e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>e3e-mail<EMAIL_ADDRESS>e4e-mail: daniel.reichelt@uni-
goettingen.de

11institutetext: Dipartimento di Fisica, Università di Genova and INFN,
Sezione di Genova, Via Dodecaneso 33, 16146, Italy 22institutetext: Institut
für Theoretische Physik, Georg-August-Universität Göttingen, Friedrich-Hund-
Platz 1, 37077 Göttingen, Germany

# Tagging the initial-state gluon

Simone Calettie1,addr1  Oleh Fedkevyche2,addr1  Simone Marzanie3,addr1  Daniel
Reichelte4,addr2

###### Abstract

We study the production of an electroweak boson in association with jets, in
processes where the jet with the highest transverse momentum is identified as
quark-initiated. The quark/gluon tagging procedure is realised by a cut on a
jet angularity and it is therefore theoretically well-defined and exhibits
infrared and collinear safety. In this context, exploiting resummed
perturbation theory, we are able to provide theoretical predictions for
transverse momentum distributions at a well-defined and, in principle,
systematically improvable accuracy. In particular, tagging the leading jet as
quark-initiated allows us to enhance the initial-state gluon contribution.
Thus these novel transverse momentum distributions are potentially interesting
observables to probe the gluonic degrees of freedom of the colliding protons.

††journal: Eur. Phys. J. C

## Introduction

Over the past decade our understanding of the internal structure of jets has
increased tremendously. Thanks to the applications of the methods of
perturbative QCD, the field has become mature and jet substructure algorithms
that are, at the same time, performant and robust have been developed.
Observables originally designed for searching new physics are now the target
of unfolded measurements that can be compared to theoretical predictions, at
the precision level, see e.g. (Aad:2020zbq, ; Aad:2019vyi, ; Aad:2019onw, ;
Aaboud:2017qwh, ; Aaboud:2019aii, ; Sirunyan:2018xdh, ; Sirunyan:2018gct, ;
CMS:2021vsp, ). Furthermore, ideas developed by the jet substructure community
have found applications in other contexts of particle physics. For instance,
after it was realised that the momentum fraction that characterises the
splitting identified by the SoftDrop algorithm (Larkoski:2014wba, ) follows a
distribution dictated by the QCD splitting functions (Larkoski:2015lea, ;
Larkoski:2017bvj, ; Tripathee:2017ybi, ; Cal:2021fla, ), this observable has
become a standard way of probing interactions with the quark/gluon plasma, see
e.g. (Chen:2021osv, ) and references therein.

This work is part of an incipient effort to find innovative ways of applying
jet substructure techniques to the broader LHC physics program and to provide
the necessary tools so that this cross-pollination can bear its fruits. In
this context, measurements of the internal structure of highly energetic jets
produced in association with a boosted electroweak boson offer an ideal
playground for such studies, because the leptonic decay of the $Z$ boson
offers a valuable trigger. These events are not only relevant for background
studies 111This process represents the main background for the associated
production of a Higgs and an electroweak boson in the boosted regime, where
the decay products of the Higgs are reconstructed in one jet. but, as we shall
detail in the following, they open up novel possibilities to employ jet
substructure for Standard Model measurements.

Substructure observables, such as jet angularities Larkoski:2014pca , measure
the pattern of the hadronic activity within a jet. For this reason, they are
often employed as tagging variables that aim to distinguish jets that have
been originated by elementary particles carrying different colour degrees of
freedom, e.g. colour singlets versus QCD partons or, even, quarks versus
gluons. Although there exist more powerful quark/gluon (_q/g_) discriminants
than a cut on a jet angularity, this procedure is theoretically well-defined,
infra-red and collinear (IRC) safe and, hence, its behaviour can be understood
with perturbative methods.

