in the computation is the Durham algorithm Catani:1992ua  with
$y_{\text{cut}}=0.1$. Using this setup, in Fig.(5) we plot the differential
cross section with respect to the rapidity of the most energetic jet.

Figure 5: Rapidity distribution of the most energetic jet in $\gamma^{*}\to 2$
jets at NLO (Durham algorithm, $y_{\text{cut}}=0.1$,
$\sqrt{s}=125\,\text{GeV}$, $\alpha_{S}(\sqrt{s})=0.11173799$). The Dual
Subtraction (DS) results are shown in red, while the Catani–Seymour (CS) ones
in blue. The error bars correspond to the statistical error given by the Monte
Carlo integration.

The excellent agreement observed in Fig.(5) among the two results validates
our procedure and is found by plotting any other differential variable.

###  6.2 $\gamma^{*}\to 3$ jets at NLO

The basic sub-process contributing to the three-jet production in $e^{+}e^{-}$
annihilation into a virtual photon is $\gamma^{*}\\!\rightarrow
q(p_{1})\,\bar{q}(p_{2})\,g(p_{3})$, while in the real sector one has the two
sub-processes $\gamma^{*}\rightarrow
q(p^{\prime}_{1})\,\bar{q}(p^{\prime}_{2})\,g(p^{\prime}_{3})\,g(p^{\prime}_{4})$
and $\gamma^{*}\rightarrow
q(p^{\prime}_{1})\,\bar{q}(p^{\prime}_{2})\,q^{\prime}(p^{\prime}_{3})\,\bar{q}^{\prime}(p^{\prime}_{4})$,
where the two quark-antiquark pairs may have equal or different flavours. For
simplicity we will consider only the gluonic channel, where an additional
gluon is radiated with respect of the virtual sector. The contribution of the
other sub-processes can be accounted for by similar considerations.

By inspecting the virtual sector, we find six emitter-spectator pairs for the
process $\gamma^{*}\rightarrow q(p_{1})\,\bar{q}(p_{2})\,g(p_{3})$, that are
$\\{q\bar{q},qg,\bar{q}g\\}$ and the three pairs where the emitter and the
spectator switch. The case of four quark production can be treated along the
same lines. Once a real kinematic configuration
$(p^{\prime}_{1},p^{\prime}_{2},p^{\prime}_{3},p^{\prime}_{4})$ is generated,
the six kinematic invariants $s_{12}$, $s_{13}$, $s_{14}$, $s_{23}$, $s_{24}$
and $s_{34}$ are analyzed and six dual counterterms are activated. To evaluate
them, six virtual configurations $(p_{1},p_{2},p_{3},q_{1})$ are built by
using six times the inverse mapping in Eq.(18), every time with the proper set
of real and virtual momenta. To each emitter-spectator pair we associate the
following dual counterterms

$\displaystyle q\bar{q}\quad$
$\displaystyle\longrightarrow\quad\left\\{\quad\begin{aligned}
&&g(p^{\prime}_{3})\,\,\text{as
radiation:}&&\quad\mathcal{C}^{(1)}_{13,2}\quad\text{if}\,\,s_{13}<s_{23},&&\quad\mathcal{C}^{(1)}_{23,1}\quad\text{if}\,\,s_{13}>s_{23}\\\
&&g(p^{\prime}_{4})\,\,\text{as
radiation:}&&\quad\mathcal{C}^{(1)}_{14,2}\quad\text{if}\,\,s_{14}<s_{24},&&\quad\mathcal{C}^{(1)}_{24,1}\quad\text{if}\,\,s_{14}>s_{24}\end{aligned}\right.$
$\displaystyle qg\quad$
$\displaystyle\longrightarrow\quad\left\\{\quad\begin{aligned}
&&g(p^{\prime}_{3})\,\,\text{as
radiation:}&&\quad\mathcal{C}^{(1)}_{13,4}\quad\text{if}\,\,s_{13}<s_{34},&&\quad\mathcal{C}^{(1)}_{43,1}\quad\text{if}\,\,s_{13}>s_{34}\\\
&&g(p^{\prime}_{4})\,\,\text{as
radiation:}&&\quad\mathcal{C}^{(1)}_{14,3}\quad\text{if}\,\,s_{14}<s_{34},&&\quad\mathcal{C}^{(1)}_{34,1}\quad\text{if}\,\,s_{14}>s_{34}\end{aligned}\right.$
$\displaystyle\bar{q}g\quad$
$\displaystyle\longrightarrow\quad\left\\{\quad\begin{aligned}
&&g(p^{\prime}_{3})\,\,\text{as
radiation:}&&\quad\mathcal{C}^{(1)}_{23,4}\quad\text{if}\,\,s_{23}<s_{34},&&\quad\mathcal{C}^{(1)}_{43,2}\quad\text{if}\,\,s_{24}>s_{34}\\\
&&g(p^{\prime}_{4})\,\,\text{as
radiation:}&&\quad\mathcal{C}^{(1)}_{24,3}\quad\text{if}\,\,s_{24}<s_{34},&&\quad\mathcal{C}^{(1)}_{34,2}\quad\text{if}\,\,s_{24}>s_{34}\end{aligned}\right.$
(161)

where the $\mathcal{C}^{(1)}_{ac,b}$ is defined in Appendix A and corresponds
to the sum of the $V^{(1)}_{ac,b}$ and $G^{(1)}_{ac,b}$ terms of our
algorithm.

In the virtual sector, we use the integrated dual counterterms of Section 3.5.
Since there are three partons, a quark, an antiquark and a gluon, the colour
algebra factorizes and one has $\mathbf{T}_{1}\mathbf{T}_{2}=C_{A}/2-C_{F}$
and $\mathbf{T}_{1}\mathbf{T}_{3}=\mathbf{T}_{2}\mathbf{T}_{3}=-C_{A}/2$.
Therefore, the integrated dual counterterms are given by

$\displaystyle\sigma^{D\,(1)}_{q,\bar{q}}$
$\displaystyle\sim\frac{\alpha_{S}}{2\pi}\frac{(4\pi)^{\varepsilon}}{\Gamma(1-\varepsilon)}\left(\frac{C_{A}}{2}-C_{F}\right)\int
d\Phi_{2}\left(\frac{\mu^{2}}{s_{12}}\right)^{\varepsilon}|A^{(0)}_{q\bar{q}g}|^{2}\left[\frac{1}{\varepsilon^{2}}+\frac{3}{2\varepsilon}+3+4\log(2)-\frac{\pi^{2}}{2}\right]$
$\displaystyle\sigma^{D\,(1)}_{\bar{q},q}$
$\displaystyle\sim\frac{\alpha_{S}}{2\pi}\frac{(4\pi)^{\varepsilon}}{\Gamma(1-\varepsilon)}\left(\frac{C_{A}}{2}-C_{F}\right)\int
d\Phi_{2}\left(\frac{\mu^{2}}{s_{12}}\right)^{\varepsilon}|A^{(0)}_{q\bar{q}g}|^{2}\left[\frac{1}{\varepsilon^{2}}+\frac{3}{2\varepsilon}+3+4\log(2)-\frac{\pi^{2}}{2}\right]$
$\displaystyle\sigma^{D\,(1)}_{q,g}$
$\displaystyle\sim-\frac{\alpha_{S}}{2\pi}\frac{(4\pi)^{\varepsilon}}{\Gamma(1-\varepsilon)}\frac{C_{A}}{2}\int
d\Phi_{2}\left(\frac{\mu^{2}}{s_{13}}\right)^{\varepsilon}|A^{(0)}_{q\bar{q}g}|^{2}\left[\frac{1}{\varepsilon^{2}}+\frac{3}{2\varepsilon}+3+4\log(2)-\frac{\pi^{2}}{2}\right]$
$\displaystyle\sigma^{D\,(1)}_{g,q}$
$\displaystyle\sim-\frac{\alpha_{S}}{2\pi}\frac{(4\pi)^{\varepsilon}}{\Gamma(1-\varepsilon)}\frac{C_{A}}{2}\int
d\Phi_{2}\left(\frac{\mu^{2}}{s_{13}}\right)^{\varepsilon}|A^{(0)}_{q\bar{q}g}|^{2}\left[\frac{1}{\varepsilon^{2}}+\frac{11}{6\varepsilon}+\frac{55}{18}+\frac{14}{3}\log(2)-\frac{\pi^{2}}{2}\right]$
$\displaystyle\sigma^{D\,(1)}_{\bar{q},g}$
$\displaystyle\sim-\frac{\alpha_{S}}{2\pi}\frac{(4\pi)^{\varepsilon}}{\Gamma(1-\varepsilon)}\frac{C_{A}}{2}\int
d\Phi_{2}\left(\frac{\mu^{2}}{s_{23}}\right)^{\varepsilon}|A^{(0)}_{q\bar{q}g}|^{2}\left[\frac{1}{\varepsilon^{2}}+\frac{3}{2\varepsilon}+3+4\log(2)-\frac{\pi^{2}}{2}\right]$
$\displaystyle\sigma^{D\,(1)}_{g,\bar{q}}$
$\displaystyle\sim-\frac{\alpha_{S}}{2\pi}\frac{(4\pi)^{\varepsilon}}{\Gamma(1-\varepsilon)}\frac{C_{A}}{2}\int
d\Phi_{2}\left(\frac{\mu^{2}}{s_{23}}\right)^{\varepsilon}|A^{(0)}_{q\bar{q}g}|^{2}\left[\frac{1}{\varepsilon^{2}}+\frac{11}{6\varepsilon}+\frac{55}{18}+\frac{14}{3}\log(2)-\frac{\pi^{2}}{2}\right]$
(162)

where $|A^{(0)}_{q\bar{q}g}|^{2}$ is the Born amplitude for the process
$\gamma^{*}\rightarrow q\bar{q}g$ and we have neglected terms of order
$\mathcal{O}(\varepsilon)$.

Figure 6: Rapidity distribution of the most energetic jet in $\gamma^{*}\to 3$
jets (Durham algorithm, $y_{\text{cut}}=0.05$, $\sqrt{s}=125\,\text{GeV}$,
$\alpha_{S}(\sqrt{s})=0.11173799$). Only the NLO corrections are plotted. The
Dual Subtraction (DS) results are shown in red, while the Catani–Seymour (CS)
ones in blue. The error bars correspond to the statistical error given by the
Monte Carlo integration.

