$\displaystyle{\rm d}\sigma_{NNLO}^{T,c}$ $\displaystyle=$
$\displaystyle\sum_{j}X_{ijk}^{0}\bigg{[}{\cal X}_{3}^{0}(s_{ik})-{\cal
X}_{3}^{0}(s_{ai})-{\cal X}_{3}^{0}(s_{kb})-{\cal X}_{3}^{0}(s_{IK})+{\cal
X}_{3}^{0}(s_{aI})+{\cal X}_{3}^{0}(s_{Kb})$ (7.43) $\displaystyle-$
$\displaystyle{\cal S}(s_{ik};s_{ik})+{\cal S}(s_{ai};s_{ik})+{\cal
S}(s_{kb};s_{ik})+{\cal S}(s_{IK};s_{ik})-{\cal S}(s_{aI};s_{ik})-{\cal
S}(s_{Kb};s_{ik})\bigg{]}$ $\displaystyle\times$ $\displaystyle|{\cal
M}_{m}^{0}(p_{1},\ldots,p_{m})|^{2}\;,$

where ${\cal S}$ is the integrated soft-eikonal factor. With the analytic
expressions for the integrated antennae and integrated soft-factors [175] we
obtain by construction a counterterm ${\rm d}\sigma_{NNLO}^{T,c}$ which is
free from explicit $1/\epsilon$-poles and has no phase space soft or collinear
singularities. In order to achieve this constraint it is necessary to add
genuine new terms of the type $X_{3}^{0}{\cal X}_{3}^{0}(s_{IK})$ to cancel
the poles of the wide-angle soft term. Such contributions must be integrated
analytically over the three-parton antenna phase space and added in integrated
form to the double-virtual $m$-parton contribution.

The full real-virtual subtraction term is given as a sum of all subtraction
terms constructed above:

${\rm d}\sigma_{NNLO}^{T}={\rm d}\sigma_{NNLO}^{T,a}+{\rm
d}\sigma_{NNLO}^{T,b}+{\rm d}\sigma_{NNLO}^{T,c}\;,$ (7.44)

which correctly approximates the real-virtual one-loop matrix element in all
single unresolved regions and simultaneously subtracts all of its
$1/\epsilon$-explicit poles as guaranteed by the KLN theorem.

####  7.2.3 Double-virtual contribution

The double virtual contribution involves the two-loop $m$-parton matrix
elements which have no implicit IR divergence in any regions of the
appropriate $m$-parton phase space. Therefore, to make this contribution
finite, all that remains is to introduce the integrated forms of the
appropriate antenna subtraction terms such that the explicit IR-poles of the
two-loop contribution are cancelled. We begin by reviewing the universal
structure of infrared singularities in on-shell QCD amplitudes at two-loop
order in Catani’s two-loop factorisation formula [185],

$\displaystyle{\cal P}oles\big{(}{\cal M}_{m}^{2}(1,\ldots,n)\big{)}$
$\displaystyle=$ $\displaystyle
2\boldsymbol{I}_{m}^{(1)}(\epsilon;1,\ldots,m)\;{\cal M}_{m}^{1}(1,\ldots,m)$
(7.45) $\displaystyle-$ $\displaystyle
2\boldsymbol{I}_{m}^{(1)}(\epsilon;1,\ldots,m)^{2}{\cal
M}_{m}^{0}(1,\ldots,m)$ $\displaystyle+$ $\displaystyle
2e^{-\epsilon\gamma}\frac{\Gamma(1-2\epsilon)}{\Gamma(1-\epsilon)}\bigg{(}\frac{\beta_{0}}{\epsilon}+K\bigg{)}\boldsymbol{I}_{m}^{(1)}(2\epsilon;1\ldots,m)\,{\cal
M}_{m}^{0}(1,\ldots,m)$ $\displaystyle+$ $\displaystyle
2\boldsymbol{H}^{(2)}(\epsilon){\cal M}_{m}^{0}(1,\ldots,m).$

The poles of the two amplitude are organised according to the
$\boldsymbol{I}_{m}^{(1)}$-operator given in [185] and hard function,
$\boldsymbol{H}^{(2)}$ and the constant $K$, which depend on the particle
content and order in $M$ under consideration. In the following we will obtain
the integrated antenna subtraction terms in a form which is in one-to-one
correspondence with (7.45) making the analytic cancellation of all explicit
$1/\epsilon$-poles in the double-virtual contribution transparent.

The first double virtual subtraction term is the integrated counterpart of the
contribution introduced at the real-virtual level ${\rm
d}\sigma_{NNLO}^{T,b}$. In that contribution, we can perform the analytic
integration over the factorised singly unresolved radiative phase space ${\rm
d}\Phi_{X_{ijk}}$ of the antenna function proportional to the one-loop matrix
element, obtaining,

$\displaystyle{\rm d}\sigma_{NNLO}^{U,a}=\sum_{ik}{\cal
X}_{3}^{0}(s_{ik})\;|M_{m}^{1}(1,\ldots,m)|^{2}\;$ $\displaystyle=$
$\displaystyle\boldsymbol{J}_{m}^{(1)}(\epsilon;1,\ldots,m)\;|{\cal
M}_{m}^{1}(1,\ldots,m)|^{2}\;,$ (7.46)

where in the last equality we have defined an IR-singular operator
$\boldsymbol{J}_{m}^{(1)}$ containing a string of integrated three-parton
antennae which contain the IR-poles of the integrated real-radiation
contribution, in analogy with the $\boldsymbol{I}_{m}^{(1)}$-operator which
describes the IR-poles of the virtual matrix elements.

