{I}^{(1)}_{ij}\left(\epsilon,\mu_{r}^{2}\right)\right]2\text{Re}\left[\mathcal{I}^{(1)}_{lk}\left(\epsilon,\mu_{r}^{2}\right)\right]\braket{\mathcal{A}_{n+2}^{0}}{(\bm{T}_{i}\cdot\bm{T}_{j})(\bm{T}_{k}\cdot\bm{T}_{l})}{\mathcal{A}_{n+2}^{0}}$
$\displaystyle-\dfrac{\beta_{0}}{\epsilon}\sum_{(i,j)}2\text{Re}\left[\mathcal{I}^{(1)}_{ij}\left(\epsilon,\mu_{r}^{2}\right)\right]\braket{\mathcal{A}_{n+2}^{0}}{\bm{T}_{i}\cdot\bm{T}_{j}}{\mathcal{A}_{n+2}^{0}}$
$\displaystyle+\sum_{(i,j)}2\text{Re}\left[\mathcal{I}^{(2)}_{ij}\left(\epsilon,\mu_{r}^{2}\right)\right]\braket{\mathcal{A}_{n+2}^{0}}{\bm{T}_{i}\cdot\bm{T}_{j}}{\mathcal{A}_{n+2}^{0}}\Big{.}\Big{\\}}.$

Therefore, the poles of the double-virtual contribution to the cross section
are given by:

$\displaystyle\mathcal{P}oles\left({\hat{\sigma}^{VV}_{NNLO}}\right)=\mathcal{N}^{VV}_{NNLO}\int\text{d}\Phi_{n+2}(p_{3},\dots,p_{n+2};p_{1},p_{2})\,J_{n+2}^{(n+2)}(\left\\{p\right\\}_{n})$
(68) $\displaystyle\times$
$\displaystyle\mathcal{P}oles\Big{\\{}\Big{.}\sum_{(i,j)}2\text{Re}\left[\mathcal{I}^{(1)}_{ij}\left(\epsilon,\mu_{r}^{2}\right)\right]\left[\braket{\mathcal{A}_{n+2}^{1}}{\bm{T}_{i}\cdot\bm{T}_{j}}{\mathcal{A}_{n+2}^{0}}+\braket{\mathcal{A}_{n+2}^{0}}{\bm{T}_{i}\cdot\bm{T}_{j}}{\mathcal{A}_{n+2}^{1}}\right]$
$\displaystyle-\dfrac{1}{2}\sum_{(i,j)}\sum_{(k,l)}2\text{Re}\left[\mathcal{I}^{(1)}_{ij}\left(\epsilon,\mu_{r}^{2}\right)\right]2\text{Re}\left[\mathcal{I}^{(1)}_{lk}\left(\epsilon,\mu_{r}^{2}\right)\right]\braket{\mathcal{A}_{n+2}^{0}}{(\bm{T}_{i}\cdot\bm{T}_{j})(\bm{T}_{k}\cdot\bm{T}_{l})}{\mathcal{A}_{n+2}^{0}}$
$\displaystyle-\dfrac{\beta_{0}}{\epsilon}\sum_{(i,j)}2\text{Re}\left[\mathcal{I}^{(1)}_{ij}\left(\epsilon,\mu_{r}^{2}\right)\right]\braket{\mathcal{A}_{n+2}^{0}}{\bm{T}_{i}\cdot\bm{T}_{j}}{\mathcal{A}_{n+2}^{0}}$
$\displaystyle+\sum_{(i,j)}2\text{Re}\left[\mathcal{I}^{(2)}_{ij}\left(\epsilon,\mu_{r}^{2}\right)\right]\braket{\mathcal{A}_{n+2}^{0}}{\bm{T}_{i_{g}}\cdot\bm{T}_{j}}{\mathcal{A}_{n+2}^{0}}\Big{.}\Big{\\}},$

where

$\mathcal{N}^{VV}_{NNLO}=\left(\dfrac{\alpha_{s}\overline{C}(\epsilon)}{2\pi}\right)^{2}\mathcal{N}_{LO}\,.$
(69)

###  3.3 Integrated dipoles from antenna functions

We are now ready to discuss the construction of integrated dipoles in colour
space. We distinguish two types of integrated dipoles: identity-preserving
(IP) and identity-changing (IC). The former reproduce the infrared singularity
structure of virtual corrections and are naturally cast as operators in colour
space. The latter address identity-changing initial-state collinear
singularities.

####  3.3.1 One-loop integrated dipoles

We begin by discussing one-loop integrated dipoles, originally presented in
Currie:2013vh . They are obtained combining integrated three-parton tree-level
antenna functions $\mathcal{X}_{3}^{0}$ and NLO mass factorization kernels
$\Gamma_{ab}^{(1)}$.

##### Identity-preserving dipoles

We define the following one-loop singularity dipole operator in colour space
for an $(n+2)$-parton process:

$\displaystyle\mathbfcal{J}^{(1)}(\epsilon)$ $\displaystyle=\sum_{(i,j)\geq
3}(\bm{T}_{i}\cdot\bm{T}_{j})\,\mathcal{J}_{2}^{(1)}(i,j)+\sum_{i\neq
1,2}(\bm{T}_{1}\cdot\bm{T}_{i})\,\mathcal{J}_{2}^{(1)}(1,i)$ (70)
$\displaystyle\,+\sum_{i\neq
1,2}(\bm{T}_{2}\cdot\bm{T}_{i})\,\mathcal{J}_{2}^{(1)}(2,i)+(\bm{T}_{1}\cdot\bm{T}_{2})\,\mathcal{J}_{2}^{(1)}(1,2)\,.$

The first sum runs over all pairs of partons in the final state, the second
and the third sums include all pairs with an initial-state parton
(respectively $1$ or $2$) and a final-state one and the last term addresses
the configuration where both partons are in the initial state. The scalar
functions $\mathcal{J}_{2}^{(1)}(i,j)$ are identity-preserving colour stripped
one-loop integrated dipoles. These can be classified according to the flavour
of the partons $(i,j)$ and their kinematical configuration: final-final (FF),
initial-final (IF) or initial-initial (II). They can be further decomposed as:

$\displaystyle\mathcal{J}_{2}^{(1)}(q,\bar{q})$ $\displaystyle=$
$\displaystyle J_{2}^{(1)}(q,\bar{q}),$ (71)
$\displaystyle\mathcal{J}_{2}^{(1)}(i,g)$ $\displaystyle=$ $\displaystyle
J_{2}^{(1)}(i,g)+\dfrac{N_{f}}{N_{c}}\hat{J_{2}}^{(1)}(i,g),\quad i=q,g\,.$
(72)

We list the IP quark-antiquark, quark-gluon and gluon-gluon one-loop colour-
stripped integrated dipoles in Tables 1–3. We note that IF and II integrated
antenna functions exhibit a dependence on the momentum fractions $x_{1}$ and
$x_{2}$, which is omitted in the previous expressions and in the remainder of
the paper for simplicity. We use the following shorthand notation:
$\delta_{i}=\delta(1-x_{i})$. We introduced, where needed, dedicated
subscripts which accompany the kinematical configuration labels IF and II, to
distinguish between different choices of initial-state partons.

|  Integrated dipoles
---|---
FF |  $J_{2}^{(1)}(i_{q},j_{\bar{q}})=\mathcal{A}_{3}^{0}(s_{ij})$
IF |  $J_{2}^{(1)}(1_{q},i_{\bar{q}})=\mathcal{A}_{3,q}^{0}(s_{1i})-\Gamma^{(1)}_{qq}\left(x_{1}\right)\delta_{2}$
II |  $J_{2}^{(1)}(1_{q},2_{\bar{q}})=\mathcal{A}_{3,q\bar{q}}^{0}(s_{12})-\Gamma^{(1)}_{qq}\left(x_{1}\right)\delta_{2}-\Gamma^{(1)}_{qq}\left(x_{2}\right)\delta_{1}$
Table 1: Identity-preserving quark-antiquark one-loop colour-stripped integrated dipoles. |  Integrated dipoles
---|---
FF |  $J_{2}^{(1)}(i_{q},j_{g})=\frac{1}{2}\mathcal{D}_{3}^{0}(s_{ij})$
$\hat{J_{2}}^{(1)}(i_{q},j_{g})=\frac{1}{2}\mathcal{E}_{3}^{0}(s_{ij})$
IFq |  $J_{2}^{(1)}(1_{q},i_{g})=\frac{1}{2}\mathcal{D}_{3,q}^{0}(s_{1i})-\Gamma^{(1)}_{qq}\left(x_{1}\right)\delta_{2}$
$\hat{J_{2}}^{(1)}(1_{q},i_{g})=\frac{1}{2}\mathcal{E}_{3,q}^{0}(s_{1i})$
IFg |  $J_{2}^{(1)}(1_{g},i_{q})=\mathcal{D}_{3,g,gq}^{0}(s_{1i})-\frac{1}{2}\Gamma^{(1)}_{gg}\left(x_{1}\right)\delta_{2}$
$\hat{J_{2}}^{(1)}(1_{g},i_{q})=-\frac{1}{2}\widehat{\Gamma}^{(1)}_{gg}\left({x_{1}}\right)\delta_{2}$
II |  $J_{2}^{(1)}(1_{q},2_{g})=\mathcal{D}_{3,qg}^{0}(s_{12})-\Gamma^{(1)}_{qq}\left(x_{1}\right)\delta_{2}-\frac{1}{2}\Gamma^{(1)}_{gg}\left(x_{2}\right)\delta_{1}$
$\hat{J_{2}}^{(1)}(1_{q},2_{g})=-\frac{1}{2}\widehat{\Gamma}^{(1)}_{gg}\left({x_{2}}\right)\delta_{1}$
Table 2: Identity-preserving quark-gluon one-loop colour-stripped integrated dipoles. The subscripts indicate different choices of initial-state partons. |  Integrated dipoles
---|---
FF |  $J_{2}^{(1)}(i_{g},j_{g})=\frac{1}{3}\mathcal{F}_{3}^{0}(s_{ij})$
$\hat{J_{2}}^{(1)}(i_{g},j_{g})=\mathcal{G}_{3}^{0}(s_{ij})$
IF |  $J_{2}^{(1)}(1_{g},i_{g})=\frac{1}{2}\mathcal{F}_{3,g}^{0}(s_{1i})-\frac{1}{2}\Gamma^{(1)}_{gg}\left(x_{1}\right)\delta_{2}$
$\hat{J_{2}}^{(1)}(1_{g},i_{g})=\frac{1}{2}\mathcal{G}_{3,g}^{0}(s_{1i})-\frac{1}{2}\widehat{\Gamma}^{(1)}_{gg}\left({x_{1}}\right)\delta_{2}$
II |  $J_{2}^{(1)}(1_{g},2_{g})=\mathcal{F}_{3,gg}^{0}(s_{12})-\frac{1}{2}\Gamma^{(1)}_{gg}\left(x_{1}\right)\delta_{2}-\frac{1}{2}\Gamma^{(1)}_{gg}\left(x_{2}\right)\delta_{1}$
$\hat{J_{2}}^{(1)}(1_{g},2_{g})=-\frac{1}{2}\widehat{\Gamma}^{(1)}_{gg}\left({x_{1}}\right)\delta_{2}-\frac{1}{2}\widehat{\Gamma}^{(1)}_{gg}\left({x_{2}}\right)\delta_{1}$
Table 3: Identity-preserving gluon-gluon one-loop colour-stripped integrated
dipoles.

##### Cancellation of infrared singularities at one-loop

The structure of the IP integrated dipoles is chosen in such a way that the
singularities carried by the mass factorization kernels cancel with poles in
the integrated IF and II antenna functions associated with initial-state
collinear divergences. The remaining $\epsilon$-poles exactly match the ones
of the virtual matrix element, once the operator in (70) is evaluated on the
corresponding LO amplitude in colour space. In particular, at one loop the
following relation holds:

$\mathcal{P}oles\left[\mathcal{J}_{2}^{(1)}(i,j)\right]=\mathcal{P}oles\left[\text{Re}\left(\mathcal{I}^{(1)}_{ij}\left(\epsilon,\mu_{r}^{2}\right)\right)\right],$
(73)

where the integrated dipole on the left-hand-side can be in the FF, IF or II
configuration. The equation above establishes a correspondence between the
operators defined in (70) and (44), which will be crucial for the formulation
of the colourful antenna subtraction method.

##### Identity-changing dipoles

For the IF and II configurations, we also introduce identity-changing
integrated dipoles, obtained from IC antenna functions and splitting kernels.
Contrary to the IP ones, they are not defined as operators in colour space,
for reasons that will be explained in Section 4. Their structure can however
be inferred from the one of splitting kernels at NLO in (8):

$\mathcal{J}_{ab;cd}^{(1)}(x_{1},x_{2})=\mathcal{J}_{2,a\to
c}^{(1)}(x_{1})\delta_{db}\delta(1-x_{2})+\mathcal{J}_{2,b\to
d}^{(1)}(x_{2})\delta_{ca}\delta(1-x_{1}).$ (74)

We can distinguish between the quark-to-gluon and gluon-to-quark dipoles,
which reflect the decomposition of the splitting kernels associated to the
corresponding IC splitting:

$\displaystyle\mathcal{J}_{2,q\to g}^{(1)}(g,i)$ $\displaystyle=$
$\displaystyle\left(\dfrac{N_{c}^{2}-1}{N_{c}}\right)J_{2,q\to g}^{(1)}(g,i),$
(75) $\displaystyle\mathcal{J}_{2,g\to q}^{(1)}(q,i)$ $\displaystyle=$
$\displaystyle J_{2,g\to q}^{(1)}(q,i)\,,$ (76)

where the first argument denotes the parton species which enters the hard
scattering, while the second argument $i=q,g$ is a spectator parton which is
not directly involved in the initial-state collinear emission and can be
either in the final (IF) or initial (II) state. For identity-changing
configurations we need to consider that the spin-averaging factor for a gluon
and a quark differ in $d$-dimensions. To properly take this into account and
compensate for the mismatch within antenna functions and splitting kernels, we
introduce the following factors:

$S_{g\to q}=\dfrac{2-2\epsilon}{2}=1-\epsilon,\quad S_{q\to
g}=\dfrac{2}{2-2\epsilon}=\dfrac{1}{1-\epsilon}.$ (77)

The expressions for the IC quark-antiquark, quark-gluon and gluon-gluon one-
loop integrated dipoles are given in Tables 4–6.

|  Integrated dipoles
---|---
IF |  $J_{2,g\to q}^{(1)}(1_{q},i_{\bar{q}})=-\frac{1}{2}\mathcal{A}_{3,g}^{0}(s_{1i})-S_{g\to q}\Gamma^{(1)}_{qg}\left(x_{1}\right)\delta_{2}$
II |  $J_{2,g\to q}^{(1)}(1_{q},2_{\bar{q}})=-\mathcal{A}_{3,gq}^{0}(s_{12})-S_{g\to q}\Gamma^{(1)}_{qg}\left(x_{1}\right)\delta_{2}$
Table 4: Identity-changing quark-antiquark one-loop colour-stripped integrated dipoles. The subscripts indicate different choices of initial-state partons. |  Integrated dipoles
---|---
IFg→q |  $J_{2,g\to q}^{(1)}(1_{q},i_{g})=-\mathcal{D}_{3,g\to q}^{0}(s_{1i})-S_{g\to q}\Gamma^{(1)}_{qg}\left(x_{1}\right)\delta_{2}$
IFq→g |  $J_{2,q\to g}^{(1)}(1_{g},i_{q})=-\mathcal{E}_{3,q^{\prime}}^{0}(s_{1i})-S_{q\to g}\Gamma^{(1)}_{gq}\left(x_{1}\right)\delta_{2}$
IIg→q |  $J_{2,g\to q}^{(1)}(1_{q},2_{g})=-\mathcal{D}_{3,gg}^{0}(s_{12})-S_{g\to q}\Gamma^{(1)}_{qg}\left(x_{1}\right)\delta_{2}$
IIq→g |  $J_{2,q\to g}^{(1)}(1_{g},2_{q})=-\mathcal{E}_{3,q^{\prime}q}^{0}(s_{12})-S_{q\to g}\Gamma^{(1)}_{gq}\left(x_{1}\right)\delta_{2}$
Table 5: Identity-changing quark-gluon one-loop colour-stripped integrated dipoles. |  Integrated dipoles
---|---
IF |  $J_{2,q\to g}^{(1)}(1_{g},i_{g})=-\mathcal{G}_{3,q}^{0}(s_{1i})-S_{q\to g}\Gamma^{(1)}_{gq}\left(x_{1}\right)\delta_{2}$
II |  $J_{2,q\to g}^{(1)}(1_{g},2_{g})=-\mathcal{G}_{3,qg}^{0}(s_{12})-S_{q\to g}\Gamma^{(1)}_{gq}\left(x_{1}\right)\delta_{2}$
Table 6: Identity-changing gluon-gluon one-loop colour-stripped integrated
dipoles.