In the following, we study $Z$+jet production, requiring that the jet with the
highest transverse momentum has been identified as quark-initiated. As
preliminarily explored in (Amoroso:2020lgh, ), by tagging the final-state, we
indirectly bias particular partonic sub-processes. If the leading jet is
tagged as quark-initiated, then, at leading order, the subprocess that
features a quark and gluon in the initial state is enhanced. Thus, this
procedure could potentially provide us with a new handle on the determination
of the gluon parton distribution function (PDF). The main result of this study
is the calculation of the transverse momentum distribution of the $Z$ boson in
events where the leading jet is identified as quark-initiated. Because of IRC
safety of the tagging procedure, we are able to compute this distribution at a
well-defined and, in principle improvable, accuracy. Therefore, this
observable could be directly included in standard fits of PDFs. The results
presented here account for the resummation of the tagging parameter at the
_next-to-leading logarithmic_ (NLL) accuracy, matched to fixed order
predictions at $O\left(\alpha_{s}^{2}\right)$ with respect to the Born
contribution, henceforth denoted as next-to-leading order (NLO) 222The fixed-
order counting is slightly different than what is usually employed in standard
transverse-momentum distribution. This has to do with the fact that, in order
to obtain a non-vanishing value of any angularity, the jet must have at least
two constituents. Thus, in what follows we will refer to the lowest-order
($2\to 2$) scattering as Born approximation, while (N)LO will be reserved for
the contributions with one (two) additional emission(s)..

Figure 1: Receiver Operating Characteristic (ROC) curves for different values
of the angularity exponent $\alpha=0.5,1,2$, from left to right obtained with
MC simulation using P

YTHIA

. In each plot, different curves refer to ungroomed jets and SoftDrop jets
with $z_{\text{cut}}=0.1$ and $\beta=0,1$. For comparison we also show the ROC
curves corresponding to a random classifier (dashed grey) and the CS ones
(dashed black). Straight lines represent the target performance, as detailed
in the text. At the bottom of each plot we show the ratio between each ROC
curve obtained at hadron-level and the corresponding parton-level result. The
dashed portion of these lines indicate the region sensitive to splittings with
relative transverse momentum below 1 GeV, where NP effects are expected to be
sizeable. This region of phase-space is sensitive to shower cut-off effects,
which cause the observed kinks.

## Enhancing the gluon contribution

We start by considering the channel fractions $f_{ij}$ that measure, to a
given order in perturbation theory, the contribution to the $Z$ (or jet)
transverse momentum distributions from the subprocess initiated by partons $i$
and $j$, which, for brevity, we indicate as $\sigma_{ij}^{a}$, where $a$ could
be the $Z$ boson or the leading jet $J$:

$f_{ij}^{a}=\frac{\sigma_{ij}^{a}}{\sigma^{a}_{qq}+\sigma^{a}_{qg}+\sigma_{gg}^{a}},\quad
a=Z,J,$ (1)

where $qq$ and $qg$ include any combination of quarks and anti-quarks. Our aim
is to study how tagging a quark-jet in the final-state changes the relative
contributions of the various partonic subprocesses. To this purpose, we define

$\widetilde{f}_{ij}^{a}=\frac{\widetilde{\sigma}_{ij}^{a}}{\widetilde{\sigma}^{a}_{qq}+\widetilde{\sigma}^{a}_{qg}+\widetilde{\sigma}_{gg}^{a}},\quad
a=Z,J,$ (2)

where the tilde indicates “after tagging”. Furthermore, in what follows we
will be interested in the fractions of the event (before and after tagging)
which feature at least one gluon in the initial state. To this purpose, we
define

$f_{g}^{a}=f_{qg}^{a}+f_{gg}^{a},\quad\text{and}\quad\widetilde{f}_{g}^{a}=\widetilde{f}_{qg}^{a}+\widetilde{f}_{gg}^{a}.$
(3)

We will refer to this fraction as _gluon channel purity_.

We begin our discussion in a simplified setting, which is nevertheless enough
to capture the essential physics points. We will then validate our conclusions
using Monte Carlo (MC) parton shower simulations. If we consider the Born
approximation, then we have no $gg$ contribution and, because the $Z$ boson
and the jet are back-to-back, we obtain $f_{ij}^{Z}=f_{ij}^{J}$. Fragmentation
of the hard partons can lead to transverse momentum imbalance, however, we can
limit ourselves to the leading logarithmic (LL) regime, where all parton
splittings happen in the soft and collinear limit, with no recoil. 333We will
lift this approximation when dealing with actual simulations. However, we can
always choose a set of kinematical cuts that favours the back-to-back
configuration. Therefore, dropping the superscript $a$, we have

$f_{g}=\frac{\sigma_{qg}}{\sigma_{qq}+\sigma_{qg}}.$ (4)

We note that, within our approximation, the fraction of events with a
(properly defined) final-state quark can be considered as a proxy for the
gluon channel purity. We find that, for $p_{t\,J}\geq 100$ GeV, $f_{g}\simeq
0.85$ and it exhibits a rather mild dependence on the transverse momentum.
Next, we note that in our approximation we simply have