In Fig.(6) we show the differential cross section with respect to the rapidity
of the most energetic jet using again $\sqrt{s}=125\,\text{GeV}$ as the energy
in the center-of-mass frame, $\alpha_{S}(\sqrt{s})=0.11173799$ (starting from
$\alpha_{S}(m_{Z})=0.117$) and the Durham jet algorithm with
$y_{\text{cut}}=0.05$. Also in this case, we observe a perfect agreement with
the same computation performed using Catani–Seymour dipoles. We refrain to
include other plots because the same level of agreement is observed for any
other differential distribution.

###  6.3 $H\to b\,\bar{b}$ at NLO

In the present Section, we will make of use the formulas for the massive case
derived in Section 4 to study the Higgs boson decay into bottom quarks. The
virtual sector process is $H\rightarrow b(p_{1})\,\bar{b}(p_{2})$, while in
the real sector an addition gluon is emitted, $H\rightarrow
b(p^{\prime}_{1})\,\bar{b}(p^{\prime}_{2})\,g(p^{\prime}_{3})$. Therefore, the
counting of the dual counterterms is the same as for the process
$\gamma^{*}\rightarrow 2$ jets analyzed in Section 6.1.

Figure 7: Rapidity distribution of the most energetic jet in $H\to b\,\bar{b}$
at NLO. (Durham algorithm, $y_{\text{cut}}=0.01$, $m_{b}=4.78\,\text{GeV}$,
$m_{H}=125.09\,\text{GeV}$, $\alpha_{S}(m_{H})=0.11263619$). The Dual
Subtraction (DS) results are shown in red, while the Catani–Seymour (CS) ones
in blue. The error bars correspond to the statistical error given by the Monte
Carlo integration.

Once a real phase space configuration is generated, the proper counterterm in
Eq.(135) is selected comparing the $s_{13}$ and $s_{23}$ invariants. As for
the virtual contribution, we have renormalized the mass of the wave–function
of the bottom quark on-shell. The virtual singularities are canceled by the
integrated dual counterterms of Section 4.3 with
$\mathbf{T}_{1}\mathbf{T}_{2}=-C_{F}$.

The differential distribution of the rapidity of the most energetic jet is
plotted in Fig.(7). We have used $m_{H}=125.09\,\text{GeV}$ for the Higgs
boson mass and $\alpha_{S}(m_{H})=0.11263619$ corresponding to
$\alpha_{S}(m_{Z})=0.118$. The on-shell mass of the bottom quark has been set
to $m_{b}=4.78\,\text{GeV}$. The jet algorithm used in the computation is the
Durham algorithm with $y_{\text{cut}}=0.01$. A fairly good agreement with the
computation performed using Catani–Seymour dipoles is observed.

###  6.4 Drell-Yan pair production plus 0 or 1 jet at NLO

In this Section we consider proton-proton collisions where a $W^{+}$ boson is
produced. We start by considering the Born process with no further radiation.
In the virtual sector the basic sub-process is
$q(p_{1})\,\bar{q}(p_{2})\rightarrow W^{+}(p_{3})$, while in the real sector
we have $q(p^{\prime}_{1})\,\bar{q}(p^{\prime}_{2})\rightarrow
W^{+}(p^{\prime}_{3})\,g(p^{\prime}_{4})$. Since there are two partons in the
initial state, we apply the method shown in Section 5. For the sake of
comparison and for simplicity, we limit here to the quark-antiquark initiated
sub-processes.

Figure 8: Rapidity distribution of the $W^{+}$ boson in proton-proton
collisions at $8\,\text{TeV}$ center-of-mass energy ($q\bar{q}$-channel only).
The renormalization and the factorization scale are both equal to
$\mu=80\,\text{GeV}$, while $\alpha_{S}(\mu)=0.12264887$,
$m_{W}=80.398\,\text{GeV}$ and $G_{F}=1.16639\cdot 10^{-}5\,\text{GeV}^{-2}$.
The Dual Subtraction (DS) results are shown in red, while the Catani–Seymour
(CS) ones in blue. The error bars correspond to the statistical error given by
the Monte Carlo integration.

In the Monte Carlo implementation, we generate a real phase space
configuration $(p^{\prime}_{3},p^{\prime}_{4})$ and compare the kinematic
invariants $s_{14}$ and $s_{24}$. If $s_{14}<s_{24}$, we apply the mapping in
Eq.(5), with
$(p^{\prime}_{a},p^{\prime}_{b},p^{\prime}_{c})=(p^{\prime}_{1},p^{\prime}_{2},p^{\prime}_{3})$,
to obtain a virtual configuration $(p_{1},p_{2},q_{1})$. Then, we evaluate the
integrand of the dual counterterm $\sigma^{D\,(1)}_{qg,\bar{q}}$ in Eq.(5) and
we add it to the real cross section. If $s_{14}>s_{24}$, we use the mapping in
Eq.(5) with
$(p^{\prime}_{a},p^{\prime}_{b},p^{\prime}_{c})=(p^{\prime}_{2},p^{\prime}_{1},p^{\prime}_{3})$,
instead. In the virtual sector we use the integrated dual counterterms of
Section 5, with $\mathbf{T}_{1}\mathbf{T}_{2}=-C_{F}$.

In Fig.(8) we show the differential cross section with respect to the rapidity
of the $W^{+}$ boson. We consider proton-proton collisions at $8\,\text{TeV}$
in the center-of-mass frame and we have used the MSTW2008NLO set of parton
distribution functions with $\alpha_{S}(m_{Z})=0.12018$. The renormalization
and the factorization scale have been set equal to
$\mu_{R}=\mu_{F}=\mu=80\,\text{GeV}$ and we have used
$m_{W}=80.398\,\text{GeV}$ for the $W^{+}$ boson mass, $G_{F}=1.16639\cdot
10^{-}5\,\text{GeV}^{-2}$ for the Fermi constant and
$\alpha_{S}(\mu)=0.12264887$ for the strong coupling constant. Also in this
case, we observe an excellent agreement among the computation performed using
dual subtractions and the one obtained with Catani–Seymour dipoles.

Figure 9: Charged lepton (left) and hardest jet (right) transverse momentum
distributions for $W^{+}$ plus 1 jet production in proton-proton collisions at
$14\,\text{TeV}$ center-of-mass energy ($q\bar{q}$-channel only). The
renormalization and the factorization scale are both equal to
$\mu=80\,\text{GeV}$. The Dual Subtraction (DS) results are shown in red,
while the Catani–Seymour (CS) ones in blue. The error bars correspond to the
statistical error given by the Monte Carlo integration.

The case in which there is radiation also at Born level, beyond the initial-
initial counterterms, requires the computation also of initial-final and final
inital counterterms. For each possible emitter spectator pair the two
kinematic invariants are computed and compared. The counterterm to subtract
correspond to the smaller of the two invariants. In Fig.(9) we show the lepton
and leading jet transverse momentum distributions respectively. Jets are
formed using the antikt algorithm with resolution paramenter $R=0.5$. Only the
$q\bar{q}$ channel is included.

###  6.5 Higgs boson production in gluon fusion plus 0 or 1 jet at NLO

We consider the production of an Higgs boson in proton-proton collision in the
framework of the effective field theory where the heavy top quark degree of
freedom that mediates the interaction among the Higgs boson and gluons has
been integrated out. At the Born and virtual sector the basic sub-process is
then $g(p_{1})\,g(p_{2})\rightarrow H(p_{3})$, while in the real sector we
have $g(p^{\prime}_{1})\,g(p^{\prime}_{2})\rightarrow
H(p^{\prime}_{3})\,g(p^{\prime}_{4})$. For simplicity, we limit here to a
theory with only gluons, so that these are the only two partonic subprocesses
for a NLO calculation. Since the Born process has a colourless final state and
presents two partons in the initial state, we apply the method shown in
Section 5.

Figure 10: Rapidity distribution of the Higgs boson in proton-proton
collisions at $14\,\text{TeV}$ center-of-mass energy ($gg$-channel only). The
renormalization and the factorization scale are both equal to
$\mu=m_{H}/2=62.5\,\text{GeV}$. The Dual Subtraction (DS) results are shown in
red, while the Catani–Seymour (CS) ones in blue. The error bars correspond to
the statistical error given by the Monte Carlo integration performed with the
same setup.

The mapping and the rest of the procedure is precisely the one already
discussed for the case of the W boson production with the only difference that
for the present case we use the formulas for a gluon emitting a gluon. In
Fig.(10) we show the rapidity distribution of the Higgs boson produced at
LHC14. We have set $m_{H}=125\,$GeV for the Higgs boson mass and used the
${\rm NNPDF31\\_nnlo\\_as\\_0118\\_luxqed}$ set of parton distribution
functions with $\alpha_{S}(m_{Z})=0.11800216$. The renormalization and the
factorization scale have been set equal to $\mu_{R}=\mu_{F}=m_{H}/2$. Once
again, we observe excellent agreement among the computation performed using
dual subtractions and the one obtained with Catani–Seymour dipoles.

Figure 11: Higgs boson rapidity (left) and transverse momentum (right)
distributions for $H$ plus 1 jet production in proton-proton collisions at
$14\,\text{TeV}$ center-of-mass energy ($gg$-channel only). The Dual
Subtraction (DS) results are shown in red, while the Catani–Seymour (CS) ones
in blue. The error bars correspond to the statistical error given by the Monte
Carlo integration.

Finally, we consider the process in which one hard radiation is present
already at Born level. In this case the basic subprocess is
$g(p_{1})\,g(p_{2})\rightarrow H(p_{3})g(p_{4})$ and the real subprocess
contains one further radiated gluon. Using the formulas for initial-initial,
initial-final and final-initial gluon radiation from a gluon we cancel locally
all the singularities obtaining fully differential predictions. In Fig.(11) we
show the Higgs boson rapidity (left) and transverse momentum distributions
(right) respectively. Again we observe a perfect agreement with the same
computation performed using Catani–Seymour dipoles.

##  7 Conclusion

Starting from the analysis of the divergences of one-loop amplitudes reduced
to scalar integrals, in the present paper we have built a subtraction
formalism exploiting the LTD theorem. Beyond the singularities of the virtual
diagrams, we have also to take into account the ones brought by the
wave–function renormalization. An integrand–level representation for these
counterterms suitable for massless and massive fermions can be found in
Sborlini:2016gbr  and Sborlini:2016hat , respectively. In the present work we
provide an integrand–level representation of the gluon wave–function
renormalization constant. Furthermore, our construction is naturally extended
to the case of initial state singularities. Starting from the dual
representation of the divergent part of virtual contribution, we cancelled the
singularities of the real contribution by simply feeding the former with the
mapped momenta of the latter and multiplying times the momentum fraction $x$.