The second double virtual subtraction term is the integrated counterpart of
the contributions introduced at the real-virtual level ${\rm
d}\sigma_{NNLO}^{T,c}$ and double-real level ${\rm d}\sigma_{NNLO}^{S,d}$,
which combine and yield,

$\displaystyle{\rm d}\sigma_{NNLO}^{U,b}$ $\displaystyle=$
$\displaystyle\sum_{ik}\sum_{ml}{\cal X}_{3}^{0}(s_{ik})\otimes{\cal
X}_{3}^{0}(s_{ml})\;|{\cal M}_{m}^{0}(1,\ldots,m)|^{2}\;$ (7.47)
$\displaystyle=$
$\displaystyle\frac{1}{2}\boldsymbol{J}_{m}^{(1)}(\epsilon;1,\ldots,m)\otimes\boldsymbol{J}_{m}^{(1)}(\epsilon;1,\ldots,m)\;|{\cal
M}_{m}^{0}(1,\ldots,m)|^{2}\;,$

where we explicitly introduced the square of the
$\boldsymbol{J}_{m}^{(1)}$-operator introduced in (7.46).

Finally, the third double virtual subtraction term is the integrated
counterpart of the contributions introduced at the real-virtual level ${\rm
d}\sigma_{NNLO}^{T,b}$ including the terms proportional to the one-loop
$X_{3}^{1}$ antenna, and the contribution introduced at the double-real level
${\rm d}\sigma_{NNLO}^{S,b}$ involving the four-parton antenna $X_{4}^{0}$,

$\displaystyle{\rm d}\sigma_{NNLO}^{U,c}$ $\displaystyle=$
$\displaystyle\sum_{ik}\Big{(}{\cal X}_{4}^{0}(s_{ik})+{\cal
X}_{3}^{1}(s_{ik})-\frac{1}{2}{\cal X}_{3}^{0}(s_{ik}){\cal
X}_{3}^{0}(s_{ik})\Big{)}|{\cal M}_{m}^{0}(1,\ldots,m)|^{2}\;$ (7.48)
$\displaystyle=$
$\displaystyle\boldsymbol{J}_{m}^{(2)}(\epsilon;1,\ldots,m)\;|{\cal
M}_{m}^{0}(1,\ldots,m)|^{2}\;.$

In Eq. (7.48) we introduced an IR-singular operator containing the double
unresolved integrated antenna string $\boldsymbol{J}_{m}^{(2)}$.

The full double-virtual subtraction term is given as a sum of all subtraction
terms constructed above:

${\rm d}\sigma_{NNLO}^{U}={\rm d}\sigma_{NNLO}^{U,a}+{\rm
d}\sigma_{NNLO}^{U,b}+{\rm d}\sigma_{NNLO}^{U,c}\;,$ (7.49)

where in particular we can observe that ${\rm d}\sigma_{NNLO}^{U,a}$ and ${\rm
d}\sigma_{NNLO}^{U,b}$ are in one-to-one correspondence with the first two
lines in Eq. (7.45), while the contribution ${\rm d}\sigma_{NNLO}^{U,c}$
subtracts the remaining IR singularities of the two-loop amplitude in the
bottom two lines in (7.45).

#### Application: $N^{2}$ contribution to $q\bar{q}\to gg$ at NNLO

In this section we present the double-virtual antenna subtraction term for the
$N^{2}$ contribution to dijet production at hadron colliders at NNLO. Focusing
on the $q_{1}\bar{q}_{2}\to g_{3}g_{4}$ channel, the subtraction term reads,

$\displaystyle{\rm d}\sigma_{q\bar{q},NNLO}^{U}$
$\displaystyle=\sum\limits_{\begin{subarray}{c}P(i,j)\end{subarray}}$
$\displaystyle\bigg{\\{}\boldsymbol{J}_{4}^{(1)}(\hat{\bar{1}}_{q},i_{g},j_{g},\hat{\bar{2}}_{\bar{q}})\
\biggl{(}B_{4}^{1}(\hat{\bar{1}}_{q},i_{g},j_{g},\hat{\bar{2}}_{\bar{q}})-\frac{b_{0}}{\epsilon}\
B_{4}^{0}(\hat{\bar{1}}_{q},i_{g},j_{g},\hat{\bar{2}}_{\bar{q}})\biggr{)}$
(7.50) $\displaystyle+$ $\displaystyle\frac{1}{2}\
\boldsymbol{J}_{4}^{(1)}(\hat{\bar{1}}_{q},i_{g},j_{g},\hat{\bar{2}}_{\bar{q}})\otimes\boldsymbol{J}_{4}^{(1)}(\hat{\bar{1}}_{q},i_{g},j_{g},\hat{\bar{2}}_{\bar{q}})\
B_{4}^{0}(\hat{\bar{1}}_{q},i_{g},j_{g},\hat{\bar{2}}_{\bar{q}})$
$\displaystyle+$
$\displaystyle\boldsymbol{J}_{4}^{(2)}(\hat{\bar{1}}_{q},i_{g},j_{g},\hat{\bar{2}}_{\bar{q}})\
B_{4}^{0}(\hat{\bar{1}}_{q},i_{g},j_{g},\hat{\bar{2}}_{\bar{q}})\biggr{\\}}\
J_{2}^{(2)}(p_{i},p_{j})\;,$