$\displaystyle\widetilde{f}_{g}$
$\displaystyle=\frac{\varepsilon_{q}\sigma_{qg}}{\varepsilon_{g}\sigma_{qq}+\varepsilon_{q}\sigma_{qg}}=\left(1+\frac{1-f_{g}}{f_{g}}\frac{\varepsilon_{g}}{\varepsilon_{q}}\right)^{-1},$
(5)

where $\varepsilon_{q}$ is the efficiency of the tagger to correctly label
quark jets and $\varepsilon_{g}$ the false-positive rate, which measures how
often gluon jets are wrongly labelled as quarks. We note that a perfect quark-
tagger with $\varepsilon_{q}=1$ and $\varepsilon_{g}=0$ returns tagged events
that have been originated by one gluon in the initial state, i.e.
$\widetilde{f}_{g}=1$. On the other hand, with an efficiency of 50%, which
corresponds to tossing a coin, we recover Eq. (4).

A rather common class of _q/g_ taggers exhibits at LL a property known as
Casimir scaling (CS) (Larkoski:2014pca, ), namely the tagging efficiencies are
related by $\varepsilon_{g}=\left(\varepsilon_{q}\right)^{C_{A}/C_{F}},$ where
the exponent is given by the ratio of the Casimir operators in the appropriate
colour representation, $C_{F}$ for quarks and $C_{A}$ for gluons. This
property emerges because the LL distributions of both quarks and gluons
exhibit the same Sudakov-like behaviour, with a coefficient determined by the
appropriate colour factor. Thus, for a CS tagger, we find

$\widetilde{f}_{g}^{\text{CS}}=\left(1+\frac{1-f_{g}}{f_{g}}{\varepsilon_{q}}^{\frac{C_{A}}{C_{F}}-1}\right)^{-1}.$
(6)

For instance, a CS tagger with $\varepsilon_{q}=0.65$, yields
$\widetilde{f}_{g}^{a}\simeq 0.9$. This can be pushed to 0.95 if the tighter
working point $\varepsilon_{q}=0.35$ is considered. Taggers that obey CS are
not the most performant, but they are interesting for us because they are
under good theoretical control. The efficiencies $\varepsilon_{i}$ can be
computed in QCD using resummed perturbation theory to a well-defined and, in
principle improvable, theoretical accuracy. Furthermore, the inclusion of
higher logarithmic corrections generally leads to an improvement with respect
to strict CS, so that, depending on the specifics of the tagging procedure,
the target gluon channel purity $\widetilde{f}_{g}^{a}\simeq 0.95$ can be
achieved at a reasonable working point. In the following we construct a CS
tagger that is based on a particular class of jet substructure observables
known as jet angularities.

## Jet angularities as a quark/gluon tagger

Jet angularities (Larkoski:2014pca, ) are defined as

$\lambda_{\alpha}=\sum_{i\in\text{jet}}\frac{p_{t,i}}{p_{t\,J}}\left(\frac{\Delta_{i}}{R_{0}}\right)^{\alpha},$
(7)

where the sum runs over the constituents of the hardest jet in the event and
$\Delta_{i}=\sqrt{(y_{i}-y_{J})^{2}+(\phi_{i}-\phi_{J})^{2}}$ is the distance
in the azimuth-rapidity plane of particle $i$ from the jet axis. We define
jets with the anti-$k_{t}$ clustering algorithm (Cacciari:2008gp, ) with
radius $R_{0}$ and standard $E$-scheme for recombination. IRC safety requires
$\alpha>0$, while angularities with $\alpha\leq 1$ are sensitive to recoil
against soft emissions (Larkoski:2013eya, ). In order to circumvent this
issue, when $\alpha\leq 1$, the jet axis is obtained using the Winner-Take-All
(WTA) recombination scheme (Larkoski:2014uqa, ). We also consider groomed
jets. In this case, we recluster the jet with the Cambridge-Aachen algorithm
(Dokshitzer:1997in, ; Wobisch:1998wt, ) and apply the SoftDrop grooming
algorithm with parameters $z_{\text{cut}}$ and $\beta$ (Larkoski:2014wba, ).
The angularity is then computed on the constituents of the groomed jet, with
the WTA prescription adopted for angularities with $\alpha\leq 1$.