We now expand a bit more on this last point. Let us consider a process in
which a colourless final state is produced and, to further simplify the
discussion, let’s take the production of a pair or leptons from $q\bar{q}$
annihilation into a virtual photon, so ignoring the contribution of the $Z$
boson. Normally, when calculating higher–order contributions in a proton-
proton collision one starts from a leading order process producing a final
state with a given invariant mass $s_{ij}$. This might seems a complication in
our case, but for the sake of the discussion, let’s consider that our
experiment is capable to see pairs of leptons only in a well defined narrow
window of the invariant mass of the pair at the time. Once a value for the
invariant mass is chosen we compute the Born, the virtual and the real
contribution to the scattering process. It is well known that requiring a
fixed final state invariant mass for the lepton pair the singularities will
not cancel. Living on different phase spaces, $s_{ij}$ for the virtual and
$s_{ab}>s_{ij}$ for the real, they cannot match. Thanks to the remarkable
property of factorization, this has been solved through a redefinition of the
initial state in the form of a renormalization of the parton distribution
functions. Let us now take a slightly different point of view. We could define
the corrections, both real and virtual, to be initiated by the same center of
mass energy $s_{ab}$. The real part is precisely as before, but the virtual
part is not contributing at all if $s_{ab}$ is greater then $s_{ij}$. By the
way, the divergences cancel if we consider $s_{ab}=s_{ij}$. The only
difference now is that, if we have $s_{ab}>s_{ij}$ and cut the loop amplitude
in such a way to produce a gluon that is not soft, then the invariant mass of
the leptons in the Born system will be lower then $s_{ab}$ and could be tuned
to be the value $s_{ij}$ of the Born process. By doing so (and also
multiplying times $x$) we observe the cancellation of the singularities. Of
course, this way of cutting the loops produces a result that does not
correspond to the original loop integral living on $s_{ab}$, but it represents
a generalization of the loop integral living on $s_{ij}$ for $x<1$. In the
present work we have exploited this feature that integrating such cut virtual
loops, originally living on $s_{ab}$, suffices to cancel both the loop
divergences at $s_{ij}$ and the collinear initial state singularities. This
fact provides a general strategy to use Loop–Tree Duality to compute cross
sections with hadrons in the initial state. In fact our procedure can be
applied also to the case of full amplitudes that are not reduced to scalar
integrals. In practice, for the cases studied in the previous section, such as
Drell-Yan and Higgs production in gluon fusion, the divergent part of the
virtual contribution that we have extracted, differs very little (by finite
terms and just one IR finite bubble) from the whole renormalized virtual
contribution.
Acting such generalized virtual as a counterterm that cancels the IR
divergences of the real part, there seems to be no room for an explicit
presence of the factorization scale. Of course, we want to take benefit of
parton evolution so the question is what is the factorization scale to be used
in PDF’s if we use a dual version of the (extended and cut) virtual
contribution in the sense discussed above. This issue will be discussed
elsewhere but we anticipate here a simple analytic strategy to answer to this
question. It consists in deducing the corresponding factorization scale
through the comparison among results obtained by performing the NLO
computation in the two ways, using the cut extended loops on one side and the
dipole subtractions plus the universal collinear counterterms (which bring the
explicit presence of the factorization scale in the hard cross section) on the
other.
On a technical ground, the structure of the dual counterterms is very close to
the Catani–Seymour dipoles. In fact, even if we count a lower number of
counter events per real phase space point with respect to the Catani–Seymour
construction, in the latter it is possible to reduce the number of active
dipoles by restricting the phase space region where they are computed to the
neighbourhood of the singularities. On a more formal ground, the dual
subtraction builds a direct link for the cancellation among virtual and real
singularities.
We conclude by adding a few comments. The extension of the formalism presented
here to the case of NNLO corrections is an interesting path that will be
addressed elsewhere. We note that many ingredients are already available
Bierenbaum:2012th ; Aguilera-Verdugo:2019kbz ; Aguilera-Verdugo:2020nrp ;
TorresBobadilla:2020ekr , including examples of full NNLO computation using
LTD, momentum mappings and master integrals. Further applications of the work
of the present paper would be the implementation of novel shower and matching
schemes based on the dual subtractions discussed here. These subjects are
beyond the aim of the present work and are left for future investigations.

##  Appendix A Counting of the dual counterterms

The computational scheme proposed in the current work exploits the dual
representation of the loop amplitude with $m$ partons in the final state in
order to cancel soft and collinear singularities of the real matrix element
living on the $(m+1)$-parton phase space. The subtractions are extracted from
the virtual sector on the base of the number of emitter-spectator pairs in its
$m$-parton phase space. Our starting point is represented by a set of dual
counterterms $\mathcal{C}_{i,j}$ (with $i,j=1,\dots,m$ and $i\neq j$), each
associated with two functions $V_{i,j}$ and $G_{i,j}$ that depend on the
emitter-spectator pair $(p_{i},p_{j})$. These are constructed by application
of LTD to the scalar three- and two-point functions, $C_{0}(p_{i},p_{j})$ and
$B_{0}(p_{i})$, respectively, and the wave–function renormalization
counterterm $\Delta Z(p_{i})$. In particular, $\mathcal{C}_{ij}$ has the
general structure in terms of the dual $V$ and $G$

$\mathcal{C}_{ij}(p_{i},p_{j})\equiv\,_{m}\langle
1,\dots,m|\mathbf{T}_{i}\mathbf{T}_{j}\left(V_{i,j}(p_{i},p_{j})+G_{i,j}(p_{i},p_{j})\right)|1,\dots,m\rangle_{m}$
(163)

The dual cross section is obtained by summing over all the possible emitter-
spectator pairs

$\sigma^{D}\equiv\mathcal{N}_{in}\sum_{\\{m\\}}\int
d\Phi_{m}\,\frac{1}{S_{\\{m\\}}}\sum_{i}\sum_{j\neq
i}\mathcal{C}_{i,j}(p_{1},\dots,p_{m})$ (164)

where $\mathcal{N}_{in}$ includes all the non-QCD factors, $\sum_{\\{m\\}}$
indicates a sum over all the $m$-parton configurations and $S_{\\{m\\}}$ is
the Bose symmetry factor for identical partons. The sum over the different
emitters on the right-hand side of Eq.(164) can be split into sums over
different flavours

$\sum_{i}\sum_{j\neq i}\dots=\sum_{i=q}\sum_{j\neq
i}\dots+\sum_{i=\bar{q}}\sum_{j\neq i}\dots+\sum_{i=g}\sum_{j\neq
i}\dots\,\,.$ (165)

Then, we need to map the emitter-spectator pairs of the virtual sector into
the dipoles of the real sector. Each pair $(i,j)$ can be mapped in more than a
single dipole $(a,c,b)$, where parton $ac$ is the emitter, $c$ is the
radiation and $b$ is the spectator. Therefore, to distinguish the case in
which the pair $(i,j)$ is linked to a specific dipole $(a,c,b)$, we use the
following notation for the momentum mappings of Section 3.1

$\displaystyle\Phi_{m}$ $\displaystyle=\mathbf{M}^{i,j}_{a,c,b}(\Phi_{m+1})$
$\displaystyle q_{k}$ $\displaystyle=\mathbf{M}^{k}_{a,c,b}(\Phi_{m+1})\,\,.$
(166)

The application of Eq.(A) in the corresponding dual counterterm leads to

$\mathcal{C}_{ac,b}\equiv\mathcal{C}_{i,j}(\mathbf{M}^{i,j}_{a,c,b}(\Phi_{m+1}))\,\,.$
(167)

We now want to use Eq.(167) and count the number of counterterms through the
number of dipoles in the real sector, by transforming each sum on the right-
hand side of Eq.(165) in the following way

$\sum_{i=f}\sum_{j\neq
i}=\frac{\\#(f)_{m}}{\\#\\{a,c\\}_{m+1}}\sum_{a<c}\sum_{b\neq a,c}$ (168)

where $\\#(f)_{m}$ denotes the number of partons with flavour $f$ in the
$m$-parton configuration and $\\#\\{a,c\\}_{m+1}$ the number of pairs
$\\{a,c\\}$ in the $(m+1)$-parton configuration such that the emitter parton
$ac$ has flavour $f$. Given an $m$-parton configuration, all the possible
$(m+1)$-parton processes can be obtained either by increasing by one the
number $m_{g}$ of gluons, or by decreasing by one $m_{g}$ and adding a quark-
antiquark pair. Let us focus on the first case, which includes all the dual
counterterms beside the quark contributions to the gluon wave–function
renormalization. The latters are required in the second class of
$(m+1)$-parton processes, treated at the end of this Appendix. Therefore, for
the moment we assume that the counterterms $\mathcal{C}_{g,j}$ only include
gluon and ghost contribution to the gluon renormalization. We have

$\displaystyle\frac{\\#(q_{f})_{m}}{\\#\\{q_{f},g\\}_{m+1}}$
$\displaystyle=\frac{m_{f}}{m_{f}(m_{g}+1)}=\frac{1}{(m_{g}+1)}$
$\displaystyle\frac{\\#(\bar{q}_{f})_{m}}{\\#\\{\bar{q}_{f},g\\}_{m+1}}$
$\displaystyle=\frac{\bar{m}_{f}}{\bar{m}_{f}(m_{g}+1)}=\frac{1}{(m_{g}+1)}$
$\displaystyle\frac{\\#(g)_{m}}{\\#\\{g,g\\}_{m+1}}$
$\displaystyle=\frac{m_{g}}{m_{g}(m_{g}+1)/2}=\frac{2}{(m_{g}+1)}$ (169)

$m_{q}$ and $m_{\bar{q}}$ being the number of quark and antiquarks of a given
flavour, respectively. The counting in Eq.(168) is possible only if the
integral of $\mathcal{C}_{i,j}$ over the $m$-particle phase space does not
change when we exchange parton $i$ with a parton of the same flavour. This
condition is certainly fulfilled if we use the $q_{1}$-cut method and so
always set on-shell the loop momentum propagator connecting the external
partons ($q_{1}$). However, particular attention must be payed to this point
if we want to use the $q_{1}$-$q_{2}$-method. In fact, if $a$ and $a^{\prime}$
had the same flavour and we used $\sigma^{D\,(1)}_{ac,b}$ to describe the
radiation collinear to $a$ and $\sigma^{D\,(2)}_{bc,a^{\prime}}$ for the
emission collinear to $a^{\prime}$, then the dual subtractions would differ
among each other. As a solution to this problem we can perform, for each dual
subtraction, the following symmetrization