The IC integrated dipoles are free of explicit infrared singularities:

$\mathcal{P}oles\left[\mathcal{J}_{2,a\to b}^{(1)}(i,j)\right]=0,$ (78)

because the only $\epsilon$-poles which are present in integrated IC antenna
functions have an initial-state collinear origin and completely cancel against
the analogous ones in the IC splitting kernels.

####  3.3.2 Two-loop integrated dipoles

The characteristic new ingredients of two-loop integrated dipoles are
integrated four-parton tree-level antenna functions $\mathcal{X}_{4}^{0}$,
three-parton one-loop antenna functions $\mathcal{X}_{3}^{1}$, convolutions of
two three-parton tree-level antenna functions
$\mathcal{X}_{3}^{0}\otimes\mathcal{X}_{3}^{0}$ and two-loop mass
factorization kernels $\overline{\Gamma}_{ab}^{(2)}$.

Also at two-loop we distinguish between identity-preserving and identity-
changing integrated dipoles, however a significant difference with respect to
the one-loop case has to be highlighted. Explicit integrated results are
available at NLO, not only for all the three-parton tree-level antenna
functions, but also for any sub-antenna, given the simplicity of the
analytical integration. This means that both at the unintegrated and
integrated level one has complete control over which infrared-divergent
configurations are considered. This is not true at NNLO, where the
unavailability of individual integrated sub-antennae forces one to include
additional infrared singularities within the identity-preserving two-loop
integrated dipoles. These singularities have to be properly identified and
subtracted. This is achieved by introducing spurious terms which target the
undesired structures. We identify three classes of such terms:

  * •

flip-flopping contributions: address IC initial-state collinear limits like
$g\to q\to g$ or $q\to g\to q$, where the identity of the initial-state
emitter formally coincides with the original one. These contributions are
denoted in the following by the subscript $\mathrm{f/f}$;

  * •

IC corrective contributions: target genuine IC limits which need to be removed
from IP structures, denoted with the subscript $\mathrm{IC}\,\mathrm{corr}.$;

  * •

unphysical triple-collinear limits: limits existing in the supersymmetric
matrix elements used to extract antenna functions, which are not present in
QCD. They are treated introducing suitable insertion operators.

We will comment further about each class in the paragraphs below. Expressions
for the two-loop colour-stripped integrated dipoles used for the construction
of NNLO subtraction terms for di-jet production can be found in Chen:2022tpk .
In the following we review them and discuss in more detail how to generalize
their construction for any process. To simplify the expressions, we omit the
scale $s_{ij}$ dependence of the integrated antenna functions.

We mention that the complete development of designer antenna functions Braun-
White:2023sgd ; Braun-White:2023zwd  at NNLO will bring significant
simplifications to some of the structures presented below. Indeed, the
assemblage of antenna functions from specific infrared limits, rather than
from physical matrix elements, naturally prevents the introduction of spurious
divergences and therefore the necessity of the corrective terms introduced
above.

##### Identity-preserving dipoles

In analogy with (70), we can define a two-loop insertion operator in colour
space:

$\displaystyle\mathbfcal{J}^{(2)}(\epsilon)$
$\displaystyle=N_{c}\sum_{(i,j)\geq
3}(\bm{T}_{i}\cdot\bm{T}_{j})\,\mathcal{J}_{2}^{(2)}(i_{g},j_{g})+N_{c}\sum_{i\neq
1,2}(\bm{T}_{1}\cdot\bm{T}_{i})\,\mathcal{J}_{2}^{(2)}(\hat{1}_{g},i_{g})$
(79) $\displaystyle\,+N_{c}\sum_{i\neq
1,2}(\bm{T}_{2}\cdot\bm{T}_{i})\,\mathcal{J}_{2}^{(2)}(\hat{2}_{g},i_{g})+N_{c}\,(\bm{T}_{1}\cdot\bm{T}_{2})\,\mathcal{J}_{2}^{(2)}(\hat{1}_{g},\hat{2}_{g})\,.$

where the functions accompanying the charge dipoles are identity-preserving
two-loop colour-stripped integrated dipoles. They can be decomposed according
to:

$\displaystyle\mathcal{J}_{2}^{(2)}\left(q,\bar{q}\right)$ $\displaystyle=$
$\displaystyle
J_{2}^{(2)}\left(q,\bar{q}\right)-\dfrac{1}{N_{c}^{2}}\tilde{J_{2}}^{(2)}\left(q,\bar{q}\right)+\dfrac{N_{f}}{N_{c}}\hat{J_{2}}^{(2)}\left(q,\bar{q}\right)$
(80) $\displaystyle\mathcal{J}_{2}^{(2)}\left(g,i\right)$ $\displaystyle=$
$\displaystyle
J_{2}^{(2)}\left(g,i\right)+\dfrac{N_{f}}{N_{c}}\,\hat{J_{2}}^{(2)}\left(g,i\right)$
(81) $\displaystyle-$
$\displaystyle\dfrac{N_{f}}{N_{c}^{3}}\hat{\tilde{J_{2}}}^{(2)}\left(g,i\right)+\dfrac{N_{f}^{2}}{N_{c}^{2}}\,\hat{\hat{J_{2}}}^{(2)}\left(g,i\right),\quad
i=g,q\,.$

##### Quark-antiquark integrated dipoles

The IP two-loop quark-antiquark colour-stripped integrated dipoles can be
constructed in a straightforward manner, since the quark-antiquark NNLO
antenna functions specifically contain only the desired physical limits in any
kinematical configuration and allow for a unique identification of the hard
radiator partons. Even if sub-antennae need to be employed, for example for
$\widetilde{A}_{4}^{0}$ Gehrmann-DeRidder:2007foh , this is only required to
properly define a colour-ordered momentum mapping. Therefore, for a given hard
quark-antiquark dipole, no spurious structure is present in the associated
integrated antenna functions. The IP quark-antiquark two-loop integrated
dipoles are listed in Table 7.

|  Integrated dipoles
---|---
FF |  $J_{2}^{(2)}\left(i_{q},j_{\bar{q}}\right)=\mathcal{A}_{4}^{0}+\mathcal{A}_{3}^{1}+\frac{b_{0}}{\epsilon}\left(\frac{|s_{ij}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{A}_{3}^{0}-\frac{1}{2}\left[\mathcal{A}_{3}^{0}\otimes\mathcal{A}_{3}^{0}\right]$
$\tilde{J_{2}}^{(2)}\left(1_{q},2_{\bar{q}}\right)=\frac{1}{2}\widetilde{\mathcal{A}}_{4}^{0}+2\mathcal{C}_{4}^{0}+\widetilde{\mathcal{A}}_{3}^{1}-\frac{1}{2}\left[\mathcal{A}_{3}^{0}\otimes\mathcal{A}_{3}^{0}\right]$
$\hat{J_{2}}^{(2)}\left(1_{q},2_{\bar{q}}\right)=\mathcal{B}_{4}^{0}+\widehat{\mathcal{A}}_{3}^{1}+\frac{b_{0,F}}{\epsilon}\left(\frac{|s_{ij}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{A}_{3}^{0}$
IF |  $J_{2}^{(2)}\left(1_{q},i_{\bar{q}}\right)=\mathcal{A}_{4,q}^{0}+\mathcal{A}_{3,q}^{1}+\frac{b_{0}}{\epsilon}\left(\frac{|s_{1i}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{A}_{3,q}^{0}-\frac{1}{2}\left[\mathcal{A}_{3,q}^{0}\otimes\mathcal{A}_{3,q}^{0}\right]$
$\phantom{J_{2}^{(2)}\left(1_{q},i_{\bar{q}}\right)}-\overline{\Gamma}^{(2)}_{qq}\left(x_{1}\right)\delta_{2}$
$\tilde{J_{2}}^{(2)}\left(1_{q},i_{\bar{q}}\right)=\frac{1}{2}\widetilde{\mathcal{A}}_{4,q}^{0}+2\mathcal{C}_{4,q}^{0}+\mathcal{C}_{4,\bar{q}}^{0}+\widetilde{\mathcal{A}}_{3,q}^{1}-\frac{1}{2}\left[\mathcal{A}_{3,q}^{0}\otimes\mathcal{A}_{3,q}^{0}\right]$
$\phantom{\tilde{J_{2}}^{(2)}\left(1_{q},i_{\bar{q}}\right)}+\widetilde{\widetilde{\overline{\Gamma}}}^{(2)}_{qq}\left(x_{1}\right)\delta_{2}$
$\hat{J_{2}}^{(2)}\left(1_{q},i_{\bar{q}}\right)=\mathcal{B}_{4,q}^{0}+\widehat{\mathcal{A}}_{3,q}^{1}+\frac{b_{0,F}}{\epsilon}\left(\frac{|s_{1i}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{A}_{3,q}^{0}-\widehat{\overline{\Gamma}}^{(2)}_{qq}\left(x_{1}\right)\delta_{2}$
II |  $J_{2}^{(2)}\left(1_{q},2_{\bar{q}}\right)=\mathcal{A}_{4,q\bar{q}}^{0}+\mathcal{A}_{3,q\bar{q}}^{1}+\frac{b_{0}}{\epsilon}\left(\frac{|s_{12}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{A}_{3,q\bar{q}}^{0}-\frac{1}{2}\left[\mathcal{A}_{3,q\bar{q}}^{0}\otimes\mathcal{A}_{3,q\bar{q}}^{0}\right]$
$\phantom{J_{2}^{(2)}\left(1_{q},2_{\bar{q}}\right)}-\overline{\Gamma}^{(2)}_{qq}\left(x_{1}\right)\delta_{2}-\overline{\Gamma}^{(2)}_{qq}\left(x_{2}\right)\delta_{1}$
$\tilde{J_{2}}^{(2)}\left(1_{q},2_{\bar{q}}\right)=\frac{1}{2}\widetilde{\mathcal{A}}_{4,q\bar{q}}^{0}+2\mathcal{C}_{4,q\bar{q}}^{0}+2\mathcal{C}_{4,\bar{q}q}^{0}+\widetilde{\mathcal{A}}_{3,q\bar{q}}^{1}-\frac{1}{2}\left[\mathcal{A}_{3,q\bar{q}}^{0}\otimes\mathcal{A}_{3,q\bar{q}}^{0}\right]$
$\phantom{\tilde{J_{2}}^{(2)}\left(1_{q},2_{\bar{q}}\right)}+\widetilde{\widetilde{\overline{\Gamma}}}^{(2)}_{qq}\left(x_{1}\right)\delta_{2}+\widetilde{\widetilde{\overline{\Gamma}}}^{(2)}_{qq}\left(x_{2}\right)\delta_{1}$
$\hat{J_{2}}^{(2)}\left(1_{q},2_{\bar{q}}\right)=\mathcal{B}_{4,q{\bar{q}}}^{0}+\widehat{\mathcal{A}}_{3,qq}^{1}+\frac{b_{0,F}}{\epsilon}\left(\frac{|s_{12}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{A}_{3,qq}^{0}$
$\phantom{\hat{J_{2}}^{(2)}\left(1_{q},2_{\bar{q}}\right)}-\widehat{\overline{\Gamma}}^{(2)}_{qq}\left(x_{1}\right)\delta_{2}-\widehat{\overline{\Gamma}}^{(2)}_{qq}\left(x_{2}\right)\delta_{1}$
Table 7: Identity-preserving quark-antiquark two-loop colour-stripped
integrated dipoles.