where $B_{4}^{1}$ and $B_{4}^{0}$ are the leading colour renormalised colour-
ordered one-loop and tree level amplitudes for $q\bar{q}\to gg$ respectively.
As demonstrated in the previous section, the IR-operators
$\boldsymbol{J}_{4}^{(1)}$ and $\boldsymbol{J}_{4}^{(2)}$ are built with
integrated antenna strings that involve colour connected particles and that
match the poles of the virtual amplitudes. For processes with coloured
particles in the initial state these operators involve integrated antennae
with hard radiators in the initial state that subtract radiation off incoming
partons leading to initial-state collinear poles. These IR-singularities
cancel with the redefinition (mass factorisation) of the parton distributions.
In the example of this section we obtain,

$\boldsymbol{J}_{4}^{(1)}(\hat{\bar{1}}_{q},i_{g},j_{g},\hat{\bar{2}}_{\bar{q}})=\boldsymbol{J}_{2}^{(1)}(\hat{\bar{1}}_{q},i_{g})+\boldsymbol{J}_{2}^{(1)}(i_{g},j_{g})+\boldsymbol{J}_{2}^{(1)}(j_{g},\hat{\bar{2}}_{\bar{q}}),$
(7.51)

which when written in terms of integrated antennae and collinear splitting
functions read [125]:

$\displaystyle\boldsymbol{J}_{2}^{(1)}(\hat{\bar{1}}_{q},i_{g})$
$\displaystyle=$ $\displaystyle\frac{1}{2}{\cal
D}_{3,q}^{0}(s_{\bar{1}i})-\Gamma_{qq}^{(1)}(x_{1})\;,$ (7.52)
$\displaystyle\boldsymbol{J}_{2}^{(1)}(i_{g},j_{g})$ $\displaystyle=$
$\displaystyle\frac{1}{3}{\cal F}_{3}^{0}(s_{ik})\;,$ (7.53)
$\displaystyle\boldsymbol{J}_{2}^{(1)}(j_{g},\hat{\bar{2}}_{q})$
$\displaystyle=$
$\displaystyle\boldsymbol{J}_{2}^{(1)}(\hat{\bar{2}}_{q},j_{g})=\frac{1}{2}{\cal
D}_{3,q}^{0}(s_{\bar{2}j})-\Gamma_{qq}^{(1)}(x_{2})\;.$ (7.54)

The analogous formula for $\boldsymbol{J}_{4}^{(2)}$ is given by,

$\displaystyle\boldsymbol{J}_{4}^{(2)}(\hat{\bar{1}}_{q},i_{g},j_{g},\hat{\bar{2}}_{\bar{q}})$
$\displaystyle=$
$\displaystyle\boldsymbol{J}_{2}^{(2)}(\hat{\bar{1}}_{q},i_{g})+\boldsymbol{J}_{2}^{(2)}(i_{g},j_{g})+\boldsymbol{J}_{2}^{(2)}(j_{g},\hat{\bar{2}}_{q})-\overline{\boldsymbol{J}}_{2}^{(2)}(\hat{\bar{1}}_{q},\hat{\bar{2}}_{\bar{q}}),$
(7.55)

where the renormalised two-parton double unresolved integrated antenna strings
are given by [125]:

$\displaystyle\boldsymbol{J}_{2}^{(2)}(\hat{\bar{1}}_{q},i_{g})$
$\displaystyle=$
$\displaystyle\frac{1}{2}{\cal{D}}_{4,q}^{0}(s_{\bar{1}i})+\frac{1}{2}{\cal{D}}_{3,q}^{1}(s_{\bar{1}i})+\frac{b_{0}}{2\epsilon}\bigg{(}\frac{|s_{\bar{1}i}|}{\mu^{2}}\bigg{)}^{-\epsilon}{\cal{D}}_{3,q}^{0}(s_{\bar{1}i})$
(7.56) $\displaystyle-$
$\displaystyle\frac{1}{4}\big{[}{\cal{D}}_{3,q}^{0}(s_{\bar{1}i})\otimes{\cal{D}}_{3,q}^{0}(s_{\bar{1}i})\big{]}(z_{1})-\overline{\Gamma}_{qq}^{(2)}(z_{1}),$
$\displaystyle\boldsymbol{J}_{2}^{(2)}(i_{g},j_{g})$ $\displaystyle=$
$\displaystyle\frac{1}{4}{\cal{F}}_{4}^{0}(s_{ij})+\frac{1}{3}{\cal{F}}_{3}^{1}(s_{ij})+\frac{b_{0}}{3\epsilon}\bigg{(}\frac{s_{ij}}{\mu^{2}}\bigg{)}^{-\epsilon}{\cal{F}}_{3}^{0}(s_{ij})$
(7.57) $\displaystyle-$
$\displaystyle\frac{1}{9}\big{[}{\cal{F}}_{3}^{0}(s_{ij})\otimes{\cal{F}}_{3}^{0}(s_{ij})\big{]},$
$\displaystyle\boldsymbol{J}_{2}^{(2)}(i_{g},\hat{\bar{2}}_{\bar{q}})$
$\displaystyle=$
$\displaystyle\boldsymbol{J}_{2}^{(2)}(\hat{\bar{2}}_{q},i_{g}),$ (7.58)
$\displaystyle\overline{\boldsymbol{{J}}}_{2}^{(2)}(\hat{\bar{1}}_{q},\hat{\bar{2}}_{\bar{q}})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\tilde{\cal{A}}_{4,q\bar{q}}^{0}(s_{\bar{1}\bar{2}})+\tilde{\cal{A}}_{3,q\bar{q}}^{1}(s_{\bar{1}\bar{2}})-\frac{1}{2}\big{[}{\cal{A}}_{3,q\bar{q}}^{0}(s_{\bar{1}\bar{2}})\otimes{\cal{A}}_{3,q\bar{q}}^{0}(s_{\bar{1}\bar{2}})\big{]}.$
(7.59)