Because of the different colour factor, angularity distributions for quark-
and gluon-initiated jets peak at different values. We can exploit this
separation and define our _q/g_ tagger through a cut on the jet angularity. In
particular, a jet with $\lambda_{\alpha}<\lambda_{\text{cut}}$ will be
labelled as a quark jet:

$\widetilde{\sigma}_{ij}^{a}=\int_{0}^{\lambda_{\text{cut}}}d\lambda_{\alpha}\frac{d{\sigma}_{ij}^{a}}{d\lambda_{\alpha}},$
(8)

where we have introduced the differential angularity distribution. The
implicit dependence of the tagged distribution on the transverse momentum, on
the angularity exponent $\alpha$ and, optionally, on the SoftDrop parameters
$z_{\text{cut}}$ and $\beta$, is understood.

From the simple CS analysis above, we have concluded that we should work with
quark efficiencies $\varepsilon_{q}\simeq 0.35$, in order to reach a purity of
initial-state gluons around 0.95. The value of the angularity cut that is
necessary to achieve this working point for the tagger clearly depends on the
angular exponent $\alpha$ in Eq. (7) as well as on the parameters
$z_{\text{cut}}$ and $\beta$ of the SoftDrop algorithm, should we wish to
employ groomed jets. Different theoretical considerations can guide us with
this choice. First of all, we would like to preserve calculability, i.e. we
want to cement our findings in perturbative field theory. Thus, we would like
our tagger to be as insensitive as possible to non-perturbative (NP)
contributions such as hadronisation corrections and the Underlying Event (UE).
Secondly, although we can calculate transverse momentum spectra in resummed
perturbation theory (Kang:2018qra, ; Kang:2018vgn, ; Caletti:2021oor, ), we
aim for perturbative stability. Thus, we favour working points for the tagger
for which $\lambda_{\text{cut}}$ is not too small.

Figure 2: The initial-state gluon purity before ($f_{g}$) and after tagging
($\widetilde{f}_{g}$) obtained with P

YTHIA

simulations, as a function of the transverse momentum of the $Z$ boson.

In order to turn the above considerations into a quantitative study we use
simulated data obtained with the MC event generator P  YTHIA 8.303
(Sjostrand:2014zea, ). The UE is simulated according to the model presented in
(Sjostrand:1985vv, ; Sjostrand:1987su, ; Sjostrand:2004pf, ) and hadronisation
effects according to the Lund string model (Andersson:1983ia, ;
Sjostrand:1984ic, ). Throughout the paper, we use the NNPDF 3.0 NLO set of
PDFs (NNPDF:2014otw, ). We consider the inclusive production of a pair of
oppositely charged muons in proton–proton collisions at $13~{}\text{TeV}$
centre-of-mass energy, requiring that the invariant mass of the muon pair to
be within 70 and 110 GeV. Jets are clustered with the anti-$k_{t}$ algorithm
with $R_{0}=0.4$ and ordered in transverse momentum. Henceforth, the jet will
be implicitly considered to be the hardest one and we will refer to the muon-
antimuon pair as the $Z$ boson. With this in mind, the fiducial volume is
defined following Ref. (CMS:2021vsp, ): $p_{t\,\mu}>26~{}\mathrm{GeV}$,
$p_{t\,Z}>30\;\text{GeV}$, $p_{t\,J}>15~{}\mathrm{GeV}$, $|\eta_{\mu}|<2.4$,
and $|y_{\text{jet}}|<1.7$. Furthermore, in order to enforce back-to-back
configurations, we impose
$\left|\frac{p_{t\,J}-p_{t\,Z}}{p_{t\,J}+p_{t\,Z}}\right|<0.3$,
$\left|\phi_{\rm jet}-\phi_{Z}\right|>2.$

For event selection and analysis we employ R  IVET (Buckley:2010ar, ;
Bierlich:2019rhm, ). Jet reconstruction is done with F  AST J  ET
(Cacciari:2011ma, ), and the SoftDrop implementation in the F  AST J  ET
`contrib` is used. We distinguish two different stages of the simulation:
“parton-level”, i.e. with parton shower effects only, and “hadron-level”, i.e.
with hadronisation and UE included. By keeping the two partonic processes of
interest separate, we compute Receiver Operating Characteristic (ROC) curves
that show the mis-tag rate (gluon efficiency) $\varepsilon_{g}$ as a function
of the signal (quark) efficiency. They are computed for different values of
the angularity exponent $\alpha=0.5,1,2$ in the ungroomed case and for
SoftDrop jets with $z_{\text{cut}}=0.1$ and $\beta=0,1$. We show hadron-level
results in Fig. 1, as well as the ratios to their parton-level counterparts,
which we take as a measure of NP contributions. The dotted portion of the
latter indicates that the efficiency $\varepsilon_{q}$ is dominated by
splittings in the non-perturbative region, as determined, for instance, in
Ref. Caletti:2021oor . We also show the target line, which fixes the gluon
efficiency $\varepsilon_{g}$ as a function of $\varepsilon_{q}$, for given
$f_{g}$ and $\widetilde{f}_{g}$,