##### Quark-gluon integrated dipoles

The IP quark-gluon dipoles are particularly affected by the presence of
undesired singularities. This is mostly due to quark-gluon NNLO antenna
functions containing any allowed unresolved limit among the involved partons,
with no possibility to clearly disentangle the hard radiators from the
unresolved partons at the integrated level. The IP quark-gluon two-loop colour
stripped integrated dipoles are given in Table 8.

|  Integrated dipoles
---|---
FF |  $J_{2}^{(2)}\left(i_{q},j_{g}\right)=\frac{1}{2}\mathcal{D}_{4}^{0}+\frac{1}{2}\mathcal{D}_{3}^{1}+\frac{1}{2}\frac{b_{0}}{\epsilon}\left(\frac{|s_{ij}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{D}_{3}^{0}-\frac{1}{4}\left[\mathcal{D}_{3}^{0}\otimes\mathcal{D}_{3}^{0}\right]$
$\hat{J_{2}}^{(2)}\left(i_{q},j_{g}\right)=\mathcal{E}_{4}^{0}+\frac{1}{2}\widehat{\mathcal{D}}_{3}^{1}+\frac{1}{2}\mathcal{E}_{3}^{1}+\frac{1}{2}\frac{b_{0,F}}{\epsilon}\left(\frac{|s_{ij}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{D}_{3}^{0}+\frac{1}{2}\frac{b_{0}}{\epsilon}\left(\frac{|s_{ij}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{E}_{3}^{0}$
$\phantom{\hat{J_{2}}^{(2)}\left(i_{q},j_{g}\right)}-\frac{1}{2}\left[\mathcal{E}_{3}^{0}\otimes\mathcal{D}_{3}^{0}\right]$
$\hat{\tilde{J_{2}}}^{(2)}\left(i_{q},j_{g}\right)=\frac{1}{2}\widetilde{\mathcal{E}}_{4}^{0}+\frac{1}{2}\widetilde{\mathcal{E}}_{3}^{1}$
$\hat{\hat{J_{2}}}^{(2)}\left(i_{q},j_{g}\right)=\frac{1}{2}\widehat{\mathcal{E}}_{3}^{1}+\frac{1}{2}\frac{b_{0,F}}{\epsilon}\left(\frac{|s_{ij}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{E}_{3}^{0}-\frac{1}{4}\left[\mathcal{E}_{3}^{0}\otimes\mathcal{E}_{3}^{0}\right]$
IFq |  $J_{2}^{(2)}\left(1_{q},i_{g}\right)=\frac{1}{2}\mathcal{D}_{4,q}^{0}+\frac{1}{2}\mathcal{D}_{3,q}^{1}+\frac{1}{2}\frac{b_{0}}{\epsilon}\left(\frac{|s_{1i}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{D}_{3,q}^{0}-\frac{1}{4}\left[\mathcal{D}_{3,q}^{0}\otimes\mathcal{D}_{3,q}^{0}\right]$
$\phantom{J_{2}^{(2)}\left(1_{q},i_{g}\right)}-\overline{\Gamma}^{(2)}_{qq}\left(x_{1}\right)\delta_{2}$
$\hat{J_{2}}^{(2)}\left(1_{q},i_{g}\right)=\mathcal{E}_{4,q}^{0}+\frac{1}{2}\widehat{\mathcal{D}}_{3,q}^{1}+\frac{1}{2}\mathcal{E}_{3,q}^{1}+\frac{1}{2}\frac{b_{0,F}}{\epsilon}\left(\frac{|s_{1i}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{D}_{3,q}^{0}$
$\phantom{\hat{J_{2}}^{(2)}\left(1_{q},i_{g}\right)}+\frac{1}{2}\frac{b_{0}}{\epsilon}\left(\frac{|s_{1i}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{E}_{3,q}^{0}-\frac{1}{2}\left[\mathcal{E}_{3,q}^{0}\otimes\mathcal{D}_{3,q}^{0}\right]-\widehat{\overline{\Gamma}}^{(2)}_{qq}\left(x_{1}\right)\delta_{2}$
$\hat{\tilde{J_{2}}}^{(2)}\left(1_{q},i_{g}\right)=\frac{1}{2}\widetilde{\mathcal{E}}_{4,q}^{0}+\frac{1}{2}\widetilde{\mathcal{E}}_{3,q}^{1}$
$\hat{\hat{J_{2}}}^{(2)}\left(1_{q},i_{g}\right)=\frac{1}{2}\widehat{\mathcal{E}}_{3,q}^{1}+\frac{1}{2}\frac{b_{0,F}}{\epsilon}\left(\frac{|s_{1i}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{E}_{3,q}^{0}-\frac{1}{4}\left[\mathcal{E}_{3,q}^{0}\otimes\mathcal{E}_{3,q}^{0}\right]$
IFg |  $J_{2}^{(2)}\left(1_{g},i_{q}\right)=\mathcal{D}_{4,g}^{0}+\frac{1}{2}\mathcal{D}_{4,g^{\prime}}^{0}+\mathcal{D}_{3,g}^{1}+\frac{b_{0}}{\epsilon}\left(\frac{|s_{1i}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{D}_{3,g}^{0}$
$\phantom{J_{2}^{(2)}\left(1_{g},i_{q}\right)}-\left[\mathcal{D}_{3,g\to
g}^{0}\otimes\mathcal{D}_{3,g\to
g}^{0}\right]-\frac{1}{2}\overline{\Gamma}^{(2)}_{gg}\left(x_{1}\right)\delta_{2}+J_{2,\mathrm{IC}\,\mathrm{corr}.}^{(2)}(q,{\bar{q}})$
$\hat{J_{2}}^{(2)}\left(1_{g}i,_{q}\right)=\mathcal{E}_{4,g}^{0}+\widehat{\mathcal{D}}_{3,g}^{1}+\frac{b_{0,F}}{\epsilon}\left(\frac{|s_{1i}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{D}_{3,g}^{0}$
$\phantom{\hat{J_{2}}^{(2)}\left(1_{g},i_{q}\right)}-\frac{1}{2}\widehat{\overline{\Gamma}}^{(2)}_{gg}\left(x_{1}\right)\delta_{2}+\hat{J}_{2,\mathrm{f/f}}^{(2)}\left(1_{g},i_{q}\right)+\hat{J}_{2,\mathrm{IC}\,\mathrm{corr}.}^{(2)}(q,{\bar{q}})$
$\hat{\tilde{J_{2}}}^{(2)}\left(1_{g},i_{q}\right)=\frac{1}{2}\widetilde{\mathcal{E}}_{4,g}^{0}+\frac{1}{2}\widehat{\widetilde{\overline{\Gamma}}}^{(2)}_{gg}\left(x_{1}\right)\delta_{2}+\hat{\tilde{J}}_{2,\mathrm{f/f}}^{(2)}\left(1_{g},i_{q}\right)$
$\hat{\hat{J_{2}}}^{(2)}\left(1_{g},i_{q}\right)=-\frac{1}{2}\widehat{\widehat{\overline{\Gamma}}}^{(2)}_{gg}\left(x_{1}\right)\delta_{2}$
II |  $J_{2}^{(2)}\left(1_{q},2_{g}\right)=\mathcal{D}_{4,qg}^{0}+\frac{1}{2}\mathcal{D}_{4,qg^{\prime}}^{0}+\mathcal{D}_{3,qg}^{1}+\frac{b_{0}}{\epsilon}\left(\frac{|s_{12}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{D}_{3,qg}^{0}$
$\phantom{J_{2}^{(2)}\left(1_{q},2_{g}\right)}-\left[\mathcal{D}_{3,qg}^{0}\otimes\mathcal{D}_{3,qg}^{0}\right]-\widehat{\overline{\Gamma}}^{(2)}_{qq}\left(x_{1}\right)\delta_{2}-\frac{1}{2}\widehat{\overline{\Gamma}}^{(2)}_{gg}\left(x_{2}\right)\delta_{1}$
$\hat{J_{2}}^{(2)}\left(1_{q},2_{g}\right)=\mathcal{E}_{4,qg}^{0}+\frac{1}{2}\widehat{\mathcal{D}}_{3,qg}^{1}+\frac{b_{0,F}}{\epsilon}\left(\frac{|s_{12}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{D}_{3,qg}^{0}-\widehat{\overline{\Gamma}}^{(2)}_{qq}\left(x_{1}\right)\delta_{2}$
$\phantom{\hat{J_{2}}^{(2)}\left(1_{q},2_{g}\right)}-\frac{1}{2}\widehat{\overline{\Gamma}}^{(2)}_{gg}\left(x_{2}\right)\delta_{1}+\hat{J}_{2,\mathrm{f/f}}^{(2)}\left(1_{q}2_{g}\right)$
$\hat{\tilde{J_{2}}}^{(2)}\left(1_{q},2_{g}\right)=\frac{1}{2}\widetilde{\mathcal{E}}_{4,qg}^{0}+\frac{1}{2}\widehat{\widetilde{\overline{\Gamma}}}^{(2)}_{gg}\left(x_{2}\right)\delta_{1}+\hat{\tilde{J}}_{2,\mathrm{f/f}}^{(2)}\left(1_{q}2_{g}\right)$
Table 8: Identity-preserving quark-gluon two-loop colour-stripped integrated
dipoles. The subscripts indicate different choices of initial-state partons.

As anticipated, we see the appearance of corrective contributions. The flip-
flopping terms remove triple-collinear limits $g\parallel
q^{\prime}\parallel{\bar{q}}^{\prime}$ with the gluon in the initial state,
which are present in the $\mathcal{E}_{4,g}^{0}$,
$\widetilde{\mathcal{E}}_{4,g}^{0}$, $\mathcal{E}_{4,qg}^{0}$ and
$\widetilde{\mathcal{E}}_{4,qg}^{0}$ integrated antenna functions.
Technically, the identity of the hard initial-state parton after integration
is preserved, but the infrared divergences associated to such an unresolved
configuration are absorbed by IC spitting kernels, rather than by virtual
corrections. Therefore, in the definition of genuinely IP integrated dipoles,
these singularities need to be removed. The flip-flopping terms are given in
Table 9.

|  Integrated dipoles
---|---
IF |  $\hat{J}_{2,\mathrm{f/f}}^{(2)}\left(1_{g},i_{q}\right)=S_{g\to q}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\mathcal{E}_{3,q^{\prime}}^{0}\right]+\frac{1}{2}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\Gamma^{(1)}_{gq}(x_{1})\right]$
|  $\hat{\tilde{J}}_{2,\mathrm{f/f}}^{(2)}\left(1_{g},i_{q}\right)=S_{g\to
q}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\mathcal{E}_{3,q^{\prime}}^{0}\right]+\frac{1}{2}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\Gamma^{(1)}_{gq}(x_{1})\right]$
II |  $\hat{J}_{2,\mathrm{f/f}}^{(2)}\left(1_{g},2_{q}\right)=S_{g\to q}\left[\Gamma^{(1)}_{qg}(x_{2})\otimes\mathcal{E}_{3,qq^{\prime}}^{0}\right]+\frac{1}{2}\left[\Gamma^{(1)}_{qg}(x_{2})\otimes\Gamma^{(1)}_{gq}(x_{2})\right]$
|  $\hat{\tilde{J}}_{2,\mathrm{f/f}}^{(2)}\left(1_{g},2_{q}\right)=S_{g\to
q}\left[\Gamma^{(1)}_{qg}(x_{2})\otimes\mathcal{E}_{3,qq^{\prime}}^{0}\right]+\frac{1}{2}\left[\Gamma^{(1)}_{qg}(x_{2})\otimes\Gamma^{(1)}_{gq}(x_{2})\right]$
Table 9: Flip-flopping contributions to identity-preserving quark-gluon two-
loop integrated dipoles.

Analogously, spurious IC singularities are present in the
$\mathcal{D}_{3,g}^{0}$, $\mathcal{D}_{3,g}^{1}$,
$\widehat{\mathcal{D}}_{3,g}^{1}$ and $\mathcal{D}_{4,g}^{0}$ antenna
functions. In particular, these gluon-initiated antenna functions contain
$\epsilon$-poles due to the final-state quark becoming collinear to the
initial-state gluon. This is an IC unresolved limit, which cannot be isolated
from the remaining IP ones due to the unavailability of dedicated integrated
sub-antennae. It is possible to remove such singularities by a suitable
combination of gluon-initiated IC quark-gluon and quark-antiquark antenna
functions. The former have the same gluon and quark involved in the dipole as
hard radiators. The latter have one quark (antiquark) coinciding with the one
in the quark-gluon dipole, and so colour-connected to the initial-state gluon,
while the other antiquark (quark) is its colour-connected partner. We note
that in such quark-antiquark antennae, no divergent behaviour is associated
with the antiquark (quark) becoming collinear to the initial-state gluon. If
the antiquark (quark) is in the final state, the introduced antenna functions
will be in the initial-final configurations, otherwise, if it is in the
initial state, the antenna functions will be in the initial-initial
configuration. The explicit expressions according to the kinematical
configurations of the quark-antiquark pair are given in Table 10.