$\displaystyle\sigma^{D\,(1)}_{ac,b}$
$\displaystyle\rightarrow\frac{1}{2}\left(\sigma^{D\,(1)}_{ac,b}+\sigma^{D\,(2)}_{bc,a}\right)$
$\displaystyle\sigma^{D\,(2)}_{ac,b}$
$\displaystyle\rightarrow\frac{1}{2}\left(\sigma^{D\,(1)}_{bc,a}+\sigma^{D\,(2)}_{ac,b}\right)$
(170)

which ensures that the dual subtractions depend only on the flavours of the
partons in the associated dipoles. Eq.(A) may be derived from a different
prescription for the application of LTD, which can be summarized as follows:
when we consider the generic contribution $\mathcal{A}$ (be it a triangle
$C_{0}$, a bubble $B_{0}$ or a wave–function renormalization counterterm
$\Delta Z$) from a given emitter-spectator pair $(i,j)$ of the virtual sector,
we first use the identity $\mathcal{A}=\mathcal{A}/2+\mathcal{A}/2$; then, for
the first term, we apply LTD as usual while, for the second term, we denote
the internal momenta e we apply LTD as if we had considered the pair $(j,i)$
instead of $(i,j)$. From a diagrammatic point of view, this means that we
associate the triangle in Fig.(3(a)) to the first $\mathcal{A}$ term and the
triangle in Fig.(3(b)) to the second one.

Using Eqs.(168) and (A) we can turn Eq.(165) into

$\sum_{i}\sum_{k\neq
i}\dots=\frac{1}{(m_{g}+1)}\left(\,\,\sum_{\begin{subarray}{c}\text{pairs}\\\
a,c=q,g\end{subarray}}\sum_{b\neq
a,c}\dots+\sum_{\begin{subarray}{c}\text{pairs}\\\
a,c=\bar{q},g\end{subarray}}\sum_{b\neq
a,c}\dots+2\sum_{\begin{subarray}{c}\text{pairs}\\\
a,c=g,g\end{subarray}}\sum_{b\neq a,c}\dots\right)$ (171)

so that, reminding the notation in Eq.(167), the dual cross section in
Eq.(164) can be rewritten as

$\sigma^{D}=\mathcal{N}_{in}\sum_{\\{m\\}}\int
d\phi_{m}\,\frac{1}{S_{\\{m\\}}}\frac{1}{(m_{g}+1)}\\\
\times\left(\,\,\sum_{\begin{subarray}{c}\text{pairs}\\\
a,c=q,g\end{subarray}}\sum_{b\neq
a,c}\mathcal{C}_{ac,b}+\sum_{\begin{subarray}{c}\text{pairs}\\\
a,c=\bar{q},g\end{subarray}}\sum_{b\neq
a,c}\mathcal{C}_{ac,b}+2\sum_{\begin{subarray}{c}\text{pairs}\\\
a,c=g,g\end{subarray}}\sum_{b\neq a,c}\mathcal{C}_{ac,b}\right)\,\,.$ (172)

The presence of a factor $2$ in front of the sum over gluon-gluon pairs on the
right-hand side of Eq.(172) has a clear interpretation: given a certain pair
of gluons in the real sector, their roles can be exchanged and, therefore, we
need to take twice the dual counterterm. In this way, indeed, in one
counterterm we can apply the mapping where one of the two gluons is the
radiation and, in the other counterterm, we can use the mapping where the role
of radiation is played by the other gluon. Furthermore, using the relation

$\frac{S_{\\{m\\}}}{S_{\\{m+1\\}}}=\frac{\dots
m_{g}!}{\dots(m_{g}+1)!}=\frac{1}{m_{g}+1}$ (173)

we can turn Eq.(172) into

$\sigma^{D}=\mathcal{N}_{in}\sum_{\\{m\\}}\int
d\phi_{m}\,\frac{1}{S_{\\{m+1\\}}}\\\
\times\left[\,\,\sum_{\begin{subarray}{c}\text{pairs}\\\
a,c=q,g\end{subarray}}\sum_{b\neq
a,c}\mathcal{C}_{ac,b}+\sum_{\begin{subarray}{c}\text{pairs}\\\
a,c=\bar{q},g\end{subarray}}\sum_{b\neq
a,c}\mathcal{C}_{ac,b}+\sum_{\begin{subarray}{c}\text{pairs}\\\
a,c=g,g\end{subarray}}\sum_{b\neq
a,c}\left(\mathcal{C}_{ac,b}+\mathcal{C}_{ca,b}\right)\right]\,\,.$ (174)

The sums in Eq.(174) present one dual counterterm for each dipole of the real
phase space. Therefore, Eq.(174) tells us how to associate every single
counterterm of the dual subtraction scheme with the proper singular
configuration of the real phase space.

Let us now move to processes with a quark-antiquark pair replacing a gluon.
These involve the counterterms $\mathcal{C}_{g,j}$, that we assume to include
only the quark contribution to the gluon renormalization, since gluon and
ghost contributions are used in processes with one additional gluon in the
final state. Before to apply Eq.(168), we first multiply and divide the
counterterms by $N_{f}$ and then use the $N_{f}$ in the numerator to introduce
a sum over the different quark flavours

$\sum_{i=g}\sum_{j\neq
i}\mathcal{C}_{i,j}=\frac{1}{N_{f}}\sum_{i=g}\Biggl{(}\,\,\underbrace{\sum_{j\neq
i}\mathcal{C}_{i,j}+\dots+\sum_{j\neq
i}\mathcal{C}_{i,j}}_{N_{f}\,\text{times}}\,\,\Biggr{)}=\frac{1}{N_{f}}\sum_{i=g}\sum_{f}\sum_{j\neq
i}\mathcal{C}_{i,j}\,\,.$ (175)

Then, we apply Eq.(168) ending up with

$\frac{1}{N_{f}}\sum_{i=g}\sum_{f}\sum_{j\neq
i}\mathcal{C}_{i,j}=\frac{1}{N_{f}}\sum_{f}\frac{\\#(g)_{m}}{\\#\\{q_{f},\bar{q}_{f}\\}_{m+1}}\sum_{\begin{subarray}{c}\text{pairs}\\\
a,c=q_{f},\bar{q}_{f}\end{subarray}}\sum_{b\neq a,c}\mathcal{C}_{ac,b}$ (176)

where

$\frac{\\#(g)_{m}}{\\#\\{q_{f},\bar{q}_{f}\\}_{m+1}}=\frac{m_{g}}{(m_{f}+1)(\bar{m}_{f}+1)}$
(177)

turns the Bose symmetry factor $S_{\\{m\\}}$ into $S_{\\{m+1\\}}$, since

$\frac{S_{\\{m\\}}}{S_{\\{m+1\\}}}=\frac{\dots
m_{f}!\bar{m}_{f}!m_{g}!}{\dots(m_{f}+1)!(\bar{m}_{f}+1)!(m_{g}-1)!}=\frac{m_{g}}{(m_{f}+1)(\bar{m}_{f}+1)}\,\,.$
(178)

Therefore, the dual cross section for processes with a quark-antiquark pair in
place of a final state gluon can be written as

$\displaystyle\sigma^{D}$ $\displaystyle=\mathcal{N}_{in}\sum_{\\{m\\}}\int
d\phi_{m}\,\frac{1}{S_{\\{m\\}}}\frac{m_{g}}{(m_{f}+1)(\bar{m}_{f}+1)}\frac{1}{N_{f}}\sum_{f}\sum_{\begin{subarray}{c}\text{pairs}\\\
a,c=q_{f},\bar{q}_{f}\end{subarray}}\sum_{b\neq a,c}\mathcal{C}_{ac,b}$
$\displaystyle=\mathcal{N}_{in}\sum_{\\{m\\}}\int
d\phi_{m}\,\frac{1}{S_{\\{m+1\\}}}\frac{1}{N_{f}}\sum_{f}\sum_{\begin{subarray}{c}\text{pairs}\\\
a,c=q_{f},\bar{q}_{f}\end{subarray}}\sum_{b\neq a,c}\mathcal{C}_{ac,b}\,\,.$
(179)

Eq.(A) associates each dual counterterm with a collinear quark-antiquark
configuration of the real phase space.

##  Appendix B Dual Counterterms for the massless case

In this section we collect all the formulas to be used for a numerical
implementation of the dual subtractions. To regulate the divergences of the
real matrix elements, we have cut the relevant loop integrals and collected
their divergences. Then, in principle, we could organize the integrated
counterpart in the form of finite remnants of the loop integrals. This could
be useful for implementations of our formalism in general purposed programs
that perform the full reduction of the virtual amplitude. On the other hand,
to make the implementation easier, in this appendix we will continue to treat
our cut loop integrals as normal countertems and proceed to present the list
of their integrated version. The formulation of our method is given as
follows: in Table 1 we list the equation number of the dual subtraction along
with the name of the corresponding jacobian factor that converts the radiation
variables to that of the loop momentum as explained in Section 3. We label
with $p_{a}$ and $p_{b}$ the partons in the initial state and $p_{i}$,
$p_{j}$, $p_{k}$ and so on for the ones in the final state. Moreover, we
remember here that the formulae for the dual dipoles are written following the
convention of in-going and out-going momenta,
$p_{a}+p_{b}=p_{i}+p_{j}+p_{k}+...\,$. Furthermore, the names of the
counterterms are written using the same conventions as in the presentation of
the Altarelli–Parisi splitting functions in Section 3.2. The prefixes specify
the emitter and spectator respectively. The suffixes have to be understood as
$(ac)c$ for a final state emitter $(ac)$ radiating the parton $c$ and it is
$a(ac)$ for an initial state emitter $(ac)$ (the hard parton) that is the
result of the splitting of the incoming parent parton $a$. So for example:
$if\\_qq$ stands for the counterterm for an initial state quark ($i$ in the
first part of the prefix and $q$ in the first part of the suffix) emitting a
gluon and becoming the hard quark of the underlying event (the second part of
the suffix $q$), and this gluon is connected to a final state spectator (as
indicated by the $f$ in the second part of the prefix); $ff\\_qg$ stands
instead for a virtual quark $(q)$ splitting into a $qg$ pair with the gluon
$(g)$ being the radiated parton connected to another final state parton. The
dual dipoles formulae do not depend from the location of the spectator
momentum.