Analytic expressions for the integrated antennae and collinear splitting
functions introduced above can be found in [94, 95, 176, 178, 179, 181]. With
these expressions we can evaluate (7.50) and expand in powers of $\epsilon$ to
obtain:

$\displaystyle{\rm
d}\sigma_{q\bar{q},NNLO}^{U}=\sum\limits_{\begin{subarray}{c}P(i,j)\end{subarray}}\Bigg{[}-\frac{9}{2\epsilon^{4}}+\frac{33}{18\epsilon^{3}}+\frac{1}{\epsilon^{2}}\Bigg{(}-\frac{5}{24}+\frac{\pi^{2}}{8}+\frac{1}{2}\bigg{(}\log^{2}\left(\frac{s_{1i}}{\mu^{2}}\right)+\log^{2}\left(\frac{s_{ij}}{\mu^{2}}\right)$
$\displaystyle+\log^{2}\left(\frac{s_{2j}}{\mu^{2}}\bigg{)}\right)+\log\left(\frac{s_{1i}}{\mu^{2}}\right)\log\left(\frac{s_{ij}}{\mu^{2}}\right)+\log\left(\frac{s_{2j}}{\mu^{2}}\right)\log\left(\frac{s_{ij}}{\mu^{2}}\right)+\log\left(\frac{s_{1i}}{\mu^{2}}\right)\log\left(\frac{s_{2j}}{\mu^{2}}\right)$
$\displaystyle-\frac{37}{12}\bigg{(}\log\left(\frac{s_{1i}}{\mu^{2}}\right)+\log\left(\frac{s_{ij}}{\mu^{2}}\right)+\log\left(\frac{s_{2j}}{\mu^{2}}\right)\bigg{)}\Bigg{)}$
$\displaystyle+\frac{1}{\epsilon}\Bigg{(}-\frac{10201}{288}+\frac{9}{4}\zeta_{3}+\frac{7\pi^{2}}{144}-\log^{2}\left(\frac{s_{1i}}{\mu^{2}}\right)-\log^{2}\left(\frac{s_{ij}}{\mu^{2}}\right)-\log^{2}\left(\frac{s_{2j}}{\mu^{2}}\right)$
$\displaystyle-2\log\left(\frac{s_{1i}}{\mu^{2}}\right)\log\left(\frac{s_{ij}}{\mu^{2}}\right)-2\log\left(\frac{s_{2j}}{\mu^{2}}\right)\log\left(\frac{s_{ij}}{\mu^{2}}\right)-2\log\left(\frac{s_{1i}}{\mu^{2}}\right)\log\left(\frac{s_{2j}}{\mu^{2}}\right)$
$\displaystyle+\left(\frac{439}{36}-\frac{\pi^{2}}{12}\right)\left(\log\left(\frac{s_{1i}}{\mu^{2}}\right)+\log\left(\frac{s_{ij}}{\mu^{2}}\right)+\log\left(\frac{s_{2j}}{\mu^{2}}\right)\right)\Bigg{)}\Bigg{]}\;B_{4}^{0}(\hat{\bar{1}}_{q},i_{g},j_{g},\hat{\bar{2}}_{\bar{q}})$
$\displaystyle+\Bigg{[}-\frac{3}{\epsilon^{2}}+\frac{1}{\epsilon}\left(-\frac{31}{6}+\left(\log\left(\frac{s_{1i}}{\mu^{2}}\right)+\log\left(\frac{s_{ij}}{\mu^{2}}\right)+\log\left(\frac{s_{2j}}{\mu^{2}}\right)\right)\right)\Bigg{]}\;B_{4}^{1}(\hat{\bar{1}}_{q},i_{g},j_{g},\hat{\bar{2}}_{\bar{q}})\;$
$\displaystyle+{\cal O}(\epsilon^{0})$ (7.60)

As expected, the initial-state collinear singularities in the integrated
antennae with initial-state hard radiators cancelled against the PDF mass-
factorisation collinear subtraction included in the definitions of
$\boldsymbol{J}_{2}^{(1)}$ and $\boldsymbol{J}_{2}^{(2)}$, and all the
remaining singularities in (7.60), cancel explicitly and analytically with the
IR-poles of the two-loop amplitude for $q\bar{q}\to gg$ as guaranteed by the
KLN theorem.