|  Integrated dipoles
---|---
IF |  $J_{2,\mathrm{IC}\,\mathrm{corr}}^{(2)}\left(1_{q},i_{{\bar{q}}}\right)=-\left[\mathcal{D}_{3,q}^{0}\otimes\mathcal{D}_{3,g\to q}^{0}\right]-\left[\Gamma^{(1)}_{gg}(x_{1})\otimes\mathcal{D}_{3,g\to q}^{0}\right]$
$\phantom{J_{2,\mathrm{IC}\,\mathrm{corr}}^{(2)}\left(1_{q},i_{{\bar{q}}}\right)}+2\left[\Gamma^{(1)}_{qq}(x_{1})\otimes\mathcal{D}_{3,g\to
q}^{0}\right]-\mathcal{A}_{4,g}^{0}-\frac{1}{2}\widetilde{\mathcal{A}}_{4,g}^{0}-\frac{1}{2}\mathcal{A}_{3,g}^{1}$
$\phantom{J_{2,\mathrm{IC}\,\mathrm{corr}}^{(2)}\left(1_{q},i_{{\bar{q}}}\right)}-\frac{1}{2}\widetilde{\mathcal{A}}_{3,g}^{1}-\frac{1}{2}\frac{b_{0}}{\epsilon}\left(\left(\frac{|s_{1i}|}{\mu_{r}^{2}}\right)^{-\epsilon}-1\right)\mathcal{A}_{3,g}^{0}+\left[\mathcal{A}_{3,q}^{0}\otimes\mathcal{A}_{3,g}^{0}\right]$
$\phantom{J_{2,\mathrm{IC}\,\mathrm{corr}}^{(2)}\left(1_{q},i_{{\bar{q}}}\right)}+\frac{1}{2}\left[\Gamma^{(1)}_{gg}(x_{1})\otimes\mathcal{A}_{3,g}^{0}\right]-\left[\Gamma^{(1)}_{qq}(x_{1})\otimes\mathcal{A}_{3,g}^{0}\right]$
$\phantom{J_{2,\mathrm{IC}\,\mathrm{corr}}^{(2)}\left(1_{q},i_{{\bar{q}}}\right)}-\frac{b_{0}}{\epsilon}\left(\mathcal{D}_{3,g}^{0}-\mathcal{D}_{3,g\to
g}^{0}\right)$
$\hat{J}_{2,\mathrm{IC}\,\mathrm{corr}}^{(2)}\left(1_{g},i_{{\bar{q}}}\right)=-\left[\mathcal{E}_{3,q}^{0}\otimes\mathcal{D}_{3,g\to
q}^{0}\right]-\left[\widehat{\Gamma}^{(1)}_{gg}(x_{1})\otimes\mathcal{D}_{3,g\to
q}^{0}\right]$
$\phantom{\hat{J}_{2,\mathrm{IC}\,\mathrm{corr}}^{(2)}\left(1_{g},i_{{\bar{q}}}\right)}-\frac{1}{2}\widehat{\mathcal{A}}_{3,g}^{1}-\frac{1}{2}\frac{b_{0,F}}{\epsilon}\left(\left(\frac{|s_{1i}|}{\mu_{r}^{2}}\right)^{-\epsilon}-1\right)\mathcal{A}_{3,g}^{0}$
$\phantom{\hat{J}_{2,\mathrm{IC}\,\mathrm{corr}}^{(2)}\left(1_{g},i_{{\bar{q}}}\right)}+\frac{1}{2}\left[\widehat{\Gamma}^{(1)}_{gg}(x_{1})\otimes\mathcal{A}_{3,g}^{0}\right]-\frac{b_{0,F}}{\epsilon}\left(\mathcal{D}_{3,g}^{0}-\mathcal{D}_{3,g\to
g}^{0}\right)$
II |  $J_{2,\mathrm{IC}\,\mathrm{corr}}^{(2)}\left(1_{q},2_{{\bar{q}}}\right)=-\left[\mathcal{D}_{3,q}^{0}\otimes\mathcal{D}_{3,g\to q}^{0}\right]-\left[\Gamma^{(1)}_{gg}(x_{1})\otimes\mathcal{D}_{3,g\to q}^{0}\right]$
$\phantom{J_{2,\mathrm{IC}\,\mathrm{corr}}^{(2)}\left(1_{q},2_{{\bar{q}}}\right)}+2\left[\Gamma^{(1)}_{qq}(x_{1})\otimes\mathcal{D}_{3,g\to
q}^{0}\right]-\mathcal{A}_{4,qg}^{0}-\mathcal{A}_{4,qg^{\prime}}^{0}$
$\phantom{J_{2,\mathrm{IC}\,\mathrm{corr}}^{(2)}\left(1_{q},2_{{\bar{q}}}\right)}-\frac{1}{2}\widetilde{\mathcal{A}}_{4,qg}^{0}-\frac{1}{2}\mathcal{A}_{3,qg}^{1}-\frac{1}{2}\widetilde{\mathcal{A}}_{3,qg}^{1}$
$\phantom{J_{2,\mathrm{IC}\,\mathrm{corr}}^{(2)}\left(1_{q},2_{{\bar{q}}}\right)}-\frac{b_{0}}{\epsilon}\left(\left(\frac{|s_{12}|}{\mu_{r}^{2}}\right)^{-\epsilon}-1\right)\mathcal{A}_{3,gq}^{0}+2\left[\mathcal{A}_{3,gq}^{0}\otimes\mathcal{A}_{3,gq}^{0}\right]$
$\phantom{J_{2,\mathrm{IC}\,\mathrm{corr}}^{(2)}\left(1_{q},2_{{\bar{q}}}\right)}+\left[\Gamma^{(1)}_{gg}(x_{1})\otimes\mathcal{A}_{3,gq}^{0}\right]-2\left[\Gamma^{(1)}_{qq}(x_{1})\otimes\mathcal{A}_{3,gq}^{0}\right]$
$\phantom{J_{2,\mathrm{IC}\,\mathrm{corr}}^{(2)}\left(1_{q},2_{{\bar{q}}}\right)}-\frac{b_{0}}{\epsilon}\left(\mathcal{D}_{3,g}^{0}-\mathcal{D}_{3,g\to
g}^{0}\right)$
$\hat{J}_{2,\mathrm{IC}\,\mathrm{corr}}^{(2)}\left(1_{q},2_{{\bar{q}}}\right)=-\left[\mathcal{E}_{3,q}^{0}\otimes\mathcal{D}_{3,g\to
q}^{0}\right]-\left[\widehat{\Gamma}^{(1)}_{gg}(x_{1})\otimes\mathcal{D}_{3,g\to
q}^{0}\right]$
$\phantom{\hat{J}_{2,\mathrm{IC}\,\mathrm{corr}}^{(2)}\left(1_{q},2_{{\bar{q}}}\right)}-\widehat{\mathcal{A}}_{3,gq}^{1}-\frac{b_{0,F}}{\epsilon}\left(\left(\frac{|s_{12}|}{\mu_{r}^{2}}\right)^{-\epsilon}-1\right)\mathcal{A}_{3,gq}^{0}$
$\phantom{\hat{J}_{2,\mathrm{IC}\,\mathrm{corr}}^{(2)}\left(1_{q},2_{{\bar{q}}}\right)}+\left[\widehat{\Gamma}^{(1)}_{gg}(x_{1})\otimes\mathcal{A}_{3,gq}^{0}\right]-\frac{b_{0,F}}{\epsilon}\left(\mathcal{D}_{3,g}^{0}-\mathcal{D}_{3,g\to
g}^{0}\right)$
Table 10: Corrective terms required to remove spurious identity-changing
singularities from integrated identity-preserving gluon-initiated quark-gluon
antenna functions.

These corrective terms display a quite involved structure. Indeed, even if the
presence of spurious singularities within integrated antenna functions is well
understood and formally under control, they inevitably complicate the
structure of IP integrated dipoles as well as of the subtraction terms. In
particular, such corrections propagate into all the layers of the subtraction
infrastructure and require careful treatment in any practical implementation.

Finally, the $D$-type tree-level four-parton antenna functions
$D_{4}^{0}(q,g_{1},g_{2},g_{3})$ (FF, IF or II) present a triple-collinear
$g_{3}\parallel q\parallel g_{1}$ limit which is an unphysical remnant due to
the cyclicity of the super-symmetric matrix element used to extract the
antenna functions Gehrmann-DeRidder:2005svg . This limit shows up at the
integrated level as spurious $\epsilon$-poles which need to be removed from
the integrated dipoles. The collinear poles in a quark-gluon dipole are
described by the hard radiation function
$\mathcal{H}^{(2)}_{gq}\left(\epsilon\right)=\mathcal{H}^{(2)}_{g}\left(\epsilon\right)+\mathcal{H}^{(2)}_{q}\left(\epsilon\right)$,
with $\mathcal{H}^{(2)}_{g}\left(\epsilon\right)$ and
$\mathcal{H}^{(2)}_{q}\left(\epsilon\right)$ given in (62) and (63). On the
contrary, for the decay of a neutralino to a gluon and a gluino $\chi\to
g\tilde{g}$, the process used to extract the antenna functions, the collinear
poles are described by
$\mathcal{H}^{(2)}_{g\tilde{g}}\left(\epsilon\right)=\mathcal{H}^{(2)}_{g}\left(\epsilon\right)+\mathcal{H}^{(2)}_{\tilde{g}}\left(\epsilon\right)$,
with the quark hard radiation function replace by the gluino one Gehrmann-
DeRidder:2005svg ; Chen:2023egx :

$\mathcal{H}^{(2)}_{\tilde{g}}\left(\epsilon\right)=\dfrac{e^{\epsilon\gamma_{E}}}{4\Gamma(1-\epsilon)\epsilon}N_{c}\left\\{\left[-\dfrac{187}{216}+\dfrac{13}{48}\pi^{2}-\dfrac{1}{2}\zeta_{3}\right]+\dfrac{N_{f}}{N_{c}}\left[-\dfrac{25}{108}+\dfrac{\pi^{2}}{24}\right]\right\\}.$
(82)

The unphysical poles which are not present in QCD matrix elements are
identified by the mismatch between the $N_{f}=0$ components of the quark and
the gluino hard radiation functions:

$\displaystyle\mathcal{H}^{(2)}_{q}\left(\epsilon\right)-\mathcal{H}^{(2)}_{\tilde{g}}\left(\epsilon\right)$
$\displaystyle=$
$\displaystyle\dfrac{e^{\epsilon\gamma_{E}}N_{c}}{4\Gamma(1-\epsilon)\epsilon}\Biggl{\\{}\left[\dfrac{29}{16}-\dfrac{\pi^{2}}{2}+4\zeta_{3}\right]+\dfrac{1}{N_{c}^{2}}\left[\dfrac{3}{16}-\dfrac{\pi^{2}}{4}+3\zeta_{3}\right]\Biggr{\\}}.$
(83)

To compensate for the discrepancy above we introduce an auxiliary quark-
antiquark two-loop integrated dipole:

$\overline{\mathcal{J}}_{2}^{(2)}(q,{\bar{q}})=\dfrac{1}{2}\bar{J_{2}}^{(2)}(q,{\bar{q}})+\dfrac{1}{2N_{c}^{2}}\tilde{\bar{J_{2}}}^{(2)}(q,{\bar{q}}),$
(84)

which satisfies:

$\overline{\mathcal{J}}_{2}^{(2)}(q,{\bar{q}})=\mathcal{H}^{(2)}_{q}\left(\epsilon\right)\Bigg{|}_{N_{f}=0}-\mathcal{H}^{(2)}_{\tilde{g}}\left(\epsilon\right)\Bigg{|}_{N_{f}=0}+\mathcal{O}(\epsilon^{0}).$
(85)

The integrated dipoles in (84) can be written in terms of integrated antenna
functions and are listed in Table 11.

|  Integrated dipoles
---|---
FF |  $\bar{J_{2}}^{(2)}(i_{q},j_{\bar{q}})=\frac{1}{2}\widetilde{\mathcal{A}}_{4}^{0}+\widetilde{\mathcal{A}}_{3}^{1}-\frac{1}{2}\left[\mathcal{A}_{3}^{0}\otimes\mathcal{A}_{3}^{0}\right]$
$\tilde{\bar{J_{2}}}^{(2)}(i_{q},j_{\bar{q}})=\tilde{J_{2}}^{(2)}(i_{q},j_{\bar{q}})$
IF |  $\bar{J_{2}}^{(2)}(1_{q},i_{\bar{q}})=\frac{1}{2}\widetilde{\mathcal{A}}_{4,q}^{0}+\widetilde{\mathcal{A}}_{3,q}^{1}-\frac{1}{2}\left[\mathcal{A}_{3,q}^{0}\otimes\mathcal{A}_{3,q}^{0}\right]$
$\tilde{\bar{J_{2}}}^{(2)}(1_{q},i_{\bar{q}})=\tilde{J_{2}}^{(2)}(i_{q},j_{\bar{q}})$
II |  $\bar{J_{2}}^{(2)}(1_{q},2_{\bar{q}})=\frac{1}{2}\widetilde{\mathcal{A}}_{4,q{\bar{q}}}^{0}+\widetilde{\mathcal{A}}_{3,q{\bar{q}}}^{1}-\frac{1}{2}\left[\mathcal{A}_{3,q{\bar{q}}}^{0}\otimes\mathcal{A}_{3,q{\bar{q}}}^{0}\right]$
$\tilde{\bar{J_{2}}}^{(2)}(1_{q},2_{\bar{q}})=\tilde{J_{2}}^{(2)}(i_{q},j_{\bar{q}})$
Table 11: Auxiliary quark-antiquark two-loop integrated dipoles needed to
removes spurious poles present in integrated quark-gluon antenna functions,
which are not present in physical matrix elements.

The combination of antenna functions which gives the auxiliary dipoles was
found previously in Gehrmann-DeRidder:2007foh  and in Currie:2013vh  during
the construction of double-real subtraction terms for $e^{+}e^{-}\to 3j$ and
$pp\to jj$, as a correction to remove the unphysical triple-collinear limits
of the $D$-type antenna functions. The presence of such a combination can be
easily tracked down due to the anomalous appearance of subleading-colour
quark-antiquark antenna functions within the leading-colour contribution to
the subtraction terms.

The newly introduced $\overline{\mathcal{J}}_{2}^{(2)}(q,{\bar{q}})$ can be
promoted to an operator in colour space, however its structure depends on the
number of fermionic lines present in the process. For a process involving a
single quark-antiquark pair $(q,{\bar{q}})$ and an arbitrary number of gluons,
we have:

$\overline{\mathbfcal{J}}^{(2)}=N_{c}\sum_{g}\left(\bm{T}_{q}+\bm{T}_{\bar{q}}\right)\bm{T}_{g}\,\overline{\mathcal{J}}_{2}^{(2)}(q,{\bar{q}}),$
(86)

which has the peculiarity of combining quark-gluon dipole charge operators
with quark-antiquark integrated dipoles. This is once again due to quark-gluon
integrated antenna functions not exactly reproducing the QCD radiation among
the physical quark-gluon dipole.

For processes involving more than one quark-antiquark pair, the appropriate
structure in colour space is more complicated. This is explained by the non-
trivial colour connections between quarks belonging to different quark-
antiquark pairs. The extension of (84) to multiple fermionic lines is first
required addressing NNLO calculations with $5$ partons at Born-level, such as
$pp\to 3j$, for the sub-processes with two fermionic lines and one gluon:
$q{\bar{q}}q^{\prime}{\bar{q}}^{\prime}g$. Configurations with two fermionic
lines and more than one gluon or three fermionic lines with at least one gluon
would appear, for example, in the computation of the NNLO correction to $pp\to
4j$ and $pp\to 5j$, respectively. Since at present such calculations are well
beyond the reach of available two-loop matrix elements and, arguably,
computational techniques, we focus on the five-parton configuration. The
appropriate colour space structure for two quark-antiquark pairs (possibly of
the same flavour), denoted with $(q,{\bar{q}})$ and
$(q^{\prime},{\bar{q}}^{\prime})$ and a gluon $g$ is then:

$\displaystyle\overline{\mathbfcal{J}}^{(2)}$ $\displaystyle=$ $\displaystyle
N_{c}\left(\bm{T}_{q}+\bm{T}_{\bar{q}}\right)\bm{T}_{g}\,\overline{\mathcal{J}}_{2}^{(2)}(q,{\bar{q}})+N_{c}\left(\bm{T}_{q^{\prime}}+\bm{T}_{{\bar{q}}^{\prime}}\right)\bm{T}_{g}\,\overline{\mathcal{J}}_{2}^{(2)}(q^{\prime},{\bar{q}}^{\prime})$
(87) $\displaystyle+$ $\displaystyle
N_{c}\left(\bm{T}_{q}-\bm{T}_{\bar{q}}\right)\bm{T}_{g}\,\left(\overline{\mathcal{J}}_{2}^{(2)}(q,{\bar{q}}^{\prime})-\overline{\mathcal{J}}_{2}^{(2)}(q^{\prime},{\bar{q}})\right)(\delta_{q,1}+\delta_{q,2})$
$\displaystyle+$ $\displaystyle
N_{c}\left(\bm{T}_{q^{\prime}}-\bm{T}_{{\bar{q}}^{\prime}}\right)\bm{T}_{g}\,\left(\overline{\mathcal{J}}_{2}^{(2)}(q,{\bar{q}}^{\prime})-\overline{\mathcal{J}}_{2}^{(2)}(q^{\prime},{\bar{q}})\right)(\delta_{q^{\prime},1}+\delta_{q^{\prime},2}),$

where the two last lines are only present if $q$ or $q^{\prime}$ are in the
initial state. We note that the structure above is in principle not unique and
other arrangements can correctly reproduce the same singularity structure,
thanks to relations among the $\overline{\mathcal{J}}_{2}^{(2)}$ integrated
dipoles, such as:

$\overline{\mathcal{J}}_{2}^{(2)}(1,2)=\overline{\mathcal{J}}_{2}^{(2)}(1,i)+\overline{\mathcal{J}}_{2}^{(2)}(2,j)-\overline{\mathcal{J}}_{2}^{(2)}(i,j),\quad
i,j\geq 3,$ (88)

and colour conservation

$\bm{T}_{i}=-\sum_{j\neq i}\bm{T}_{j}.$ (89)

The configuration of antenna functions in (87), in their unintegrated form,
correctly subtracts the singular behaviour also at the real-virtual and
double-real levels, in infrared limits involving a $q\parallel g$ collinear
pair.