Counterterm | Equation $\times$ jac | Counterterm | Equation $\times$ jac
---|---|---|---
ff_qg | Eq.(48)$\times$ jac(ff) | fi_qg | Eq.(48)$\times$ jac(fi)
ff_gq | Eq.(63)$\times$ jac(ff) | fi_gq | Eq.(63)$\times$ jac(fi)
ff_gg | Eq.(54)$\times$ jac(ff) | fi_gg | Eq.(54)$\times$ jac(fi)
if_qq | Eq.(155)$\times$ jac(if) | ii_qq | Eq.(155)$\times$ jac(ii)
if_qg | Eq.(5)$\times$ jac(if) | ii_qg | Eq.(5)$\times$ jac(ii)
if_gq | Eq.(157)$\times$ jac(if) | ii_gq | Eq.(157)$\times$ jac(ii)
if_gg | Eq.(149)$\times$ jac(if) | ii_gg | Eq.(149)$\times$ jac(ii)
Table 1: Counterterms definition as product of the cut loop integrals in the
equations times the corresponding jacobian factor.

The formulae for the dipole subtractions do not depend from the specific
momentum mappings nevertheless for their numerical implementation one choice
has to be made. Here we adopt the momentum mappings reported in
ref.Catani:1996vz  that from now on we dub CS and will refer to the equation
x.y in that paper with CS(x.y) for easy of reference444Note that sometimes we
have relabelled the suffixes of the momenta with respect to the equations of
ref.Catani:1996vz ..

###  B.1 final-final

For the case of final state emitter and spectator we will report in detail all
the formulas that we take from CS. For the other cases we will just point to
the other formulas in the same paper. The momentum mapping to transform the
three momentum $\\{i,j,k\\}$ into $\\{\tilde{ij},\tilde{k}\\}$ is defined in
the Equation CS(5.3)

$\displaystyle\tilde{p}_{k}^{\mu}$ $\displaystyle=\frac{1}{1-y}p_{k}^{\mu}$
(180) $\displaystyle\tilde{p}_{ij}^{\mu}$
$\displaystyle=p_{i}^{\mu}+p_{j}^{\mu}-\frac{y}{1-y}p_{k}^{\mu}$

in terms of the variables

$\displaystyle y$
$\displaystyle=\frac{p_{i}p_{j}}{p_{i}p_{j}+p_{j}p_{k}+p_{k}p_{i}}$ (181)
$\displaystyle z_{i}$ $\displaystyle=\frac{p_{i}p_{k}}{p_{i}p_{k}+p_{j}p_{k}}$
$\displaystyle z_{j}$ $\displaystyle=1-z_{i}\,.$

The factorization of the phase space in Eq. CS(5.17-20) is then

$d\phi(p_{i},p_{j},p_{k};Q)=d\phi(\tilde{p}_{ij},\tilde{p}_{k};Q)[dp_{j}(\tilde{p}_{ij},\tilde{p}_{k})]$
(182)

where

$\displaystyle[dp_{j}(\tilde{p}_{ij},\tilde{p}_{k})]$
$\displaystyle=\frac{d^{d}p_{j}}{(2\,\pi)^{d-1}}\delta_{+}(p_{j}^{2})\mathcal{I}(p_{j};\tilde{p}_{ij},\tilde{p}_{k})$
(183)

with

$\displaystyle\mathcal{I}(p_{j};\tilde{p}_{ij},\tilde{p}_{k})$
$\displaystyle=\Theta(1-z_{j})\Theta(1-y)\frac{(1-y)^{d-3}}{1-z_{j}}\,.$ (184)

In terms of the radiation variables introduced above Equation CS(5.20) reads

$\displaystyle[dp_{j}(\tilde{p}_{ij},\tilde{p}_{k})]=$
$\displaystyle\frac{(2\,\tilde{p}_{ij}\tilde{p}_{k})^{1-\varepsilon}}{16\,\pi^{2}}\frac{d\Omega^{d-3}}{(2\,\pi)^{1-2\varepsilon}}dz_{i}dy\Theta(z_{i}(1-z_{i}))\Theta(y(1-y))$
$\displaystyle\times(z_{i}(1-z_{i}))^{-\varepsilon}(1-y)^{1-2\,\varepsilon}y^{-\varepsilon}\,.$
(185)

As explained in Section 3, the local cancellation of the singularities among
virtual and real contribution is obtained only if one takes into account also
the jacobian factor associated to the mapping. In particular we have to divide
the formulae for the counterterms in Tab.1 by the factor
$\mathcal{I}(p_{j};\tilde{p}_{ij},\tilde{p}_{k})$ that for this case amount to
multiply them by

$\displaystyle{\rm jac(ff)}=\frac{z_{i}}{(1-y)^{1-2\varepsilon}}$ (186)

Furthermore, instead of the whole region $0<z_{i},y<1$ implied by the $\Theta$
functions in Eq.(B.1) the restriction of the phase space induced by the
$R_{1}$ function corresponds to limit the integration region to the
configurations with $p_{i}p_{j}<p_{j}p_{k}$ or in terms of the radiation
variables to the restriction

$\displaystyle 0<y<\frac{1-z_{i}}{2-z_{i}}\qquad\qquad 0<z_{i}<1\,.$ (187)

We introduce the normalization factor

$\displaystyle N_{\varepsilon}$
$\displaystyle=\frac{(4\,\pi)^{\varepsilon}}{\Gamma(1-\varepsilon)}$ (188)

and the short hand notation $l_{2}\equiv\log(2)$. We integrate the
counterterms built by the product of the dual integrands multiplied times the
corresponding jacobian factor (jac(ff) in this case), using the measure in
Eq.(B.1) and restricting the integration to the region in Eq.(187), obtaining

$\displaystyle{\rm ff\\_qg\\_int}$
$\displaystyle=\frac{\alpha_{s}}{2\,\pi}N_{\varepsilon}\left(\frac{\mu^{2}}{2\,\tilde{p}_{ij}\tilde{p}_{k}}\right)^{\varepsilon}\left(\frac{1}{\varepsilon^{2}}+\frac{3}{2\,\varepsilon}-\frac{\pi^{2}}{2}+4\,l_{2}+3\right)$
(189) $\displaystyle{\rm ff\\_gq\\_int}$
$\displaystyle=\frac{\alpha_{s}}{2\,\pi}N_{\varepsilon}\frac{N_{f}}{C_{A}}\left(\frac{\mu^{2}}{2\,\tilde{p}_{ij}\tilde{p}_{k}}\right)^{\varepsilon}\left(-\frac{2}{3\,\varepsilon}-\frac{10}{3}\,l_{2}+\frac{2}{9}\right)$
(190) $\displaystyle{\rm ff\\_gg\\_int}$
$\displaystyle=\frac{\alpha_{s}}{2\,\pi}N_{\varepsilon}\left(\frac{\mu^{2}}{2\,\tilde{p}_{ij}\tilde{p}_{k}}\right)^{\varepsilon}\left(\frac{2}{\varepsilon^{2}}+\frac{11}{3\,\varepsilon}-\pi^{2}+\frac{28}{3}\,l_{2}+\frac{55}{9}\right)$
(191)

Note that we adopt the convention to have in the prefactor the kinematic
invariant built with the momenta of the emitter and spectator partons. The
same convention is adopted also for the other cases Campbell:2004ch .
Following the approach of combining the integrated countertems with the
virtual matrix element it is more convenient to do so because this is the
invariant which is kept fixed during the $x$ integration. In the above
equations the case for ${\rm ff\\_g\bar{q}\\_int}$ is included in ${\rm
ff\\_gq\\_int}$ as a factor of 2 compensated with the $T_{R}$ factor.
Furthermore, the two parts corresponding to the simmetrization in Eq.(48)
gives identical integrated contribution. We point out that the momentum
mapping in Eqs.(180,181) is symmetric for the exchange of $p_{i}$ and $p_{j}$
and so there is just one counter event although its weight is determined by
the phase space restrictions that event by event are different for the two
contributions. The same considerations also apply to the next case that is the
final-initial.

###  B.2 final-initial

The momentum mapping to transform the three momentum $\\{i,j,a\\}$ into
$\\{\tilde{ij},\tilde{a}\\}$ is defined in the Equation CS(5.37)

$\displaystyle\tilde{p}_{a}^{\mu}$ $\displaystyle=x^{\prime}p_{a}^{\mu}$ (192)
$\displaystyle\tilde{p}_{ij}^{\mu}$
$\displaystyle=p_{i}^{\mu}+p_{j}^{\mu}-(1-x^{\prime})p_{a}^{\mu}$

in terms of the variables

$\displaystyle x^{\prime}$
$\displaystyle=\frac{p_{i}p_{a}+p_{j}p_{a}-p_{i}p_{j}}{p_{i}p_{a}+p_{j}p_{a}}$
(193) $\displaystyle z_{i}$
$\displaystyle=\frac{p_{i}p_{a}}{p_{i}p_{a}+p_{j}p_{a}}$ $\displaystyle z_{j}$
$\displaystyle=1-z_{i}\,.$

The factorization of the three parton phase space in Eq. CS(5.45-48) is then

$d\phi(p_{i},p_{j};Q+p_{a})=\int_{0}^{1}dxd\phi(\tilde{p}_{ij};Q+xp_{a})[dp_{j}(\tilde{p}_{ij};p_{a},x)]$
(194)

where

$\displaystyle[dp_{j}(\tilde{p}_{ij};p_{a},x)]$
$\displaystyle=\frac{d^{d}p_{j}}{(2\,\pi)^{d-1}}\delta_{+}(p_{j}^{2})\Theta(x(1-x))\delta(x-x^{\prime})\frac{1}{1-z_{j}}$
(195)

witch in terms of the radiation variables introduced above reads

$\displaystyle[dp_{j}(\tilde{p}_{ij};p_{a},x)]=$
$\displaystyle\frac{(2\,\tilde{p}_{ij}p_{a})^{1-\varepsilon}}{16\,\pi^{2}}\frac{d\Omega^{d-3}}{(2\,\pi)^{1-2\varepsilon}}dz_{i}\Theta(z_{i}(1-z_{i}))\Theta(x(1-x))\delta(x-x^{\prime})$
$\displaystyle\times(z_{i}(1-z_{i}))^{-\varepsilon}(1-x)^{-\varepsilon}\,.$
(196)