###  7.3 Discussion

In this report we have reviewed the main aspects of the antenna subtraction
scheme for the subtraction of infrared singularities in the calculation of jet
observables at NNLO. We introduced subtraction terms for double real radiation
at tree level and single real radiation at one loop based on antenna
functions. These antenna functions at NLO and NNLO describe the colour-ordered
radiation of unresolved partons between a pair of hard (radiator) partons, and
can be derived from physical matrix elements [94].

We have shown how all singularities in intermediate steps of perturbative QCD
calculations can be mapped to Born-like configurations exploiting the
universal factorised structure of QCD amplitudes in the IR-limits. A key
ingredient are the phase space mappings that smoothly interpolate between the
various singular limits, and the factorisation of the real-radiation phase
space, which allows for the analytic integration of the antenna functions,
decoupling it from the integration over the Born configurations. All the
integrated counterterms that are necessary to have a fully general subtraction
method for massless final-state [94] and initial-state [95, 176, 178, 179,
181] QCD have been computed.

Phenomenological results for jet cross sections and transverse momentum
distributions at NNLO at hadron colliders have been recently obtained within
this approach. The results are obtained in the NNLOJET code framework [191]
which is a parton-level event generator that provides the framework for the
implementation of jet production processes to NNLO accuracy, using the antenna
subtraction method. It contains the event generator infrastructure (Monte
Carlo phase-space integration, event handling and analysis routines) and
provides the unintegrated and integrated antenna functions and the phase-space
mappings for all kinematical configurations.

Processes included in NNLOJET up to now are $Z$ and $Z+j$ production [196,
197, 198], $W$ and $W+j$ production [199, 200], $WH+j$ production [201], $H$
and $H+j$ production [202, 203], $H+2j$ (VBF) [204], di-jet production in
hadron-hadron collisions [205, 206, 207] and in lepton-hadron collisions [208,
209], isolated $\gamma$ and $\gamma+j$ production [210], di-photon production
[211] as well as three-jet production in electron-position annihilation [212].
More recently, flavour sensitive observables at NNLO have been studied in
$pp\to HV$ with $H\to b\bar{b}$ and $V\to ll$ [213], and for $Z+b$ production
[214].

##  8 Conclusions and Outlook

The purpose of this section is twofold. In the first part, we briefly remark
on the strengths ($+$) and weaknesses ($-$) of the formalisms presented in
this write-up. Whereas in the second part, we summarise the discussion in the
closing session of the workshop.

#### FDH and DRED

  * $+$

The evaluation of the Lorentz algebra is significantly simpler than in
conventional dimensional regularisation. For an NNLO computation in dred this
is particularly true for double-real contributions and for integrated
counterterms of subtraction methods since the $\mathcal{O}(\epsilon)$ terms of
the matrix elements are not required.

  * $+$

fdh is more amenable to methods that rely on strictly four-dimensional objects
like the spinor-helicity formalism and unitarity. Similar to completely four-
dimensional regularisation approaches, however, this is not true for the
treatment of $\gamma_{5}$.

  * $+$

As $\mathcal{O}(\epsilon)$ terms cannot contain any physical information, fdh
and dred might help to improve the conceptual understanding of regularisation
and of subtraction methods. Both schemes constitute the most promising
candidates to find links between dimensional regularisation and strictly four-
dimensional approaches like fdu, fdr, and ireg.

  * $-$

The evaluation of (master) integrals is not affected. Compared to cdr we still
need the same loop and phase-space integrals.

  * $-$

The UV renormalisation is slightly more complicated than in cdr. The
procedure, however, is standardised and well understood. For an NNLO
computation in fdh or dred, the evanescent renormalisation constants at most
have to be known at one-loop order.

#### FDR

  * $+$

Both UV and IR divergences are regularised strictly in four dimensions.

  * $+$

No UV counterterms need to be computed.

  * $+$

Lowest order parts of the calculation remain the same even when embedded in
higher order computations. For instance, no ${\cal O}(\epsilon)$ terms need to
be included at two loops.

  * $+$

Being four dimensional, fdr is suitable for a fully numerical treatment.

  * $-$

One cannot rely on existing libraries and/or reduction methods. In particular,
loop and phase-space integrals have to be computed.

  * $-$

At the moment, a local cancellation algorithm for final-state IR singularities
has been implemented at NLO only.

#### FDU

  * $+$

Direct connection between multi-loop and tree-level amplitudes within a common
phase-space by means of the loop-tree duality formalism. In other words, real
contributions are mapped onto virtual ones.

  * $+$

Local cancellation of IR and UV singularities at integrand level. Namely, no
need of ad-hoc and integrated counterterms. The local cancellation allows to
perform loop and phase-space integration in four dimensions.

  * $+$

Local UV renormalisation at one- and two-loop level has been tested with
proof-of-concept calculations.

  * $+$

The integrand representation of multi-loop Feynman integrals and scattering
amplitudes in the loop-tree duality is manifestly causal, i.e. it displays
only physical information. Absence of spurious singularities.

  * $-$

The subleading UV local counterterms that implement the renormalisation scheme
are still evaluated in $d$ dimensions.

  * $-$

Contrary to calculations at NLO, the treatment of IR singularities at NNLO is
not yet fully developed.