##### Gluon-gluon integrated dipoles

The IP gluon-gluon dipoles only require minimal adjustments to the natural
arrangement of integrated antenna functions. They are given in Table 12.

|  Integrated dipoles
---|---
FF |  $J_{2}^{(2)}\left(i_{g},j_{g}\right)=\frac{1}{4}\mathcal{F}_{4}^{0}+\frac{1}{3}\mathcal{F}_{3}^{1}+\frac{1}{3}\frac{b_{0}}{\epsilon}\left(\frac{|s_{ij}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{F}_{3}^{0}-\frac{1}{9}\left[\mathcal{F}_{3}^{0}\otimes\mathcal{F}_{3}^{0}\right]$
$\hat{J_{2}}^{(2)}\left(i_{g},j_{g}\right)=\mathcal{G}_{4}^{0}+\frac{1}{3}\widehat{\mathcal{F}}_{3}^{1}+\mathcal{G}_{3}^{1}+\frac{1}{3}\frac{b_{0,F}}{\epsilon}\left(\frac{|s_{ij}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{F}_{3}^{0}+\frac{b_{0}}{\epsilon}\left(\frac{|s_{ij}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{G}_{3}^{0}$
$\phantom{\hat{J_{2}}^{(2)}\left(i_{g},j_{g}\right)}-\frac{2}{3}\left[\mathcal{G}_{3}^{0}\otimes\mathcal{F}_{3}^{0}\right]$
$\hat{\tilde{J_{2}}}^{(2)}\left(i_{g},j_{g}\right)=\frac{1}{2}\widetilde{\mathcal{G}}_{4}^{0}+\widetilde{\mathcal{G}}_{3}^{1}$
$\hat{\hat{J_{2}}}^{(2)}\left(i_{g},j_{g}\right)=\frac{1}{2}\mathcal{H}_{4}^{0}+\widehat{\mathcal{G}}_{3}^{1}+\frac{b_{0,F}}{\epsilon}\left(\frac{|s_{ij}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{G}_{3}^{0}-\left[\mathcal{G}_{3}^{0}\otimes\mathcal{G}_{3}^{0}\right]$
IF |  $J_{2}^{(2)}\left(1_{g},i_{g}\right)=\frac{1}{2}\mathcal{F}_{4,g}^{0}+\frac{1}{2}\mathcal{F}_{3,g}^{1}+\frac{1}{2}\frac{b_{0}}{\epsilon}\left(\frac{|s_{1i}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{F}_{3,g}^{0}-\frac{1}{4}\left[\mathcal{F}_{3,g}^{0}\otimes\mathcal{F}_{3,g}^{0}\right]$
$\phantom{J_{2}^{(2)}\left(1_{g},i_{g}\right)}-\frac{1}{2}\overline{\Gamma}^{(2)}_{gg}\left(x_{1}\right)\delta_{2}$
$\hat{J_{2}}^{(2)}\left(1_{g},i_{g}\right)=\mathcal{G}_{4,g}^{0}+\frac{1}{2}\widehat{\mathcal{F}}_{3,g}^{1}+\frac{1}{2}\mathcal{G}_{3,g}^{1}+\frac{1}{2}\frac{b_{0,F}}{\epsilon}\left(\frac{|s_{1i}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{F}_{3,g}^{0}$
$\phantom{\hat{J_{2}}^{(2)}\left(1_{g},i_{g}\right)}+\frac{1}{2}\frac{b_{0}}{\epsilon}\left(\frac{|s_{1i}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{G}_{3,g}^{0}-\frac{1}{2}\left[\mathcal{G}_{3,g}^{0}\otimes\mathcal{F}_{3,g}^{0}\right]-\frac{1}{2}\widehat{\overline{\Gamma}}^{(2)}_{gg}\left(x_{1}\right)\delta_{2}$
$\phantom{\hat{J_{2}}^{(2)}\left(1_{g},i_{g}\right)}+\hat{J}_{2,\mathrm{f/f}}^{(2)}\left(1_{g},i_{g}\right)$
$\hat{\tilde{J_{2}}}^{(2)}\left(1_{g},i_{g}\right)=\frac{1}{2}\widetilde{\mathcal{G}}_{4,g}^{0}+\frac{1}{2}\widetilde{\mathcal{G}}_{3,g}^{1}+\frac{1}{2}\widehat{\widetilde{\overline{\Gamma}}}^{(2)}_{gg}\left(x_{1}\right)\delta_{2}+\hat{\tilde{J}}_{2,\mathrm{f/f}}^{(2)}\left(1_{g},i_{g}\right)$
$\hat{\hat{J_{2}}}^{(2)}\left(1_{g},i_{g}\right)=\frac{1}{2}\widehat{\mathcal{G}}_{3,g}^{1}+\frac{1}{2}\frac{b_{0,F}}{\epsilon}\left(\frac{|s_{1i}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{G}_{3,g}^{0}-\frac{1}{4}\left[\mathcal{G}_{3,g}^{0}\otimes\mathcal{G}_{3,g}^{0}\right]$
$\phantom{\hat{\hat{J_{2}}}^{(2)}\left(1_{g},i_{g}\right)}-\frac{1}{2}\widehat{\widehat{\overline{\Gamma}}}^{(2)}_{gg}\left(x_{1}\right)\delta_{2}$
II |  $J_{2}^{(2)}\left(1_{g},2_{g}\right)=\mathcal{F}_{4,gg}^{0}+\frac{1}{2}\mathcal{F}_{4,gg^{\prime}}^{0}+\mathcal{F}_{3,gg}^{1}+\frac{b_{0}}{\epsilon}\left(\frac{|s_{12}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{F}_{3,gg}^{0}$
$\phantom{J_{2}^{(2)}\left(1_{g},2_{g}\right)}-\left[\mathcal{F}_{3,gg}^{0}\otimes\mathcal{F}_{3,gg}^{0}\right]-\frac{1}{2}\overline{\Gamma}^{(2)}_{gg}\left(x_{1}\right)\delta_{2}-\frac{1}{2}\overline{\Gamma}^{(2)}_{gg}\left(x_{2}\right)\delta_{1}$
$\hat{J_{2}}^{(2)}\left(1_{g},2_{g}\right)=\mathcal{G}_{4,gg}^{0}+\widehat{\mathcal{F}}_{3,gg}^{1}+\frac{b_{0,F}}{\epsilon}\left(\frac{|s_{12}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{F}_{3,gg}^{0}-\frac{1}{2}\widehat{\overline{\Gamma}}^{(2)}_{gg}\left(x_{1}\right)\delta_{2}$
$\phantom{\hat{J_{2}}^{(2)}\left(1_{g},2_{g}\right)}-\frac{1}{2}\widehat{\overline{\Gamma}}^{(2)}_{gg}\left(x_{2}\right)\delta_{1}+\hat{J}_{2,\mathrm{f/f}}^{(2)}\left(1_{g},2_{g}\right)$
$\hat{\tilde{J_{2}}}^{(2)}\left(1_{g},2_{g}\right)=\frac{1}{2}\widetilde{\mathcal{G}}_{4,gg}^{0}+\frac{1}{2}\widehat{\widetilde{\overline{\Gamma}}}^{(2)}_{gg}\left(x_{1}\right)\delta_{2}+\frac{1}{2}\widehat{\widetilde{\overline{\Gamma}}}^{(2)}_{gg}\left(x_{2}\right)\delta_{1}+\hat{\tilde{J}}_{2,\mathrm{f/f}}^{(2)}\left(1_{g},2_{g}\right)$
$\hat{\hat{J_{2}}}^{(2)}\left(1_{g},2_{g}\right)=-\frac{1}{2}\widehat{\widehat{\overline{\Gamma}}}^{(2)}_{gg}\left(x_{1}\right)\delta_{2}-\frac{1}{2}\widehat{\widehat{\overline{\Gamma}}}^{(2)}_{gg}\left(x_{2}\right)\delta_{1}$
Table 12: Identity-preserving gluon-gluon two-loop colour-stripped integrated
dipoles.

As for the $E$-type IF and II antenna functions, the $G$-type ones
$\mathcal{G}_{4,g}^{0}$, $\widetilde{\mathcal{G}}_{4,g}^{0}$,
$\mathcal{G}_{4,gg}^{0}$ and $\widetilde{\mathcal{G}}_{4,gg}^{0}$ need to have
the flip-flopping limits removed. When the final-state quark-antiquark pair
becomes collinear to an initial-state gluon, an IC $g\to q\to g$ configuration
is generated. The required flip-flopping terms are given in Table 13.

|  Integrated dipoles
---|---
IF |  $\hat{J}_{2,\mathrm{f/f}}^{(2)}\left(1_{g},i_{g}\right)=S_{g\to q}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\mathcal{G}_{3,q^{\prime}}^{0}\right]+\frac{1}{2}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\Gamma^{(1)}_{gq}(x_{1})\right]$
$\hat{\tilde{J}}_{2,\mathrm{f/f}}^{(2)}\left(1_{g},i_{g}\right)=S_{g\to
q}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\mathcal{G}_{3,q^{\prime}}^{0}\right]+\frac{1}{2}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\Gamma^{(1)}_{gq}(x_{1})\right]$
II |  $\hat{J}_{2,\mathrm{f/f}}^{(2)}\left(1_{g},2_{g}\right)=S_{g\to q}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\mathcal{G}_{3,qg}^{0}\right]+\frac{1}{2}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\Gamma^{(1)}_{gq}(x_{1})\right]$
$\phantom{\hat{J}_{2,\mathrm{f/f}}^{(2)}\left(1_{g},2_{g}\right)}+S_{g\to
q}\left[\Gamma^{(1)}_{qg}(x_{2})\otimes\mathcal{G}_{3,gq}^{0}\right]+\frac{1}{2}\left[\Gamma^{(1)}_{qg}(x_{2})\otimes\Gamma^{(1)}_{gq}(x_{2})\right]$
$\hat{\tilde{J}}_{2,\mathrm{f/f}}^{(2)}\left(1_{g},2_{g}\right)=S_{g\to
q}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\mathcal{G}_{3,qg}^{0}\right]+\frac{1}{2}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\Gamma^{(1)}_{gq}(x_{1})\right]$
$\phantom{\hat{\tilde{J}}_{2,\mathrm{f/f}}^{(2)}\left(1_{g},2_{g}\right)}+S_{g\to
q}\left[\Gamma^{(1)}_{qg}(x_{2})\otimes\mathcal{G}_{3,gq}^{0}\right]+\frac{1}{2}\left[\Gamma^{(1)}_{qg}(x_{2})\otimes\Gamma^{(1)}_{gq}(x_{2})\right]$
Table 13: Flip-flopping contributions to identity-preserving gluon-gluon two-
loop integrated dipoles.

##### Cancellation of infrared singularities at two-loop

In analogy with (73), the $\epsilon$-poles of IP two-loop colour-stripped
integrated dipoles can be related to the to the infrared singularities of two-
loop matrix elements: any pole of initial-state collinear origin cancel
against two-loop mass factorization kernels, and the remaining singularities
satisfy:

$\displaystyle\mathcal{P}oles\left[N_{c}\,\mathcal{J}_{2}^{(2)}(q,{\bar{q}})-\dfrac{\beta_{0}}{\epsilon}\mathcal{J}_{2}^{(1)}(q,{\bar{q}})\right]=$
$\displaystyle\hskip
142.26378pt\mathcal{P}oles\left[\text{Re}\left(\mathcal{I}^{(2)}_{q{\bar{q}}}\left(\epsilon,\mu_{r}^{2}\right)-\dfrac{\beta_{0}}{\epsilon}\mathcal{I}^{(1)}_{q{\bar{q}}}\left(\epsilon,\mu_{r}^{2}\right)\right)\right],$
(90)
$\displaystyle\mathcal{P}oles\left[N_{c}\,\mathcal{J}_{2}^{(2)}(g,g)-\dfrac{\beta_{0}}{\epsilon}\mathcal{J}_{2}^{(1)}(g,g)\right]=$
$\displaystyle\hskip
142.26378pt\mathcal{P}oles\left[\text{Re}\left(\mathcal{I}^{(2)}_{gg}\left(\epsilon,\mu_{r}^{2}\right)-\dfrac{\beta_{0}}{\epsilon}\mathcal{I}^{(1)}_{gg}\left(\epsilon,\mu_{r}^{2}\right)\right)\right],$
(91)
$\displaystyle\mathcal{P}oles\Bigg{[}\Bigg{.}N_{c}\left(\mathcal{J}_{2}^{(2)}(q,g)+\mathcal{J}_{2}^{(2)}(g,{\bar{q}})-2\overline{\mathcal{J}}_{2}^{(2)}(q,{\bar{q}})\right)$
$\displaystyle\hskip
170.71652pt-\dfrac{\beta_{0}}{\epsilon}\left(\mathcal{J}_{2}^{(1)}(q,g)+\mathcal{J}_{2}^{(1)}(g,{\bar{q}})\right)\Bigg{.}\Bigg{]}=$
$\displaystyle\mathcal{P}oles\left[\text{Re}\left(\mathcal{I}^{(2)}_{qg}\left(\epsilon,\mu_{r}^{2}\right)+\mathcal{I}^{(2)}_{g{\bar{q}}}\left(\epsilon,\mu_{r}^{2}\right)-\dfrac{\beta_{0}}{\epsilon}\left(\mathcal{I}^{(1)}_{gg}\left(\epsilon,\mu_{r}^{2}\right)+\mathcal{I}^{(1)}_{g{\bar{q}}}\left(\epsilon,\mu_{r}^{2}\right)\right)\right)\right].$
(92)

For quark-gluon dipoles we observe in the last equation the presence of
$\overline{\mathcal{J}}_{2}^{(2)}$ to compensate for unphysical poles.