From Eq.(195) the jacobian factor needed to transform the radiated momentum
into the loop momentum is given by

$\displaystyle{\rm jac(fi)}=z_{i}$ (197)

Furthermore, the integration region corresponding to $p_{i}p_{j}<p_{j}p_{a}$
in terms of the radiation variables is given by

$\displaystyle 0<z_{i}<x\qquad\qquad 0<x<1\,.$ (198)

We integrate the counterterms built by the product of the dual integrands
multiplied times the corresponding jacobian factor in Eq.(197), using the
measure in Eq.(B.2) and restricting the integration to the region in Eq.(198),
obtaining

$\displaystyle{\rm fi\\_qg\\_int}$
$\displaystyle=\frac{\alpha_{s}}{2\,\pi}N_{\varepsilon}\left(\frac{\mu^{2}}{2\,\tilde{p}_{ij}\tilde{p}_{a}}\right)^{\varepsilon}\left[\delta(1-x)\left(\frac{1}{\varepsilon^{2}}+\frac{3}{2\,\varepsilon}-\frac{\pi^{2}}{3}+\frac{7}{2}\right)-\frac{3}{2}\frac{1}{(1-x)}_{+}\right.$
$\displaystyle\left.-\left(\frac{2\log(1-x)}{1-x}\right)_{+}+2\log(1-x)+\frac{1}{2}(3+3x+x^{2})\right]$
(199) $\displaystyle{\rm fi\\_gq\\_int}$
$\displaystyle=\frac{\alpha_{s}}{2\,\pi}N_{\varepsilon}\frac{N_{f}}{C_{A}}\left(\frac{\mu^{2}}{2\,\tilde{p}_{ij}\tilde{p}_{a}}\right)^{\varepsilon}\left[\delta(1-x)\left(-\frac{2}{3\,\varepsilon}-\frac{10}{9}\right)+\frac{2}{3}\frac{1}{(1-x)}_{+}\right.$
$\displaystyle\left.-\frac{1}{3}(2+2x-x^{2}+2x^{3})\right]$ (200)
$\displaystyle{\rm fi\\_gg\\_int}$
$\displaystyle=\frac{\alpha_{s}}{2\,\pi}N_{\varepsilon}\left(\frac{\mu^{2}}{2\,\tilde{p}_{ij}\tilde{p}_{a}}\right)^{\varepsilon}\left[\delta(1-x)\left(\frac{2}{\varepsilon^{2}}+\frac{11}{3\,\varepsilon}-\frac{2\,\pi^{2}}{3}+\frac{67}{9}\right)-\frac{11}{3}\frac{1}{(1-x)}_{+}\right.$
$\displaystyle\left.-\left(\frac{4\log(1-x)}{1-x}\right)_{+}+4\log(1-x)+\frac{1}{3}(11+11x-x^{2}+2x^{3})\right]$
(201)

###  B.3 initial-final

The momentum mapping to transform the three momentum $\\{a,j,i\\}$ into
$\\{\tilde{aj},\tilde{i}\\}$ is defined in the Equation CS(5.62-64)

$\displaystyle\tilde{p}_{aj}^{\mu}$ $\displaystyle=x^{\prime}p_{a}^{\mu}$
(202) $\displaystyle\tilde{p}_{i}^{\mu}$
$\displaystyle=p_{i}^{\mu}+p_{j}^{\mu}-(1-x)p_{a}^{\mu}.$

in terms of the variables

$\displaystyle x^{\prime}$
$\displaystyle=\frac{p_{i}p_{a}+p_{j}p_{a}-p_{j}p_{i}}{p_{i}p_{a}+p_{j}p_{a}}$
(203) $\displaystyle u_{j}$
$\displaystyle=\frac{p_{j}p_{a}}{p_{j}p_{a}+p_{i}p_{a}}=z_{j}\,.$

This momentum mapping coincides with the previous one. This is particularly
useful for our approach because such a map connects a set of radiation
variables and a Born configuration with an initial(final) state emitter and a
final(initial) state spectator to a unique real configuration and vivecersa.
The factorization of the three parton phase space in Eq. CS(5.70-73) is the
same as in the previous case and the jacobian factor needed to transform the
radiated momentum into the loop momentum is again given by

$\displaystyle{\rm jac(if)}=1-u_{j}=z_{i}={\rm jac(fi)}$ (204)

The integration region corresponding to $p_{a}p_{j}<p_{i}p_{j}$ in terms of
the radiation variables is now given by

$\displaystyle 0<u_{j}<1-x\qquad\qquad 0<x<1\,.$ (205)

We integrate the counterterms built by the product of the dual integrands
multiplied times the corresponding jacobian factor in Eq.(204), using the
measure in Eq.(B.2) and restricting the integration to the region in Eq.(205),
obtaining

$\displaystyle{\rm if\\_qq\\_int}$
$\displaystyle=\frac{\alpha_{s}}{2\,\pi}N_{\varepsilon}\left(\frac{\mu^{2}}{2\,\tilde{p}_{aj}\tilde{p}_{i}}\right)^{\varepsilon}\left[\delta(1-x)\frac{1}{\varepsilon^{2}}-\frac{2}{\varepsilon}\frac{1}{(1-x)}_{+}+\left(\frac{4\log(1-x)}{1-x}\right)_{+}\right.$
$\displaystyle\left.+(1+x)\left(\frac{1}{\varepsilon}-2\log(1-x)\right)-\frac{1+x^{2}}{1-x}\log(x)-x(x^{2}+x+3)+1\right]$
(206) $\displaystyle{\rm if\\_qg\\_int}$
$\displaystyle=\frac{\alpha_{s}}{2\,\pi}N_{\varepsilon}\frac{T_{R}}{C_{A}}\left(\frac{\mu^{2}}{2\,\tilde{p}_{aj}\tilde{p}_{i}}\right)^{\varepsilon}\left[-\frac{1+(1-x)^{2}}{x}\left(\frac{1}{\varepsilon}-2\log(1-x)+2\log(x)+1\right)\right.$
$\displaystyle\left.+(2-x)\log(x)+x^{2}\right]$ (207) $\displaystyle{\rm
if\\_gq\\_int}$
$\displaystyle=\frac{\alpha_{s}}{2\,\pi}N_{\varepsilon}\left(\frac{\mu^{2}}{2\,\tilde{p}_{aj}\tilde{p}_{i}}\right)^{\varepsilon}\left[-(1-2x+2x^{2})\left(\frac{1}{\varepsilon}-2\log(1-x)+\log(x)\right)\right.$
$\displaystyle\left.-\left(\frac{1}{2}-2x+2x^{2}\right)\log(1-2x+2x^{2})+1-x\right]$
(208) $\displaystyle{\rm if\\_gg\\_int}$
$\displaystyle=\frac{\alpha_{s}}{2\,\pi}N_{\varepsilon}\left(\frac{\mu^{2}}{2\,\tilde{p}_{aj}\tilde{p}_{i}}\right)^{\varepsilon}\left[\delta(1-x)\frac{1}{\varepsilon^{2}}-\frac{2}{\varepsilon}\frac{1}{(1-x)}_{+}+\left(\frac{4\log(1-x)}{1-x}\right)_{+}\right.$
$\displaystyle\left.-2\left(\frac{1-x}{x}+x(1-x)-1\right)\left(\frac{1}{\varepsilon}-2\log(1-x)+\log(x)\right)-2\frac{1-2x+x^{2}}{x}\log(x)\right.$
$\displaystyle\left.-\frac{2}{1-x}\log(x)+\left(\frac{1}{2}-2x+2x^{2}\right)\log(1-2x+2x^{2})-x(x^{2}+3)\right]$
(209)

Note that in $if\\_qg\\_int$ there is no $N_{f}$ factor because it refers to
the contribution from a single kind of identified initial quark (or
antiquark). The same will happen also for the case initial-initial.

###  B.4 initial-initial

The momentum mapping to transform the three momenta $\\{a,j,b\\}$ into
$\\{\tilde{aj},\tilde{b}\\}$ is defined in the Equation CS(5.137-140)

$\displaystyle\tilde{p}_{aj}^{\mu}$ $\displaystyle=x^{\prime}p_{a}^{\mu}$
(210) $\displaystyle\tilde{p}_{b}^{\mu}$ $\displaystyle=p_{b}^{\mu}$

in terms of the variables

$\displaystyle x^{\prime}$
$\displaystyle=\frac{p_{a}p_{b}-p_{j}p_{a}-p_{j}p_{b}}{p_{a}p_{b}}$ (211)
$\displaystyle v_{j}$ $\displaystyle=\frac{p_{a}p_{j}}{p_{a}p_{b}}\,.$

Note that in this case the momenta of all the other final state particles
$k_{i}$ are transformed as follows (CS(5.139-140))

$\displaystyle\tilde{k}_{i}^{\mu}$
$\displaystyle=k_{i}^{\mu}-\frac{2\,k_{i}\cdot(K+\tilde{K})}{(K+\tilde{K})^{2}}(K+\tilde{K})^{\mu}+\frac{2\,k_{i}\cdot
K}{K^{2}}\tilde{K}^{\mu}$ (212) $\displaystyle K^{\mu}$
$\displaystyle=p_{a}^{\mu}+p_{b}^{\mu}-p_{j}^{\mu}$
$\displaystyle\tilde{K}^{\mu}$
$\displaystyle=\tilde{p}_{aj}^{\mu}+p_{b}^{\mu}\,.$

The factorization of the three parton phase space in Eq. CS(5.149-151) is then

$d\phi(p_{k},k_{1},\dots;p_{a}+p_{b})=\int_{0}^{1}dxd\phi(\tilde{k}_{1},\dots;\tilde{p}_{a}+\tilde{p}_{b})[dp_{j}(p_{a},p_{b},x)]$
(213)

where

$\displaystyle[dp_{j}(p_{a},p_{b},x)]$
$\displaystyle=\frac{d^{d}p_{j}}{(2\,\pi)^{d-1}}\delta_{+}(p_{j}^{2})\Theta(x(1-x))\delta(x-x^{\prime})$
(214)

witch in terms of the radiation variables introduced above reads

$\displaystyle[dp_{j}(p_{a},p_{b},x)]=$
$\displaystyle\frac{(2\,p_{a}p_{b})^{1-\varepsilon}}{16\,\pi^{2}}\frac{d\Omega^{d-3}}{(2\,\pi)^{1-2\varepsilon}}dv_{j}\Theta(v_{j})\Theta\left(1-\frac{v_{j}}{1-x}\right)\Theta(x(1-x))\delta(x-x^{\prime})$
$\displaystyle\times(v_{j}(1-x-v_{j}))^{-\varepsilon}\,.$ (215)