  * $-$

Processes including initial state radiation have not yet been studied within
fdu. This is because an integrand and local representation of the Altarelli-
Parisi kernels is currently missing.

  * $-$

fdu cannot profit from current techniques for the calculation of multi-loop
Feynman integrals, such as integration-by-parts identities and
differential/difference equation methods. This is because the latter modify
the local IR and UV behaviour.

#### IREG

  * $+$

Fully four-dimensional scheme for momentum integration and Clifford algebra.
Although implicit four dimensional schemes such as ireg share the same
problems with the $\gamma_{5}$ matrix a well-defined procedure can avoid
inconsistencies as shown in [7, 70].

  * $+$

The UV content of a given Feynman integral can be cast in terms of a well-
defined set of basic divergent integrals which do not need to be evaluated.
From the viewpoint of anomalies in perturbation theory it is a useful scheme.

  * $+$

Generalisation to $L$-loops is straightforward and compatible with local
subtraction theorems such as the BHPZ scheme and the Bogolyubov recursion
formula.

  * $-$

Although IR and UV divergences are clearly separated in a gauge invariant way,
and no extra fields are needed in the Lagrangian (such as epsilon-scalar
fields) compatibility with factorisation theorems are yet to be studied beyond
leading order.

  * $-$

Although a diagrammatic all order proof of gauge invariance in abelian models
can be constructed for ireg, a general all order proof for the non-abelian
case need to be constructed based on quantum action principles.

#### Local analytic Sector Subtraction

  * $+$

IR singularities are locally cancelled at NLO (validated for generic processes
with QCD partons in the final state) and NNLO (ongoing validation in the
general case).

  * $+$

Subtraction counterterms feature a minimal structure, thanks to the radiative
phase-space partition with sector functions (inspired by FKS).

  * $+$

An optimal phase-space parametrisation (via multiple CS mappings) enables the
analytic integration of the counterterms by means of standard techniques.

  * $+$

The scheme is valid for an arbitrary number of QCD partons in the final state.
It has been tested at NNLO with a proof-of-concept calculation of the
$T_{R}C_{F}$ contribution to $e^{+}e^{-}\rightarrow 2j$.

  * $-$

Several checks still need to be performed to test the cancellation of IR poles
in the general case at NNLO.

  * $-$

In order to stick to a minimal structure, a delicate tuning is needed for the
counterterm definition and for the corresponding phase-space mappings.

  * $-$

A complete implementation in a Monte Carlo code is still missing.

  * $-$

At NNLO, the scheme is currently designed only for FSR and for massless
partons. Preliminary investigation is ongoing for the extension to ISR and to
the massive case.

#### qt-subtraction

  * $+$

The method benefits from any existing calculation for “F+jet” production at
LO, NLO, NNLO, etc, to produce results for “F” production at one corresponding
higher order of the perturbative expansion: NLO, NNLO, N3LO, etc. In fact, the
singular behaviour of “F+jet” as $q_{\scriptscriptstyle\rm T}\to 0$ is well-
known from the resummation program of logarithmically-enhanced contributions
to $q_{\scriptscriptstyle\rm T}$ distributions.

  * $+$

The universality of the logarithmic contributions to the $q_{T}$ distributions
allows to construct counterterms which require minimal information about the
process, such as the Born subprocess and the finite remainder of the multi-
loop scattering amplitudes (at any corresponding order in the coupling).

  * $+$

$q_{T}$-subtraction is fully developed for production processes of colourless
particles and massive quarks. The general structure is the same for any number
of colourless particles in the final state, any number of massive quarks in
the final state, or any combination between colourless particles and massive
quarks.

  * $-$

The subtraction is non-local and the control on large cancellations is an
issue. A small resolution variable, i.e. a cutoff, leads to numerical
difficulties and slower integrations. Conversely, a greater value for the
cutoff enhances the contributions from power corrections in the resolution
variable, which are actually neglected in the general formulation of the
subtraction.

  * $-$

Real scattering amplitudes and counterterms are integrated in different phase
spaces, with the counterterm always evaluated in the corresponding Born phase
space.

  * $-$

$q_{T}$-subtraction is not fully developed for production processes of
detected massless coloured particles and/or jets.

#### Antenna subtraction

  * $+$

Local subtraction scheme with phase-space averaging. Good control on the
numerical accuracy of the final result with double-real, real-virtual and
double-virtual contributions separately finite.

  * $+$

No need to introduce phase-space slicing parameters in the calculation.

  * $+$

IR singularities are cancelled analytically, i.e., the explicit
$\epsilon$-poles in the dimension regularisation parameter of one- and two-
loop matrix elements are cancelled in analytic and local form against the
$\epsilon$-poles of the integrated antenna subtraction terms. Good control on
the correctness of the pole cancellation.

  * $+$

Fully general subtraction scheme at NNLO for massless final-state and initial
state QCD for any jet multiplicity.

  * $-$

The antenna functions are scalar objects and do not subtract angular
correlations in gluon-splittings, which vanish when the azimuthal variable of
the collinear system (with respect to the collinear axis, defined by the
collinear momentum and a light-like recoil momentum) is integrated out.
Cancellation is accomplished with the method locally, by combining phase space
points correlated by a $\pi/2$ rotation around the collinear. axis

  * $-$

Involves many re-mappings and subtraction terms as expected for a local
method. Can be improved by introducing a caching system to store the
evaluation of the phase space mappings applied to the real contributions.