|  Integrated dipoles
---|---
IFg→q |  $J_{2,g\to q}^{(2)}\left(1_{q},i_{\bar{q}}\right)=-\mathcal{A}_{4,g}^{0}-\frac{1}{2}\mathcal{A}_{3,g}^{1}-\frac{1}{2}\frac{b_{0}}{\epsilon}\left(\frac{|s_{1i}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{A}_{3,g}^{0}$
$\phantom{J_{2,g\to
q}^{(2)}\left(1_{q},i_{\bar{q}}\right)}+\frac{1}{2}\left[\mathcal{A}_{3,g}^{0}\otimes\mathcal{A}_{3,q}^{0}\right]-S_{g\to
q}\overline{\Gamma}^{(2)}_{qg}\left(x_{1}\right)\delta_{2}$
$\phantom{J_{2,g\to
q}^{(2)}\left(1_{q},i_{\bar{q}}\right)}+\frac{1}{2}\left[\mathcal{A}_{3,g}^{0}\otimes\Gamma^{(1)}_{gg}(x_{1})\right]+\frac{1}{2}S_{g\to
q}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\Gamma^{(1)}_{gg}(x_{1})\right]$
$\phantom{J_{2,g\to
q}^{(2)}\left(1_{q},i_{\bar{q}}\right)}-\frac{1}{2}\left[\mathcal{A}_{3,g}^{0}\otimes\Gamma^{(1)}_{qq}(x_{1})\right]-\frac{1}{2}S_{g\to
q}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\Gamma^{(1)}_{qq}(x_{1})\right]$
$\hat{J}_{2,g\to
q}^{(2)}\left(1_{q},i_{\bar{q}}\right)=-\frac{1}{2}\widehat{\mathcal{A}}_{3,g}^{1}-\frac{1}{2}\frac{b_{0,F}}{\epsilon}\left(\frac{|s_{1i}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{A}_{3,g}^{0}-S_{g\to
q}\widehat{\overline{\Gamma}}^{(2)}_{qg}\left(x_{1}\right)\delta_{2}$
$\phantom{\hat{J}_{2,g\to
q}^{(2)}\left(1_{q},i_{\bar{q}}\right)}+\frac{1}{2}\left[\mathcal{A}_{3,g}^{0}\otimes\widehat{\Gamma}^{(1)}_{gg}(x_{1})\right]+\frac{1}{2}S_{g\to
q}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\widehat{\Gamma}^{(1)}_{gg}(x_{1})\right]$
$\tilde{J}_{2,g\to
q}^{(2)}\left(1_{q},i_{\bar{q}}\right)=-\widetilde{\mathcal{A}}_{4,g}^{0}-\widetilde{\mathcal{A}}_{3,g}^{1}+\left[\mathcal{A}_{3,q}^{0}\otimes\mathcal{A}_{3,g}^{0}\right]-\left[\Gamma^{(1)}_{qq}(x_{1})\otimes\mathcal{A}_{3,g}^{0}\right]$
$\phantom{\tilde{J}_{2,g\to q}^{(2)}\left(1_{q},i_{\bar{q}}\right)}-S_{g\to
q}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\widehat{\Gamma}^{(1)}_{qq}(x_{1})\right]+2S_{g\to
q}\widetilde{\overline{\Gamma}}^{(2)}_{qg}\left(x_{1}\right)\delta_{2}$
IF${}_{q\to{\bar{q}}}$ |  $\tilde{J}_{2,q\to{\bar{q}}}^{(2)}\left(1_{\bar{q}},i_{q}\right)=\mathcal{C}_{4,{\bar{q}}^{\prime}}^{0}+\overline{\Gamma}^{(2)}_{q{\bar{q}}}\left(x_{1}\right)\delta_{2}$
IFq→g→q |  $J_{2,q\to g\to q}^{(2)}\left(1_{q},i_{g}\right)=\mathcal{B}_{4,q^{\prime}}^{0}+S_{q\to g}\left[\Gamma^{(1)}_{gq}(x_{1})\otimes\mathcal{A}_{3,g}^{0}\right]$
$\phantom{J_{2,q\to g\to
q}^{(2)}\left(1_{q},i_{g}\right)}+\left[\Gamma^{(1)}_{gq}(x_{1})\otimes\Gamma^{(1)}_{qg}(x_{1})\right]-\widetilde{\overline{\Gamma}}^{(2)}_{qq}\left(x_{1}\right)\delta_{2}$
Table 14: Initial-final identity-changing quark-antiquark two-loop colour-stripped integrated dipoles. |  Integrated dipoles
---|---
IIg→q |  $J_{2,g\to q}^{(2)}\left(1_{q},2_{\bar{q}}\right)=-\mathcal{A}_{4,gq}^{0}-\mathcal{A}_{4,g^{\prime}q}^{0}-\mathcal{A}_{3,gq}^{1}-\frac{b_{0}}{\epsilon}\left(\frac{|s_{12}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{A}_{3,gq}^{0}$
$\phantom{J_{2,g\to
q}^{(2)}\left(1_{q},2_{\bar{q}}\right)}+\left[\mathcal{A}_{3,gq}^{0}\otimes\mathcal{A}_{3,q{\bar{q}}}^{0}\right]-S_{g\to
q}\overline{\Gamma}^{(2)}_{qg}\left(x_{1}\right)\delta_{2}$
$\phantom{J_{2,g\to
q}^{(2)}\left(1_{q},2_{\bar{q}}\right)}+\left[\mathcal{A}_{3,gq}^{0}\otimes\Gamma^{(1)}_{gg}(x_{1})\right]+\frac{1}{2}S_{g\to
q}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\Gamma^{(1)}_{gg}(x_{1})\right]$
$\phantom{J_{2,g\to
q}^{(2)}\left(1_{q},2_{\bar{q}}\right)}-\left[\mathcal{A}_{3,gq}^{0}\otimes\Gamma^{(1)}_{qq}(x_{1})\right]-\frac{1}{2}S_{g\to
q}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\Gamma^{(1)}_{qq}(x_{1})\right]$
$\hat{J}_{2,g\to
q}^{(2)}\left(1_{q},2_{\bar{q}}\right)=-\frac{1}{2}\widehat{\mathcal{A}}_{3,gq}^{1}-\frac{b_{0,F}}{\epsilon}\left(\frac{|s_{12}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{A}_{3,gq}^{0}-S_{g\to
q}\widehat{\overline{\Gamma}}^{(2)}_{qg}\left(x_{1}\right)\delta_{2}$
$\phantom{J_{2,g\to
q}^{(2)}\left(1_{q},2_{\bar{q}}\right)}+\left[\mathcal{A}_{3,gq}^{0}\otimes\widehat{\Gamma}^{(1)}_{gg}(x_{1})\right]+\frac{1}{2}S_{g\to
q}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\widehat{\Gamma}^{(1)}_{gg}(x_{1})\right]$
$\tilde{J}_{2,g\to
q}^{(2)}\left(1_{q},2_{\bar{q}}\right)=-\widetilde{\mathcal{A}}_{4,gq}^{0}-\widetilde{\mathcal{A}}_{3,gq}^{1}$
$\phantom{\tilde{J}_{2,g\to
q}^{(2)}\left(1_{q},2_{\bar{q}}\right)}+\left[\mathcal{A}_{3,qq}^{0}\otimes\mathcal{A}_{3,gq}^{0}\right]-\left[\Gamma^{(1)}_{qq}(x_{1})\otimes\mathcal{A}_{3,gq}^{0}\right]$
$\phantom{\tilde{J}_{2,g\to
q}^{(2)}\left(1_{q},2_{\bar{q}}\right)}-\frac{1}{2}S_{g\to
q}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\widehat{\Gamma}^{(1)}_{qq}(x_{1})\right]+S_{g\to
q}\widetilde{\overline{\Gamma}}^{(2)}_{qg}\left(x_{1}\right)\delta_{2}$
II${}_{q\to q^{\prime}}$ |  $J_{2,q\to q^{\prime}}^{(2)}\left(1_{q^{\prime}},2_{\bar{q}}\right)=\mathcal{B}_{4,q^{\prime}q}^{0}+S_{q\to g}\left[\Gamma^{(1)}_{gq}(x_{1})\otimes\mathcal{A}_{3,gq}^{0}\right]$
$\phantom{J_{2,q\to
q^{\prime}}^{(2)}\left(1_{q^{\prime}},2_{\bar{q}}\right)}+\frac{1}{2}\left[\Gamma^{(1)}_{gq}(x_{1})\otimes\Gamma^{(1)}_{qg}(x_{1})\right]-\overline{\Gamma}^{(2)}_{qq^{\prime}}\left(x_{1}\right)\delta_{2}$
II${}_{q\to{\bar{q}}}$ |  $\tilde{J}_{2,q\to{\bar{q}}}^{(2)}\left(1_{\bar{q}},2_{q}\right)=\mathcal{C}_{4,{\bar{q}}^{\prime}{\bar{q}}}^{0}+\overline{\Gamma}^{(2)}_{q{\bar{q}}}\left(x_{1}\right)\delta_{2}$
II${}_{gg\to q{\bar{q}}}$ |  $J_{2,gg\to q{\bar{q}}}^{(2)}\left(1_{q},2_{\bar{q}}\right)=\mathcal{A}_{4,gg}^{0}+S_{g\to q}\left[\Gamma^{(1)}_{qg}(x_{1})\otimes\mathcal{A}_{3,qg}^{0}\right]$
$\phantom{J_{2,gg\to q{\bar{q}}}^{(2)}\left(1_{q},2_{\bar{q}}\right)}+S_{g\to
q}\left[\Gamma^{(1)}_{qg}(x_{2})\otimes\mathcal{A}_{3,gq}^{0}\right]+S_{g\to
q}^{2}\Gamma^{(1)}_{qg}\left(x_{1}\right)\Gamma^{(1)}_{qg}\left(x_{2}\right)$
$\tilde{J}_{2,gg\to
q{\bar{q}}}^{(2)}\left(1_{q},2_{\bar{q}}\right)=\widetilde{\mathcal{A}}_{4,gg}^{0}+2S_{g\to
q}\left[\Gamma^{(1)}_{gq}(x_{1})\otimes\mathcal{A}_{3,qg}^{0}\right]$
$\phantom{\tilde{J}_{2,gg\to
q{\bar{q}}}^{(2)}\left(1_{q},2_{\bar{q}}\right)}+2S_{g\to
q}\left[\Gamma^{(1)}_{qg}(x_{2})\otimes\mathcal{A}_{3,gq}^{0}\right]+2S_{g\to
q}^{2}\Gamma^{(1)}_{qg}\left(x_{1}\right)\Gamma^{(1)}_{qg}\left(x_{2}\right)$
Table 15: Initial-initial identity-changing quark-antiquark two-loop colour-stripped integrated dipoles. |  Integrated dipoles
---|---
IFq→g |  $J_{2,q\to g}^{(2)}(1_{g},i_{q})=-\mathcal{E}_{4,q^{\prime}}^{0}-\mathcal{E}_{3,q^{\prime}}^{1}-\frac{b_{0}}{\epsilon}\left(\frac{|s_{1i}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{E}_{3,q^{\prime}}^{0}$
$\phantom{J_{2,q\to g}^{(2)}(1_{g},i_{q})}+2\left[\mathcal{D}_{3,g\to
g}^{0}\otimes\mathcal{E}_{3,q^{\prime}}^{0}\right]+\left[\Gamma^{(1)}_{qq}(x_{1})\otimes\mathcal{E}_{3,q^{\prime}}^{0}\right]$
$\phantom{J_{2,q\to
g}^{(2)}(1_{g},i_{q})}-\left[\Gamma^{(1)}_{gg}(x_{1})\otimes\mathcal{E}_{3,q^{\prime}}^{0}\right]+\frac{1}{2}S_{g\to
q}\left[\Gamma^{(1)}_{qq}(x_{1})\otimes\Gamma^{(1)}_{gq}(x_{1})\right]$
$\phantom{J_{2,q\to g}^{(2)}(1_{g},i_{q})}-\frac{1}{2}S_{q\to
g}\left[\Gamma^{(1)}_{gg}(x_{1})\otimes\Gamma^{(1)}_{gq}(x_{1})\right]-S_{q\to
g}\overline{\Gamma}^{(2)}_{gq}\left(x_{1}\right)\delta_{2}$
$\hat{J}_{2,q\to
g}^{(2)}(1_{g},i_{q})=-\widehat{\mathcal{E}}_{3,q^{\prime}}^{1}-\frac{b_{0,F}}{\epsilon}\left(\frac{|s_{1i}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{E}_{3,q^{\prime}}^{0}-\left[\widehat{\Gamma}^{(1)}_{gg}(x_{1})\otimes\mathcal{E}_{3,q^{\prime}}^{0}\right]$
$\phantom{\hat{J}_{2,q\to g}^{(2)}(1_{g},i_{q})}-\frac{1}{2}S_{q\to
g}\left[\widehat{\Gamma}^{(1)}_{gg}(x_{1})\otimes\Gamma^{(1)}_{gq}(x_{1})\right]-S_{q\to
g}\widehat{\overline{\Gamma}}^{(2)}_{gq}\left(x_{1}\right)\delta_{2}$
$\tilde{J}_{2,q\to
g}^{(2)}(1_{g},i_{q})=-\widetilde{\mathcal{E}}_{4,q^{\prime}}^{0}-\widetilde{\mathcal{E}}_{3,q^{\prime}}^{1}+\left[\Gamma^{(1)}_{qq}(x_{1})\otimes\mathcal{E}_{3,q^{\prime}}^{0}\right]$
$\phantom{\tilde{J}_{2,q\to g}^{(2)}(1_{g},i_{q})}+\frac{1}{2}S_{g\to
q}\left[\Gamma^{(1)}_{qq}(x_{1})\otimes\Gamma^{(1)}_{gq}(x_{1})\right]+S_{q\to
g}\widetilde{\overline{\Gamma}}^{(2)}_{gq}\left(x_{1}\right)\delta_{2}$
IIq→g |  $J_{2,q\to g}^{(2)}(1_{g},2_{q})=-\mathcal{E}_{4,q^{\prime}q}^{0}-\mathcal{E}_{4,{\bar{q}}^{\prime}q}^{0}-\mathcal{E}_{3,q^{\prime}q}^{1}-\frac{b_{0}}{\epsilon}\left(\frac{|s_{12}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{E}_{3,q^{\prime}q}^{0}$
$\phantom{J_{2,q\to
g}^{(2)}(1_{g},2_{q})}+2\left[\mathcal{D}_{3,gq}^{0}\otimes\mathcal{E}_{3,q^{\prime}q}^{0}\right]+\left[\Gamma^{(1)}_{qq}(x_{1})\otimes\mathcal{E}_{3,q^{\prime}q}^{0}\right]$
$\phantom{J_{2,q\to
g}^{(2)}(1_{g},i_{q})}-\left[\Gamma^{(1)}_{gg}(x_{1})\otimes\mathcal{E}_{3,q^{\prime}q}^{0}\right]+\frac{1}{2}S_{g\to
q}\left[\Gamma^{(1)}_{qq}(x_{1})\otimes\Gamma^{(1)}_{gq}(x_{1})\right]$
$\phantom{J_{2,q\to g}^{(2)}(1_{g},i_{q})}-\frac{1}{2}S_{q\to
g}\left[\Gamma^{(1)}_{gg}(x_{1})\otimes\Gamma^{(1)}_{gq}(x_{1})\right]$
$\phantom{J_{2,q\to g}^{(2)}(1_{g},2_{q})}-S_{q\to
g}\overline{\Gamma}^{(2)}_{gq}\left(x_{1}\right)\delta_{2}$
$\hat{J}_{2,q\to
g}^{(2)}(1_{g},2_{q})=-\widehat{\mathcal{E}}_{3,q^{\prime}q}^{1}-\frac{b_{0,F}}{\epsilon}\left(\frac{|s_{12}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{E}_{3,q^{\prime}q}^{0}$
$\phantom{J_{2,q\to
g}^{(2)}(1_{g},i_{q})}-\left[\widehat{\Gamma}^{(1)}_{gg}(x_{1})\otimes\mathcal{E}_{3,q^{\prime}q}^{0}\right]$
$\phantom{J_{2,q\to g}^{(2)}(1_{g},i_{q})}-\frac{1}{2}S_{q\to
g}\left[\widehat{\Gamma}^{(1)}_{gg}(x_{1})\otimes\Gamma^{(1)}_{gq}(x_{1})\right]-S_{q\to
g}\widehat{\overline{\Gamma}}^{(2)}_{gq}\left(x_{1}\right)\delta_{2}$
$\tilde{J}_{2,q\to
g}^{(2)}(1_{g},2_{q})=-\widetilde{\mathcal{E}}_{4,q^{\prime}q}^{0}-\widetilde{\mathcal{E}}_{3,q^{\prime}q}^{1}+\left[\Gamma^{(1)}_{qq}(x_{1})\otimes\mathcal{E}_{3,q^{\prime}q}^{0}\right]$
$\phantom{\tilde{J}_{2,q\to g}^{(2)}(1_{g},2_{q})}+\frac{1}{2}S_{g\to
q}\left[\Gamma^{(1)}_{qq}(x_{1})\otimes\Gamma^{(1)}_{gq}(x_{1})\right]+S_{q\to
g}\widetilde{\overline{\Gamma}}^{(2)}_{gq}\left(x_{1}\right)\delta_{2}$
II${}_{q^{\prime}g\to gq}$ |  $J_{2,q^{\prime}g\to gq}^{(2)}(1_{g},2_{q})=-\mathcal{E}_{4,q^{\prime}g}^{0}+S_{q\to g}\left[\Gamma^{(1)}_{gq}(x_{1})\otimes\mathcal{D}_{3,gg\to gq}^{0}\right]$
$\phantom{J_{2,q^{\prime}g\to gq}^{(2)}(1_{g},2_{q})}+S_{g\to
q}\left[\Gamma^{(1)}_{gq}(x_{2})\otimes\mathcal{E}_{3,q^{\prime}q}^{0}\right]+\Gamma^{(1)}_{gq}\left(x_{1}\right)\Gamma^{(1)}_{qg}\left(x_{2}\right)$
$\tilde{J}_{2,q^{\prime}g\to
gq}^{(2)}(1_{g},2_{q})=-\widetilde{\mathcal{E}}_{4,q^{\prime}g}^{0}+S_{q\to
g}\left[\Gamma^{(1)}_{gq}(x_{1})\otimes\mathcal{D}_{3,gg\to gq}^{0}\right]$
$\phantom{\tilde{J}_{2,q^{\prime}g\to gq}^{(2)}(1_{g},2_{q})}+S_{g\to
q}\left[\Gamma^{(1)}_{gq}(x_{2})\otimes\mathcal{E}_{3,q^{\prime}q}^{0}\right]+\Gamma^{(1)}_{gq}\left(x_{1}\right)\Gamma^{(1)}_{qg}\left(x_{2}\right)$
Table 16: Identity-changing quark-gluon two-loop colour-stripped integrated dipoles. |  Integrated dipoles
---|---
IFq→g |  $J_{2,q\to g}^{(2)}\left(1_{g},i_{g}\right)=-\mathcal{G}_{4,q}^{0}-\mathcal{G}_{3,q}^{1}-\frac{b_{0}}{\epsilon}\left(\frac{|s_{1i}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{G}_{3,q}^{0}$
$\phantom{J_{2,q\to
g}^{(2)}\left(1_{g},i_{g}\right)}+2\left[\mathcal{G}_{3,q}^{0}\otimes\mathcal{F}_{3,g}^{0}\right]-S_{q\to
g}\overline{\Gamma}^{(2)}_{gq}\left(x_{1}\right)\delta_{2}$
$\phantom{J_{2,q\to
g}^{(2)}\left(1_{g},i_{g}\right)}+\left[\mathcal{A}_{3,q}^{0}\otimes\mathcal{G}_{3,q}^{0}\right]+\left[\Gamma^{(1)}_{qq}(x_{1})\otimes\mathcal{G}_{3,q}^{0}\right]-\left[\Gamma^{(1)}_{gg}(x_{1})\otimes\mathcal{G}_{3,q}^{0}\right]$
$\phantom{J_{2,q\to g}^{(2)}\left(1_{g},i_{g}\right)}+\frac{1}{2}S_{q\to
g}\left[\Gamma^{(1)}_{qq}(x_{1})\otimes\Gamma^{(1)}_{gq}(x_{1})\right]$
$\phantom{J_{2,q\to g}^{(2)}\left(1_{g},i_{g}\right)}-\frac{1}{2}S_{q\to
g}\left[\Gamma^{(1)}_{gg}(x_{1})\otimes\Gamma^{(1)}_{gq}(x_{1})\right]$
$\hat{J}_{2,q\to
g}^{(2)}(1_{g},i_{g})=-\mathcal{H}_{4,q}^{0}-\widehat{\mathcal{G}}_{3,q}^{1}-\frac{b_{0,F}}{\epsilon}\left(\frac{|s_{1i}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{G}_{3,q}^{0}$
$\phantom{\hat{J}_{2,q\to g}^{(2)}(1_{g},i_{g})}+S_{q\to
g}\left[\Gamma^{(1)}_{gq}(x_{1})\otimes\mathcal{G}_{3,g}^{0}\right]-\frac{1}{2}S_{q\to
g}\left[\Gamma^{(1)}_{gq}(x_{1})\otimes\widehat{\Gamma}^{(1)}_{gg}\left({1}\right)\right]$
$\phantom{\hat{J}_{2,q\to g}^{(2)}(1_{g},i_{g})}-S_{q\to
g}\widehat{\overline{\Gamma}}^{(2)}_{gq}\left(x_{1}\right)\delta_{2}$
$\tilde{J}_{2,q\to
g}^{(2)}\left(1_{g},i_{g}\right)=-\widetilde{\mathcal{G}}_{4,q}^{0}-\widetilde{\mathcal{G}}_{3,q}^{1}-\left[\mathcal{G}_{3,q}^{0}\otimes\Gamma^{(1)}_{qq}(x_{1})\right]$
$\phantom{\tilde{J}_{2,q\to
g}^{(2)}\left(1_{g},i_{g}\right)}+\frac{1}{2}S_{q\to
g}\left[\Gamma^{(1)}_{qq}(x_{1})\otimes\Gamma^{(1)}_{gq}(x_{1})\right]+S_{q\to
g}\widetilde{\overline{\Gamma}}^{(2)}_{gq}\left(x_{1}\right)\delta_{2}$
IIq→g |  $J_{2,q\to g}^{(2)}\left(1_{g},2_{g}\right)=-\mathcal{G}_{4,qg}^{0}-\mathcal{G}_{4,qg^{\prime}}^{0}-\mathcal{G}_{3,qg}^{1}-\frac{b_{0}}{\epsilon}\left(\frac{|s_{12}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{G}_{3,qg}^{0}$
$\phantom{J_{2,q\to
g}^{(2)}\left(1_{g},2_{g}\right)}+2\left[\mathcal{G}_{3,qg}^{0}\otimes\mathcal{F}_{3,gg}^{0}\right]-S_{q\to
g}\overline{\Gamma}^{(2)}_{gq}\left(x_{1}\right)\delta_{2}$
$\phantom{J_{2,q\to
g}^{(2)}\left(1_{g},2_{g}\right)}+\left[\mathcal{A}_{3,qg}^{0}\otimes\mathcal{G}_{3,qg}^{0}\right]+\left[\Gamma^{(1)}_{qq}(x_{1})\otimes\mathcal{G}_{3,qg}^{0}\right]-\left[\Gamma^{(1)}_{gg}(x_{1})\otimes\mathcal{G}_{3,qg}^{0}\right]$
$\phantom{J_{2,q\to g}^{(2)}\left(1_{g},2_{g}\right)}+\frac{1}{2}S_{q\to
g}\left[\Gamma^{(1)}_{qq}(x_{1})\otimes\Gamma^{(1)}_{gq}(x_{1})\right]$
$\phantom{J_{2,q\to g}^{(2)}\left(1_{g},2_{g}\right)}-\frac{1}{2}S_{q\to
g}\left[\Gamma^{(1)}_{gg}(x_{1})\otimes\Gamma^{(1)}_{gq}(x_{1})\right]$
$\hat{J}_{2,q\to
g}^{(2)}\left(1_{g},2_{g}\right)=-\widehat{\mathcal{G}}_{3,qg}^{1}-\frac{b_{0,F}}{\epsilon}\left(\frac{|s_{12}|}{\mu_{r}^{2}}\right)^{-\epsilon}\mathcal{G}_{3,qg}^{0}-S_{q\to
g}\widehat{\overline{\Gamma}}^{(2)}_{gq}\left(x_{1}\right)\delta_{2}$
$\phantom{\hat{J}_{2,q\to
g}^{(2)}\left(1_{g},2_{g}\right)}-\frac{1}{2}\left[\mathcal{G}_{3,qg}^{0}\otimes\widehat{\Gamma}^{(1)}_{gg}(x_{1})\right]-\frac{1}{2}S_{q\to
g}\left[\Gamma^{(1)}_{gq}(x_{1})\otimes\widehat{\Gamma}^{(1)}_{gg}(x_{1})\right]$
$\tilde{J}_{2,q\to
g}^{(2)}\left(1_{g},2_{g}\right)=-\widetilde{\mathcal{G}}_{4,qg}^{0}-\widetilde{\mathcal{G}}_{3,qg}^{1}-\left[\mathcal{G}_{3,qg}^{0}\otimes\Gamma^{(1)}_{qq}(x_{1})\right]$
$\phantom{\tilde{J}_{2,q\to
g}^{(2)}\left(1_{g},2_{g}\right)}+\frac{1}{2}S_{q\to
g}\left[\Gamma^{(1)}_{qq}(x_{1})\otimes\Gamma^{(1)}_{gq}(x_{1})\right]+S_{q\to
g}\widetilde{\overline{\Gamma}}^{(2)}_{gq}\left(x_{1}\right)\delta_{2}$
II${}_{q{\bar{q}}\to gg}$ |  $J_{2,q{\bar{q}}\to gg}^{(2)}\left(1_{g},2_{g}\right)=\mathcal{H}_{4,qq^{\prime}}^{0}+S_{q\to g}\left[\Gamma^{(1)}_{gq}(x_{2})\otimes\mathcal{G}_{3,gq}^{0}\right]$
$\phantom{J_{2,q\to g}^{(2)}(1_{g},i_{q})}+S_{q\to
g}\left[\Gamma^{(1)}_{gq}(x_{1})\otimes\mathcal{G}_{3,qg}^{0}\right]+S_{q\to
g}^{2}\Gamma^{(1)}_{gq}\left(x_{1}\right)\Gamma^{(1)}_{gq}\left(x_{2}\right)$
Table 17: Identity-changing gluon-gluon two-loop colour-stripped integrated
dipoles.