From Eq.(214) the jacobian factor needed to transform the radiated momentum
into the loop momentum is given by

$\displaystyle{\rm jac(ii)}=1$ (216)

Furthermore, the integration region corresponding to $p_{a}p_{j}<p_{b}p_{j}$
in terms of the radiation variables is given by

$\displaystyle 0<v_{j}<\frac{1-x}{2}\qquad\qquad 0<x<1\,.$ (217)

We integrate the dual counterterms using the measure in Eq.(B.4) and
restricting the integration to the region in Eq.(217), obtaining

$\displaystyle{\rm ii\\_qq\\_int}$
$\displaystyle=\frac{\alpha_{s}}{2\,\pi}N_{\varepsilon}\left(\frac{\mu^{2}}{2\,\tilde{p}_{aj}\tilde{p}_{b}}\right)^{\varepsilon}\left[\delta(1-x)\left(\frac{1}{\varepsilon^{2}}-\frac{\pi^{2}}{6}\right)-\frac{2}{\varepsilon}\frac{1}{(1-x)}_{+}+\left(\frac{4\log(1-x)}{1-x}\right)_{+}\right.$
$\displaystyle\left.+(1+x)\left(\frac{1}{\varepsilon}-2\log(1-x)-l_{2}\right)-\frac{1+x^{2}}{1-x}\log(x)+\frac{1-x}{2}\right]$
(218) $\displaystyle{\rm ii\\_qg\\_int}$
$\displaystyle=\frac{\alpha_{s}}{2\,\pi}N_{\varepsilon}\frac{T_{R}}{C_{A}}\left(\frac{\mu^{2}}{2\,\tilde{p}_{aj}\tilde{p}_{b}}\right)^{\varepsilon}\left[-\frac{1+(1-x)^{2}}{x}\left(\frac{1}{\varepsilon}-2\log(1-x)+\log(x)+l_{2}\right)\right.$
$\displaystyle\left.-\frac{3}{4}\frac{(1-x)^{2}}{x}-3\frac{1-x}{x}\right]$
(219) $\displaystyle{\rm ii\\_gq\\_int}$
$\displaystyle=\frac{\alpha_{s}}{2\,\pi}N_{\varepsilon}\left(\frac{\mu^{2}}{2\,\tilde{p}_{aj}\tilde{p}_{b}}\right)^{\varepsilon}\left[-(1-2\,x+2\,x^{2})\left(\frac{1}{\varepsilon}-2\log(1-x)+\log(x)+l_{2}\right)\right.$
$\displaystyle\left.+2x^{2}(l_{2}-\log(1+2\,x-x^{2}))+\frac{1-x}{2}\right]$
(220) $\displaystyle{\rm ii\\_gg\\_int}$
$\displaystyle=\frac{\alpha_{s}}{2\,\pi}N_{\varepsilon}\left(\frac{\mu^{2}}{2\,\tilde{p}_{aj}\tilde{p}_{b}}\right)^{\varepsilon}\left[\delta(1-x)\left(\frac{1}{\varepsilon^{2}}-\frac{\pi^{2}}{6}\right)-\frac{2}{\varepsilon}\frac{1}{(1-x)}_{+}+\left(\frac{4\log(1-x)}{1-x}\right)_{+}\right.$
$\displaystyle\left.-2\left(\frac{1-x}{x}+x(1-x)-1\right)\left(\frac{1}{\varepsilon}-2\log(1-x)+\log(x)+l_{2}\right)-\frac{2}{1-x}\log(x)\right.$
$\displaystyle\left.+2\,x^{2}\log(1+2\,x-x^{2})-2\,(1+x+x^{2})\,l_{2}-\frac{3}{4}\frac{(1-x)^{2}}{x}-\frac{1-x}{x}\right]$
(221)

###### Acknowledgements.

The authors gratefully acknowledge Germán Rodrigo for a critical reading of
the manuscript, Gábor Somogyi for useful communications and also Jonathan
Ronca, German Sborlini and William J. Torres Bobadilla for useful discussions.
Our work is supported by INFN.