### Closing discussion

The closing event of the workshop was a discussion session in which all the
attendants participated and shared opinions. Here, we present a summary of the
main topics and questions that were mentioned in that session, preserving the
original ordering in which the discussion took place.

As stated in the general introduction, the purpose of the present workshop was
to deepen into technical aspects of modern high-precision computations for
QFTs. This implies covering several topics, several techniques and subtleties,
that might hide conceptual and/or computational issues. These issues range
from subtle definitions (e.g., _what does locality mean?_) to deeper
conceptual problems (e.g., _does it make sense going till N kLO in
perturbation theory without having quantitative control on the size of various
corrections of non-perturbative origin?_). Since the time was limited, the
discussion focused on three points: theoretical errors, factorisation breaking
and $\gamma^{5}$ issues.

#### About theoretical errors

It is a fact that experiments are reaching an impressive level of accuracy,
due to the increase in the data collection and its treatment. So, one question
is: _how can theory keep the pace and produce predictions within the required
precision? How can we control the theoretical errors in a reliable way?_

Nowadays there is a shared opinion within the HEP community that more legs and
more loops will lead to an error reduction, but there are not yet established
precise procedures to properly quantify the error estimation (see, e.g., Refs.
[215, 216] and related references therein).

Moreover, regarding theoretical errors in collider observables, we need to
include the non-negligible impact of PDFs. Essentially, the predictions are
being affected by perturbative and non-perturbative contributions, and both of
them are potential sources of errors that need to be kept under control. Thus,
another important question is: _how does the theoretical framework affect our
skills to extract predictions from QFTs?_ PDF extractions relies not only on
highly-accurate experimental data but also on highly-precise partonic
predictions (coming from multi-loop multi-leg computations). A non-trivial
interplay between perturbative uncertainties at partonic level and ensuing PDF
errors is always present. So, it is not possible to claim a certain accuracy
if there is not a rigorous control on the errors present in all the
ingredients of the calculation (see. e.g., Refs. [217, 218] and related
references therein).

#### Factorization breaking

Most of the time, we focus our methods on trying to compute NLO, NNLO and even
higher-order corrections to hard-scattering processes at hadron colliders. We
rely on the validity of the factorisation theorem, but only a few times we are
(fully) aware of the potential limitations.

Our ability to compute predictions for high-energy colliders strongly relies
on the parton model and factorisation formulae, which isolate the dominant
(i.e., ‘leading twist’) non-perturbative nature of the colliding proton inside
process-independent PDFs. If collinear factorisation is spoiled at some
perturbative orders, then PDFs would carry an implicit dependence on the
process. Correlations among initial- and final-state partons will survive, and
this will break the possibility of using the factorisation theorem.

We recall that a general (process and observable independent) proof of the
factorisation theorem to all perturbative orders is still lacking. In Ref.
[219], the violation of strict collinear factorisation at the scattering
amplitude level was pointed out. In the collinear limit, the scattering
amplitude ${\cal M}$ factorises according to

$\displaystyle{\cal M}_{n}(p_{1},\ldots,p_{l},p_{l+1},\ldots,p_{n})\to{\rm
Sp}(p_{1},\ldots,p_{l},\tilde{P})\times{\cal
M}_{n-l+1}(\tilde{P},p_{l+1},\ldots,p_{n})\,,$ (8.1)

where $\\{p_{1},\ldots,p_{l}\\}$ ($\\{p_{l+1},\ldots,p_{n}\\}$) are the
collinear (non-collinear) momenta and $\tilde{P}$ denotes the light-like
vector carrying the total momenta of the collinear partons. The factor ${\rm
Sp}$ embodies the contributions that are singular in the collinear region. In
a naive picture, the singular factor ${\rm Sp}$ is expected to be universal
(process-independent), namely it can only depend on the momenta and flavours
of the collinear partons. This picture is valid at the tree level and, more
generally, in the time-like region, but, including loop corrections, it was
proven [219] that color and momentum correlations among collinear and non-
collinear partons are present in ${\rm Sp}$ in the space-like region, i.e., in
the case of collinear emission from initial-state colliding partons. The
contributions that break strict collinear factorisation originate from
absorptive interactions that takes place in the far past (long before the
occurrence of the hard scattering) between the initial-state colliding
partons. Owing to their absorptive origin, these contributions are imaginary
at the lowest perturbative order and, therefore, they cancel at the squared
amplitude level up to NNLO (and also N3LO in pure QCD processes [219, 220]).

The occurrence of such cancellation mechanism of factorization breaking terms
is not guaranteed at high perturbative orders. Further studies and
investigations [219, 220, 221, 222, 223] of these factorization breaking terms
(and of their precise structure) are certainly relevant in view of the
conceptual and computational importance of the factorisation theorem
(‘assumption’) for extracting predictions within QFT. Even if collinear-
factorisation breaking contributions eventually do not spoil the validity of
the factorisation theorem, their presence definitely introduce technical
complications in the cancellation mechanism of IR divergences for multileg
hard-scattering observables in hadron collisions. Such complications have to
be overcome to extend IR-subtraction methods beyond the NNLO.