##### Identity-changing dipoles

Identity-changing integrated dipoles, as for the one-loop case, are not
treated in colour space. We have two possible structures, one is associated to
configurations where at least one initial state parton actually changes
identity, while the other addresses flip-flopping contributions for which,
after integration, the species of the initial-state partons is formally the
same as before. The first type reads:

$\displaystyle\mathcal{J}_{ab;cd}^{(2)}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle\mathcal{J}_{2,a\to
c}^{(2)}(x_{1})\delta_{db}\delta(1-x_{2})+\mathcal{J}_{2,b\to
d}^{(2)}(x_{2})\delta_{ca}\delta(1-x_{1})$ (93) $\displaystyle+$
$\displaystyle\mathcal{J}_{2,ab\to cd}^{(2)}(x_{1},x_{2}),$

where either a single initial-state leg changes identity (first two terms) or
both legs change identity (last term). The flip-flopping integrated dipoles
are given by:

$\displaystyle\mathcal{J}_{ab;cd;ab}^{(2)}(x_{1},x_{2})$ $\displaystyle=$
$\displaystyle\mathcal{J}_{2,a\to c\to
a}^{(2)}(x_{1})\delta_{db}\delta(1-x_{2})+\mathcal{J}_{2,b\to d\to
b}^{(2)}(x_{2})\delta_{ca}\delta(1-x_{1}).$ (94)

The integrated dipoles in (93) and (94) have a colour decomposition which
mirrors the one of two-loop IC splitting kernels:

$\displaystyle\mathcal{J}_{2,q\to g}^{(2)}(g,i)$ $\displaystyle=$
$\displaystyle\left(\dfrac{N_{c}^{2}-1}{N_{c}}\right)\Big{[}\Big{.}N_{c}J_{2,q\to
g}^{(2)}(g,i)+\dfrac{1}{N_{c}}\tilde{J}_{2,q\to g}^{(2)}(g,i)$ (95)
$\displaystyle\hskip 156.49014pt+N_{f}\hat{J}_{2,q\to
g}^{(2)}(g,i)\Big{.}\Big{]}\,,$ $\displaystyle\mathcal{J}_{2,g\to
q}^{(2)}(q,i)$ $\displaystyle=$ $\displaystyle N_{c}J_{2,g\to
q}^{(2)}(q,i)+\dfrac{1}{N_{c}}\tilde{J}_{2,g\to
q}^{(2)}(q,i)+N_{f}\hat{J}_{2,g\to q}^{(2)}(q,i)\,,$ (96)
$\displaystyle\mathcal{J}_{2,q\to{\bar{q}}}^{(2)}({\bar{q}},i)$
$\displaystyle=$
$\displaystyle\left(\dfrac{N_{c}^{2}-1}{N_{c}}\right)\left[J_{2,q\to{\bar{q}}}^{(2)}({\bar{q}},i)+\dfrac{1}{N_{c}}\tilde{J}_{2,q\to{\bar{q}}}^{(2)}({\bar{q}},i)\right]\,,$
(97) $\displaystyle\mathcal{J}_{2,q\to q^{\prime}}^{(2)}(q^{\prime},i)$
$\displaystyle=$
$\displaystyle\left(\dfrac{N_{c}^{2}-1}{N_{c}}\right)J_{2,q\to
q^{\prime}}^{(2)}(q^{\prime},i)\,,$ (98) $\displaystyle\mathcal{J}_{2,gg\to
q{\bar{q}}}^{(2)}(q,{\bar{q}})$ $\displaystyle=$ $\displaystyle
N_{c}\,J_{2,gg\to
q{\bar{q}}}^{(2)}(q,{\bar{q}})+\dfrac{1}{N_{c}}\tilde{J}_{2,gg\to
q{\bar{q}}}^{(2)}(q,{\bar{q}})\,,$ (99)
$\displaystyle\mathcal{J}_{2,qq^{\prime}\to gg}^{(2)}(g,g)$ $\displaystyle=$
$\displaystyle\left(\dfrac{N_{c}^{2}-1}{N_{c}}\right)J_{2,q{\bar{q}}\to
gg}^{(2)}(g,g)\,,$ (100) $\displaystyle\mathcal{J}_{2,q^{\prime}g\to
gq}^{(2)}(g,q)$ $\displaystyle=$ $\displaystyle N_{c}\,J_{2,q^{\prime}g\to
gq}^{(2)}(g,q)+\dfrac{1}{N_{c}}\tilde{J}_{2,q^{\prime}g\to gq}^{(2)}(g,q)\,,$
(101) $\displaystyle\mathcal{J}_{2,q\to g\to q}^{(2)}(q,i)$ $\displaystyle=$
$\displaystyle\left(\dfrac{N_{c}^{2}-1}{N_{c}}\right)J_{2,q\to g\to
q}^{(2)}(q,i)\,.$ (102)

where $i=q,g$. For configurations where only one initial-state parton changes
identity, the second hard parton in the dipole acts as a mere spectator and no
unresolved limits are associated to it. This means that in any practical
implementation one has the freedom to choose different spectators for the same
IC limits. In particular, in any IC unresolved configuration resulting in a
hard quark, such as $g\to q$, $q\to q^{\prime}$, $q\to{\bar{q}}$ and $q\to
g\to q$, it is always possible to choose the fermionic partner of the hard
quark as a spectator. For this reason, there is no need to define IC
integrated dipoles for these configurations with a hard gluon as spectator. In
the following we present the expressions for the two-loop IC integrated
dipoles. We note that typically for a specific process only some of the listed
dipoles are actually needed. However, we provide expressions for all of them
for the sake of generality. The IC quark-antiquark, quark-gluon and gluon-
gluon two-loop integrated dipoles are listed in Tables 14–17.