## References

  * (1) F. Englert and R. Brout, _Broken Symmetry and the Mass of Gauge Vector Mesons_ , _Phys. Rev. Lett._ 13 (1964) 321–323.
  * (2) P. W. Higgs, _Spontaneous Symmetry Breakdown without Massless Bosons_ , _Phys. Rev._ 145 (1966) 1156–1163.
  * (3) ATLAS collaboration, G. Aad et al., _Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC_ , _Phys. Lett. B_ 716 (2012) 1–29, [1207.7214].
  * (4) CMS collaboration, S. Chatrchyan et al., _Observation of a New Boson at a Mass of 125 GeV with the CMS Experiment at the LHC_ , _Phys. Lett. B_ 716 (2012) 30–61, [1207.7235].
  * (5) S. Catani and M. H. Seymour, _A General algorithm for calculating jet cross-sections in NLO QCD_ , _Nucl. Phys._ B485 (1997) 291–419, [hep-ph/9605323].
  * (6) S. Catani, S. Dittmaier, M. H. Seymour and Z. Trocsanyi, _The Dipole formalism for next-to-leading order QCD calculations with massive partons_ , _Nucl. Phys._ B627 (2002) 189–265, [hep-ph/0201036].
  * (7) S. Frixione, Z. Kunszt and A. Signer, _Three jet cross-sections to next-to-leading order_ , _Nucl. Phys._ B467 (1996) 399–442, [hep-ph/9512328].
  * (8) S. Catani and M. Grazzini, _An NNLO subtraction formalism in hadron collisions and its application to Higgs boson production at the LHC_ , _Phys.Rev.Lett._ 98 (2007) 222002, [hep-ph/0703012].
  * (9) R. Boughezal, C. Focke, X. Liu and F. Petriello, _$W$ -boson production in association with a jet at next-to-next-to-leading order in perturbative QCD_, _Phys. Rev. Lett._ 115 (2015) 062002, [1504.02131].
  * (10) J. Gaunt, M. Stahlhofen, F. J. Tackmann and J. R. Walsh, _N-jettiness Subtractions for NNLO QCD Calculations_ , _JHEP_ 09 (2015) 058, [1505.04794].
  * (11) A. Gehrmann-De Ridder, T. Gehrmann and E. N. Glover, _Antenna subtraction at NNLO_ , _JHEP_ 0509 (2005) 056, [hep-ph/0505111].
  * (12) A. Daleo, T. Gehrmann and D. Maitre, _Antenna subtraction with hadronic initial states_ , _JHEP_ 0704 (2007) 016, [hep-ph/0612257].
  * (13) A. Gehrmann-De Ridder, T. Gehrmann and E. W. N. Glover, _Gluon-gluon antenna functions from Higgs boson decay_ , _Phys. Lett._ B612 (2005) 49–60, [hep-ph/0502110].
  * (14) A. Gehrmann-De Ridder, T. Gehrmann and E. W. N. Glover, _Quark-gluon antenna functions from neutralino decay_ , _Phys. Lett._ B612 (2005) 36–48, [hep-ph/0501291].
  * (15) A. Daleo, A. Gehrmann-De Ridder, T. Gehrmann and G. Luisoni, _Antenna subtraction at NNLO with hadronic initial states: initial-final configurations_ , _JHEP_ 01 (2010) 118, [0912.0374].
  * (16) T. Gehrmann and P. F. Monni, _Antenna subtraction at NNLO with hadronic initial states: real-virtual initial-initial configurations_ , _JHEP_ 12 (2011) 049, [1107.4037].
  * (17) R. Boughezal, A. Gehrmann-De Ridder and M. Ritzmann, _Antenna subtraction at NNLO with hadronic initial states: double real radiation for initial-initial configurations with two quark flavours_ , _JHEP_ 02 (2011) 098, [1011.6631].
  * (18) A. Gehrmann-De Ridder, T. Gehrmann and M. Ritzmann, _Antenna subtraction at NNLO with hadronic initial states: double real initial-initial configurations_ , _JHEP_ 10 (2012) 047, [1207.5779].
  * (19) J. Currie, E. Glover and S. Wells, _Infrared Structure at NNLO Using Antenna Subtraction_ , _JHEP_ 1304 (2013) 066, [1301.4693].
  * (20) G. Somogyi, Z. Trocsanyi and V. Del Duca, _Matching of singly- and doubly-unresolved limits of tree-level QCD squared matrix elements_ , _JHEP_ 06 (2005) 024, [hep-ph/0502226].
  * (21) G. Somogyi, Z. Trócsányi and V. Del Duca, _A Subtraction scheme for computing QCD jet cross sections at NNLO: Regularization of doubly-real emissions_ , _JHEP_ 0701 (2007) 070, [hep-ph/0609042].
  * (22) G. Somogyi and Z. Trócsányi, _A Subtraction scheme for computing QCD jet cross sections at NNLO: Regularization of real-virtual emission_ , _JHEP_ 0701 (2007) 052, [hep-ph/0609043].
  * (23) G. Somogyi and Z. Trocsanyi, _A Subtraction scheme for computing QCD jet cross sections at NNLO: Integrating the subtraction terms. I._ , _JHEP_ 08 (2008) 042, [0807.0509].
  * (24) U. Aglietti, V. Del Duca, C. Duhr, G. Somogyi and Z. Trocsanyi, _Analytic integration of real-virtual counterterms in NNLO jet cross sections. I._ , _JHEP_ 09 (2008) 107, [0807.0514].
  * (25) G. Somogyi, _Subtraction with hadronic initial states at NLO: An NNLO-compatible scheme_ , _JHEP_ 05 (2009) 016, [0903.1218].
  * (26) P. Bolzoni, S.-O. Moch, G. Somogyi and Z. Trocsanyi, _Analytic integration of real-virtual counterterms in NNLO jet cross sections. II._ , _JHEP_ 08 (2009) 079, [0905.4390].
  * (27) P. Bolzoni, G. Somogyi and Z. Trocsanyi, _A subtraction scheme for computing QCD jet cross sections at NNLO: integrating the iterated singly-unresolved subtraction terms_ , _JHEP_ 01 (2011) 059, [1011.1909].
  * (28) V. Del Duca, G. Somogyi and Z. Trocsanyi, _Integration of collinear-type doubly unresolved counterterms in NNLO jet cross sections_ , _JHEP_ 06 (2013) 079, [1301.3504].
  * (29) G. Somogyi, _A subtraction scheme for computing QCD jet cross sections at NNLO: integrating the doubly unresolved subtraction terms_ , _JHEP_ 04 (2013) 010, [1301.3919].
  * (30) M. Czakon, _A novel subtraction scheme for double-real radiation at NNLO_ , _Phys.Lett._ B693 (2010) 259–268, [1005.0274].
  * (31) M. Czakon, _Double-real radiation in hadronic top quark pair production as a proof of a certain concept_ , _Nucl.Phys._ B849 (2011) 250–295, [1101.0642].
  * (32) M. Czakon and D. Heymes, _Four-dimensional formulation of the sector-improved residue subtraction scheme_ , _Nucl. Phys._ B890 (2014) 152–227, [1408.2500].
  * (33) M. Czakon, A. van Hameren, A. Mitov and R. Poncelet, _Single-jet inclusive rates with exact color at $\mathcal{O}$ (${\alpha}_{s}^{4}$)_, _JHEP_ 10 (2019) 262, [1907.12911].
  * (34) F. Caola, K. Melnikov and R. Röntsch, _Nested soft-collinear subtractions in NNLO QCD computations_ , _Eur. Phys. J._ C77 (2017) 248, [1702.01352].
  * (35) F. Caola, M. Delto, H. Frellesvig and K. Melnikov, _The double-soft integral for an arbitrary angle between hard radiators_ , _Eur. Phys. J._ C78 (2018) 687, [1807.05835].
  * (36) M. Delto and K. Melnikov, _Integrated triple-collinear counter-terms for the nested soft-collinear subtraction scheme_ , _JHEP_ 05 (2019) 148, [1901.05213].
  * (37) F. Caola, K. Melnikov and R. Röntsch, _Analytic results for color-singlet production at NNLO QCD with the nested soft-collinear subtraction scheme_ , _Eur. Phys. J. C_ 79 (2019) 386, [1902.02081].
  * (38) F. Caola, K. Melnikov and R. Röntsch, _Analytic results for decays of color singlets to $gg$ and $q\bar{q}$ final states at NNLO QCD with the nested soft-collinear subtraction scheme_, _Eur. Phys. J. C_ 79 (2019) 1013, [1907.05398].
  * (39) M. Cacciari, F. A. Dreyer, A. Karlberg, G. P. Salam and G. Zanderighi, _Fully Differential Vector-Boson-Fusion Higgs Production at Next-to-Next-to-Leading Order_ , _Phys. Rev. Lett._ 115 (2015) 082002, [1506.02660].
  * (40) L. Magnea, E. Maina, G. Pelliccioli, C. Signorile-Signorile, P. Torrielli and S. Uccirati, _Local Analytic Sector Subtraction at NNLO_ , _JHEP_ 12 (2018) 107, [1806.09570].
  * (41) L. Magnea, E. Maina, G. Pelliccioli, C. Signorile-Signorile, P. Torrielli and S. Uccirati, _Factorisation and Subtraction beyond NLO_ , _JHEP_ 12 (2018) 062, [1809.05444].
  * (42) L. Magnea, G. Pelliccioli, C. Signorile-Signorile, P. Torrielli and S. Uccirati, _Analytic integration of soft and collinear radiation in factorised QCD cross sections at NNLO_ , 2010.14493.
  * (43) F. Herzog, _Geometric IR subtraction for final state real radiation_ , _JHEP_ 08 (2018) 006, [1804.07949].
  * (44) S. Catani, T. Gleisberg, F. Krauss, G. Rodrigo and J.-C. Winter, _From loops to trees by-passing Feynman’s theorem_ , _JHEP_ 09 (2008) 065, [0804.3170].
  * (45) I. Bierenbaum, S. Catani, P. Draggiotis and G. Rodrigo, _A Tree-Loop Duality Relation at Two Loops and Beyond_ , _JHEP_ 10 (2010) 073, [1007.0194].
  * (46) G. F. R. Sborlini, F. Driencourt-Mangin, R. Hernandez-Pinto and G. Rodrigo, _Four-dimensional unsubtraction from the loop-tree duality_ , _JHEP_ 08 (2016) 160, [1604.06699].
  * (47) R. J. Hernandez-Pinto, G. F. R. Sborlini and G. Rodrigo, _Towards gauge theories in four dimensions_ , _JHEP_ 02 (2016) 044, [1506.04617].
  * (48) G. F. R. Sborlini, F. Driencourt-Mangin and G. Rodrigo, _Four-dimensional unsubtraction with massive particles_ , _JHEP_ 10 (2016) 162, [1608.01584].
  * (49) F. Driencourt-Mangin, G. Rodrigo, G. F. Sborlini and W. J. Torres Bobadilla, _Universal four-dimensional representation of $H\to\gamma\gamma$ at two loops through the Loop-Tree Duality_, _JHEP_ 02 (2019) 143, [1901.09853].
  * (50) S. Buchta, G. Chachamis, P. Draggiotis, I. Malamos and G. Rodrigo, _On the singular behaviour of scattering amplitudes in quantum field theory_ , _JHEP_ 11 (2014) 014, [1405.7850].
  * (51) S. Buchta, G. Chachamis, P. Draggiotis and G. Rodrigo, _Numerical implementation of the loop–tree duality method_ , _Eur. Phys. J. C_ 77 (2017) 274, [1510.00187].
  * (52) S. Buchta, _Theoretical foundations and applications of the Loop-Tree Duality in Quantum Field Theories_.  PhD thesis, Valencia U., 2015.  1509.07167.
  * (53) S. Ramírez-Uribe, R. J. Hernández-Pinto, G. Rodrigo, G. F. Sborlini and W. J. Torres Bobadilla, _Universal opening of four-loop scattering amplitudes to trees_ , 2006.13818.
  * (54) J. J. Aguilera-Verdugo, R. J. Hernandez-Pinto, G. Rodrigo, G. F. Sborlini and W. J. Torres Bobadilla, _Causal representation of multi-loop amplitudes within the loop-tree duality_ , 2006.11217.
  * (55) J. J. Aguilera-Verdugo, F. Driencourt-Mangin, R. J. Hernández-Pinto, J. Plenter, S. Ramirez-Uribe, A. E. Renteria Olivo, G. Rodrigo, G. F. Sborlini, W. J. Torres Bobadilla and S. Tracz, _Open Loop Amplitudes and Causality to All Orders and Powers from the Loop-Tree Duality_ , _Phys. Rev. Lett._ 124 (2020) 211602, [2001.03564].
  * (56) C. Anastasiou, R. Haindl, G. Sterman, Z. Yang and M. Zeng, _Locally finite two-loop amplitudes for off-shell multi-photon production in electron-positron annihilation_ , 2008.12293.
  * (57) Z. Capatti, V. Hirschi, D. Kermanschah, A. Pelloni and B. Ruijl, _Numerical Loop-Tree Duality: contour deformation and subtraction_ , _JHEP_ 04 (2020) 096, [1912.09291].
  * (58) Z. Capatti, V. Hirschi, D. Kermanschah, A. Pelloni and B. Ruijl, _Manifestly Causal Loop-Tree Duality_ , 2009.05509.
  * (59) S. Seth and S. Weinzierl, _Numerical integration of subtraction terms_ , _Phys. Rev. D_ 93 (2016) 114031, [1605.06646].
  * (60) A. Denner and S. Dittmaier, _Reduction schemes for one-loop tensor integrals_ , _Nucl. Phys. B_ 734 (2006) 62–115, [hep-ph/0509141].
  * (61) Z. Kunszt, A. Signer and Z. Trocsanyi, _Singular terms of helicity amplitudes at one loop in QCD and the soft limit of the cross-sections of multiparton processes_ , _Nucl. Phys. B_ 420 (1994) 550–564, [hep-ph/9401294].
  * (62) W. Giele and E. Glover, _Higher order corrections to jet cross-sections in e+ e- annihilation_ , _Phys. Rev. D_ 46 (1992) 1980–2010.
  * (63) G. Altarelli and G. Parisi, _Asymptotic Freedom in Parton Language_ , _Nucl. Phys. B_ 126 (1977) 298–318.
  * (64) A. Bassetto, M. Ciafaloni and G. Marchesini, _Jet Structure and Infrared Sensitive Quantities in Perturbative QCD_ , _Phys. Rept._ 100 (1983) 201–272.
  * (65) S. Catani, S. Dittmaier and Z. Trocsanyi, _One loop singular behavior of QCD and SUSY QCD amplitudes with massive partons_ , _Phys. Lett. B_ 500 (2001) 149–160, [hep-ph/0011222].
  * (66) S. Keller and E. Laenen, _Next-to-leading order cross-sections for tagged reactions_ , _Phys. Rev. D_ 59 (1999) 114004, [hep-ph/9812415].
  * (67) S. Catani, L. Trentadue, G. Turnock and B. Webber, _Resummation of large logarithms in e+ e- event shape distributions_ , _Nucl. Phys. B_ 407 (1993) 3–42.
  * (68) I. Bierenbaum, S. Buchta, P. Draggiotis, I. Malamos and G. Rodrigo, _Tree-Loop Duality Relation beyond simple poles_ , _JHEP_ 03 (2013) 025, [1211.5048].
  * (69) J. J. Aguilera-Verdugo, F. Driencourt-Mangin, J. Plenter, S. Ramírez-Uribe, G. Rodrigo, G. F. Sborlini, W. J. Torres Bobadilla and S. Tracz, _Causality, unitarity thresholds, anomalous thresholds and infrared singularities from the loop-tree duality at higher orders_ , _JHEP_ 12 (2019) 163, [1904.08389].
  * (70) J. J. Aguilera-Verdugo, R. J. Hernandez-Pinto, G. Rodrigo, G. F. Sborlini and W. J. Torres Bobadilla, _Mathematical properties of nested residues and their application to multi-loop scattering amplitudes_ , 2010.12971.
  * (71) W. Torres Bobadilla et al., _May the four be with you: Novel IR-subtraction methods to tackle NNLO calculations_ , 2012.02567.
  * (72) J. M. Campbell, R. K. Ellis and F. Tramontano, _Single top production and decay at next-to-leading order_ , _Phys. Rev. D_ 70 (2004) 094012, [hep-ph/0408158].