####  $\gamma^{5}$ problems

One of the main topics of this workshop regarded the subtleties that appear
when extending a theory to $d$ dimensions, whatever $d$ means (as in the
context of dreg). However, there are problems that also arise when $d=4$: this
is the case of $\gamma^{5}$. Of course, in the standard four-dimensional
space-time there is a well established recipe to mathematically define
$\gamma^{5}$. Moreover, in any even-dimensional Minkowskian manifold analogous
objects to $\gamma^{5}$ are properly defined.

Regularisation involving $\gamma_{5}$ is problematic. In dimensional schemes
the problems are well-known (see e.g. the review [224]), and recent references
have focused on comparing different $\gamma_{5}$-prescriptions up to the two-
loop level [6] and on determining gauge invariance-restoring counterterms for
the Breitenlohner/Maison/’t Hooft/Veltman prescription of $\gamma_{5}$ [225].
Quite surprisingly, non-dimensional schemes are not exempted of issues in the
presence of $\gamma_{5}$ [70, 71]. The reason boils down to requiring very
basic properties such as shift invariance and numerator-denominator
consistency to be respected, showing that virtually any regularization scheme
will need to deal with $\gamma_{5}$-problems [7].

Therefore, consistent definition of $\gamma_{5}$, together with full
understanding of its properties with respect to symmetries, gauge invariance
and anomaly cancellation, is crucial for higher-order calculations. This is
especially important in the context of high-precision predictions taking into
account electroweak corrections.

#### Further open questions

After the exciting discussion session, many issues and questions remained
opened. In particular, we would like to highlight:

  * •

_How to re-define a QFT in such a way that no distinction among real and
virtual corrections is done?_

  * •

Even if we manage to combine the real and virtual contributions from the very
beginning, still threshold singularities might survive. _How to tackle them
and develop efficient techniques to integrate through thresholds?_

Recent studies at NNLO point to computational frameworks in which IR, UV and
threshold singularities are treated in purely four dimensions. In fact, NLO
calculations in a four-dimensional framework started to be carried out long
time ago by Soper [226] and subsequent related works. More recently, studies
that aim at achieving a complete cancellation of singularities at the
integrand level were presented in Refs. [27, 28, 29, 38] at NLO, and
preliminary results that involve two-loop scattering amplitudes were presented
in Ref. [37]. The studies in Ref. [227], based on the knowledge of the
infrared structure of scattering amplitudes [228, 229], point towards the same
direction. Furthermore, novel techniques for the evaluation of multi-loop
Feynman integrals, inspired by the loop-tree duality approach [34, 32], have
shown to display a causal representation depending only on physical
singularities (see Refs. [46, 39, 40, 47, 42, 41, 45] and related references
therein).

Alternatives approaches based on analytic and semi-numerical techniques for
NNLO calculations are summarised in the recent review [230].

## Acknowledgments

In this work, we summarise all discussions originated as a result of the
WorkStop/ThinkStart 3.0: paving the way to alternative NNLO strategies that
took place on 4.–6. November 2019 at the Galileo Galilei Institute for
Theoretical Physics (GGI). We gratefully acknowledge the support of GGI and
the COST Action CA16201 PARTICLEFACE. We wish to thank to W.M. Marroquín and
M. Morandini for their help in organising the workshop.

P. Banerjee acknowledges support by the European Union’s Horizon 2020 research
and innovation programme under the Marie Skłodowska-Curie grant agreement No
701647.

A.L. Cherchiglia, B. Hiller and M.Sampaio acknowledge support from Fundação
para a Ciência e Tecnologia (FCT) through the projects UID/FIS/04564/2020 and
CERN/FIS-COM/0035/2019.

The work of L. Cieri has received funding from the European Union’s Horizon
2020 research and innovation programme under the Marie Sklodowska-Curie grant
agreement No 754496.

The work of F. Driencourt-Mangin, G. Rodrigo, G. Sborlini and W.J. Torres
Bobadilla is supported by the Spanish Government (Agencia Estatal de
Investigación), ERDF funds from European Commission (Grant No.
FPA2017-84445-P), Generalitat Valenciana (Grant No. PROMETEO/2017/053) and
from the Spanish Government (FJCI-2017-32128).

T. Engel acknowledges support by the Swiss National Science Foundation (SNF)
under contract 200021_178967.

C. Gnendiger, R. Pittau, A. Signer and D. Stöckinger wish to thank B. Page for
his help in establishing (2.60).

The work of R.J. Hernández-Pinto is supported by CONACyT through the Project
No. A1-S-33202 (Ciencia Basica) and Sistema Nacional de Investigadores.

G.Pelliccioli. was supported by the Bundesministerium für Bildung und
Forschung (BMBF, German Federal Ministry for Education and Research) under
contract no. 05H18WWCA1.

J. Pires was supported by Fundação para a Ciência e Tecnologia (FCT, Portugal)
through the contract UIDP/50007/2020 and project CERN/FIS-PAR/0024/2019.

The work of R. Pittau has been supported by the Spanish Government grant
PID2019-106087GB-C21 and by the Junta de Andalucía project P18-FR-4314 (fondos
FEDER).

M. Sampaio acknowledges a research grant from CNPq (Conselho Nacional de
Desenvolvimento Científico e Tecnológico - 303482/2017-6).

C. Signorile-Signorile was supported by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) under grant no. 396021762 - TRR 257.

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