Contrary to the one-loop IC dipoles, two-loop ones are not free from
singularities. This is understood considering that, with two unresolved
partons, along with initial-state collinear singularities which cancel against
the mass factorization kernels, additional soft or final-state collinear poles
can be present. As we will see in more detail in Section 5, the singularities
of the IC two-loop integrated dipoles cancel against explicit poles in other
structures present in the two-loop mass factorization counterterm for
identity-changing splittings. This cancellation yields an IC double-virtual
subtraction term which is correctly free from explicit singularities.

##  4 Colourful antenna subtraction at NLO

In the following, we describe the colourful antenna subtraction approach at
NLO, aiming to introduce the underlying concepts and preparing for the full
NNLO formulation.

The NLO correction to a partonic sub-process initiated by partons $a$ and $b$
reads as follow.

$\text{d}{\hat{\sigma}_{ab,\mathrm{NLO}}}=\int_{n}\left[\text{d}{\hat{\sigma}^{V}_{ab,\mathrm{NLO}}}-\text{d}{\hat{\sigma}^{T}_{ab,\mathrm{NLO}}}\right]+\int_{n+1}\left[\text{d}{\hat{\sigma}^{R}_{ab,\mathrm{NLO}}}-\text{d}{\hat{\sigma}^{S}_{ab,\mathrm{NLO}}}\right],$
(103)

with

$\text{d}{\hat{\sigma}^{T}_{ab,\mathrm{NLO}}}=-\int_{1}\text{d}{\hat{\sigma}^{S}_{ab,\mathrm{NLO}}}-\text{d}{\hat{\sigma}^{MF}_{ab,\mathrm{NLO}}}.$
(104)

Typically, the construction of the subtraction terms would begin by analysing
the divergent behaviour of the real emission correction and by individuating
the proper combination of antenna functions and reduced LO squared amplitudes
which reproduces the implicit infrared singularities. Hence, the real
subtraction term $\text{d}{\hat{\sigma}^{S}_{ab,\mathrm{NLO}}}$, up to an
overall process-dependent normalisation, has the following form Currie:2013vh
:

$\displaystyle\text{d}{\hat{\sigma}^{S}_{ab,\mathrm{NLO}}}$
$\displaystyle\sim$
$\displaystyle\sum_{c,c^{\prime}}\sum_{i,j,k}\text{d}\Phi_{n+1}(p_{3},\dots,p_{n+3};p_{1},p_{2})\,$
(105) $\displaystyle\hskip
42.67912pt\times\,X_{3}^{0}(i,k,j)\,a_{n+2}^{0}(c,c^{\prime};\\{.,\widetilde{ik},.,\widetilde{kj},.\\})J_{n}^{(n)}(.,p_{\widetilde{ik}},.,p_{\widetilde{kj}},.).$

In the notation of Section 3, $a_{n+2}^{0}(c,c^{\prime},\\{\dots\\})$
indicates a generic $(n+2)$-parton interference between colour ordering $c$
and $c^{\prime}$. The first sum runs over the possible colour structures and
orderings, while the inner sum covers permutations of external momenta. In
(105), as well as analogous expressions in the remainder of the paper, each
term in the sum can come with a different numerical factor in front of it.
Such factors are not generalizable and not particularly relevant for the
description of the structure of the subtraction terms. The symbol $\sim$
indicates their omissions.

An analytical integration of (105) over the phase space of the unresolved
radiation (antenna phase space) yields:

$\displaystyle\int_{1}\text{d}{\hat{\sigma}^{S}_{ab,\mathrm{NLO}}}$
$\displaystyle\sim$
$\displaystyle\sum_{c,c^{\prime}}\sum_{i,j,k}\text{d}\Phi_{n}(.,p_{\widetilde{ik}},.,p_{\widetilde{kj}},.;p_{1},p_{2})\,$
(106) $\displaystyle\hskip
31.2982pt\times\,\mathcal{X}_{3}^{0}(s_{(\widetilde{ik})(\widetilde{kj})})\,a_{n+2}^{0}(c,c^{\prime};\\{.,\widetilde{ik},.,\widetilde{kj},.\\})J_{n}^{(n)}(.,p_{\widetilde{ik}},.,p_{\widetilde{kj}},.)\,.$

Finally, the full virtual subtraction term (6) is obtained by adding all mass
factorization counterterms appropriate for the incoming partons.

In the context of the antenna subtraction method, the construction of real-
emission subtraction terms for a generic process may become cumbersome and
require a considerable amount of process-specific work. In particular, beyond
leading-colour, one cannot directly rely on colour connections of squared
amplitudes as the guiding principle for the construction of the subtraction
terms, especially at high multiplicities. The colourful antenna subtraction
approach is designed to systematise the construction of the real-emission
subtraction term. The main idea behind it consists of exploiting the
predictability of the singularity structure of virtual amplitudes in colour
space to straightforwardly construct the virtual subtraction term in a
completely general way. Subsequently, the real subtraction term is derived
with a systematic procedure exploiting the correspondence between integrated
and unintegrated antenna functions. To properly introduce how this is done we
first need to recast the NLO mass factorization counterterm in colour space.

###  4.1 NLO mass factorization in colour space

We notice that equation (7) has both identity-preserving and identity-changing
contributions, respectively when $(c,d)=(a,b)$ and $(c,d)\neq(a,b)$. For a
given LO partonic matrix element, initiated by partons $a$ and $b$, the
virtual infrared singularities in (53) and the identity-preserving
contributions in (7) factorize upon the same set of LO partonic colour
correlations. Nevertheless, it is possible to express the identity-preserving
mass factorization kernels in colour space, in analogy to the singularity
structure of one-loop amplitudes. We define:

$\mathbf{\Gamma}^{(1)}_{ab;ab}\left(x_{1},x_{2}\right)=\mathbf{\Gamma}^{(1)}_{aa,\text{full}}\left(x_{1}\right)\delta(1-x_{2})+\mathbf{\Gamma}^{(1)}_{bb,\text{full}}\left(x_{2}\right)\delta(1-x_{1}),$
(107)

where, as usual, bold symbols are used to indicate that the splitting kernels
have been promoted to be operators in colour space:

$\mathbf{\Gamma}^{(1)}_{cc,\text{full}}\left(x_{i}\right)=-\Gamma^{(1)}_{cc,\text{full}}\left(x_{i}\right)\dfrac{1}{C_{c}}\sum_{j\neq
i}\bm{T}_{i}\cdot\bm{T}_{j},\quad c=a(b),\quad i=1(2).$ (108)

This colour operator is proportional to the identity in colour space when it
acts on a colour-singlet vector, due to colour conservation:

$\mathbf{\Gamma}^{(1)}_{ab;ab}\left(x_{1},x_{2}\right)\ket{\mathcal{A}_{n+2}^{\ell}}=\Gamma^{(1)}_{ab;ab}\left(x_{1},x_{2}\right)\ket{\mathcal{A}_{n+2}^{\ell}},$
(109)

which restores the original result. We can then rewrite the identity-
preserving (IP) sector of the mass factorization counterterms in (7) as:

$\displaystyle\text{d}{\hat{\sigma}^{MF,\mathrm{IP}}_{ab,\mathrm{NLO}}}$
$\displaystyle=$
$\displaystyle-\mathcal{N}^{V}_{NLO}\int{\dfrac{\text{d}x_{1}}{x_{1}}}{\dfrac{\text{d}x_{2}}{x_{2}}}\int\text{d}\Phi_{n}(p_{3},\dots,p_{n+2};x_{1}p_{1},x_{2}p_{2})\,J_{n}^{(n)}(\left\\{p\right\\}_{n})$
(110)
$\displaystyle\times\braket{\mathcal{A}_{n+2}^{0}}{\mathbf{\Gamma}^{(1)}_{ab;ab}\left(x_{1},x_{2}\right)}{\mathcal{A}_{n+2}^{0}}\,,$

which is analogous to (53).

While it is possible to apply a similar treatment for the identity-changing
mass factorization kernels in principle, it is not particularly convenient to
rewrite them as operators in colour space. Hence, we choose to maintain the
identity-changing mass factorization counterterm as colour scalars:

$\text{d}{\hat{\sigma}^{MF,\mathrm{IC}}_{ab,\mathrm{NLO}}}=-\left(\dfrac{\alpha_{s}\overline{C}(\epsilon)}{2\pi}\right)\sum_{(c,d)\neq(a,b)}\int{\dfrac{\text{d}x_{1}}{x_{1}}}{\dfrac{\text{d}x_{2}}{x_{2}}}\Gamma^{(1)}_{ab;cd}(x_{1},x_{2})\,\text{d}{\hat{\sigma}_{cd,\mathrm{LO}}}\,.$
(111)

###  4.2 NLO virtual subtraction term

The virtual subtraction term at NLO, ${\hat{\sigma}^{T}_{ab,\mathrm{NLO}}}$
can be separated into IP and IC components:

${\hat{\sigma}^{T}_{ab,\mathrm{NLO}}}={\hat{\sigma}^{T,\mathrm{IP}}_{ab,\mathrm{NLO}}}+{\hat{\sigma}^{T,\mathrm{IC}}_{ab,\mathrm{NLO}}}\,.$
(112)

The IP component has to:

  * •

remove the explicit poles of the virtual matrix element;

  * •

contain the IP mass factorization counterterm
$\text{d}{\hat{\sigma}^{MF,\mathrm{IP}}_{ab,\mathrm{NLO}}}$.

We know that the first requirement can be addressed with (53), while
$\text{d}{\hat{\sigma}^{MF,\mathrm{IP}}_{ab,\mathrm{NLO}}}$ is given above in
(110). By (70), (73) and the explicit formulae given in Section 3.3.1. One can
see that the one-loop integrated dipoles that were introduced in Section 3.3.1
address both tasks, yielding the construction of the NLO IP virtual
subtraction term as

$\displaystyle\text{d}{\hat{\sigma}^{T,\mathrm{IP}}_{ab,\mathrm{NLO}}}$
$\displaystyle=$
$\displaystyle\mathcal{N}^{V}_{NLO}\int{\dfrac{\text{d}x_{1}}{x_{1}}}{\dfrac{\text{d}x_{2}}{x_{2}}}\text{d}\Phi_{n}(p_{3},\dots,p_{n+2};x_{1}p_{1},x_{2}p_{2})J_{n}^{(n)}(\left\\{p\right\\}_{n})$
(113) $\displaystyle\times
2\braket{\mathcal{A}_{n+2}^{0}}{\mathbfcal{J}^{(1)}(\epsilon)}{}\,.$

The equation above explains the important role of the integrated dipoles and
justifies their definition. First of all, we can describe in a general way the
singularity structure of one-loop matrix elements, naturally including also
the IP mass factorization contribution. Moreover, we achieved this by means of
integrated antenna functions, which will be crucial for the next step. The
explicit form of (113) is obtained computing
$\braket{\mathcal{A}_{n+2}^{0}}{\bm{T}_{i}\cdot\bm{T}_{j}}{\mathcal{A}_{n+2}^{0}}$
and dressing them with the associated colour stripped integrated dipoles. We
remark that (113) is a completely process-independent result, which is valid
for any multiplicity and retains the full $N_{c}$ dependence.

The IC component of the virtual subtraction term is analogously obtained
exploiting IC one-loop integrated dipoles. These were defined as scalars in
colour space in (74), since IC infrared singularities factorize onto full LO
squared matrix elements. Hence,
$\text{d}{\hat{\sigma}^{T,\mathrm{IC}}_{ab,\mathrm{NLO}}}$ is given by:

$\text{d}{\hat{\sigma}^{T,\mathrm{IC}}_{ab,\mathrm{NLO}}}=\left(\dfrac{\alpha_{s}\overline{C}(\epsilon)}{2\pi}\right)\sum_{(c,d)\neq(a,b)}\int{\dfrac{\text{d}x_{1}}{x_{1}}}{\dfrac{\text{d}x_{2}}{x_{2}}}\mathcal{J}_{ab;cd}^{(1)}(x_{1},x_{2})\,\text{d}{\hat{\sigma}_{cd,\mathrm{LO}}}\,.$
(114)

The contribution above contains integrated initial-state collinear limits and
the associated mass factorization kernels.

###  4.3 NLO real subtraction term

In the colourful antenna approach, we aim for a systematic generation of
$\text{d}{\hat{\sigma}^{S}_{ab,\mathrm{NLO}}}$, starting from the
$\text{d}{\hat{\sigma}^{T}_{ab,\mathrm{NLO}}}$ derived above. Each term in
$\text{d}{\hat{\sigma}^{T}_{ab,\mathrm{NLO}}}$ must have an unintegrated
counterpart in $\text{d}{\hat{\sigma}^{S}_{ab,\mathrm{NLO}}}$. Since
$\text{d}{\hat{\sigma}^{T}_{ab,\mathrm{NLO}}}$ is completely written in terms
of integrated antenna functions, there is a one-to-one relation with their
unintegrated counterparts:

$\mathcal{X}_{3}^{0}(s_{ij})\,a_{n+2}^{0}(c,c^{\prime},\\{.,i,.,j,.\\})\,\leftrightarrow\,X_{3}^{0}(i,u,j)a_{n+2}^{0}(c,c^{\prime},\\{.,\widetilde{iu},.,\widetilde{uj},.\\}),$
(115)

where $\mathcal{X}_{3}^{0}(s_{ij})$ is the integrated antenna function
obtained by integrating the tree-level three-parton antenna function