$\varepsilon_{g}=\frac{f_{g}(1-\widetilde{f}_{g})}{\widetilde{f}_{g}(1-f_{g})}\varepsilon_{q},$
(9)

which is easily derived from Eq. (5). The slope shown in Fig. 1 is determined
by the original $f_{g}=0.85$ and target gluon purity $\widetilde{f}_{g}=0.95$.
The intersections of each ROC curve with the target line set our tagger
working points. Corresponding values of $\lambda_{\text{cut}}$ are reported in
the figure. The choice of the tagger working points is clearly not unique. For
instance, we could have optimised the signal-to-background ratio by choosing
on each ROC curve the point that is closest to the $(1,0)$ corner.

The analysis of Ref. (Caletti:2021oor, ) tells us that larger values of
$\alpha$ are under better theoretical control. However, as it is clear from
Fig. 1, lower values of $\alpha$ have better performance, essentially because
of their increased sensitivity to the collinear region. Thus, the choice
$\alpha=1$ appears to be a good compromise between performance and robustness.
We note that the use of SoftDrop does not always provide us with improvements
on the size of the NP contributions. This might be related to the fact that in
order to obtain the same efficiency $\varepsilon_{q}$ we need to cut groomed
jets at lower values of $\lambda_{\text{cut}}$, where NP physics may be more
prominent. Furthermore, we note that we are working with a rather small jet
radius, which prevents large contributions from the UE. We expect (light)
grooming to be beneficial, should one consider larger jet radii. Finally,
configurations with aggressive SoftDrop, i.e. $\beta=0$, typically result in
worse performance, because important information is groomed away. Thus, in
what follows, we shall focus on the $\alpha=1$ case either with no grooming or
with $\beta=1$, $z_{\text{cut}}=0.1$.

We now quantify the gain in the initial-state gluon purity that we obtain
after tagging. Fig. 2 shows the fractions $\tilde{f}_{g}^{Z}$ as a function of
the $Z$ boson transverse momentum for the selected taggers, with $f_{g}^{Z}$
also shown for comparison. We first note that the performance of both taggers
is very good, leading to gluon channel purities that exceed our 95% target.
Analogous conclusions can also be drawn if we plot our results as a function
of $p_{t\,J}$. However, as we will argue shortly, in this context, the
transverse momentum distribution of the $Z$ boson is a more robust observable.
We also notice that the gluon channel purities decrease with $p_{t\,Z}$. This
is due to the fact that our taggers are defined looking at their efficiencies
with $p_{t\,J}>100$ GeV and, therefore, we expect them to work better at the
lower end of the transverse momentum spectrum. This loss in performance could
be cured by adjusting the cut on the angularity as a function of $p_{t\,J}$.
However, in this first study, we prefer to keep our framework simple.

Figure 3: Transverse momentum distribution of the $Z$ boson in $Z$+jet events,
with the leading jet tagged as quark-initiated, according to our operational
definition, detailed in the text. The untagged distribution is also shown for
comparison. The NLO+NLL′ calculation is supplemented with a NP correction
factor, shown at the bottom.

## Transverse momentum distributions

We now provide theoretical predictions for our observables of interest, namely
transverse-momentum distributions, in the presence of tagging. Our calculation
includes the resummation of logarithms of $\lambda_{\text{cut}}$ at NLL
accuracy and is matched to NLO. Thanks to a flavour-dependent matching
procedure, _cf._ also Banfi:2006hf ; Baron:2020xoi , we are able to achieve
NLO+NLL′ accuracy. We also include a bin-by-bin NP correction factor obtained
with MC simulations. 444See Ref. Caletti:2021oor  for details about the
calculation and its numerical implementation in the resummation plugin
Gerwick:2014gya ; Baberuxki:2019ifp  to the S  HERPA Gleisberg:2008ta ;
Sherpa:2019gpd  framework, including perturbative uncertainties, obtained by
varying the perturbative (renormalisation, factorisation and resummation)
scales and NP corrections. We use COMIX Gleisberg:2008fv  in conjunction with
OpenLoops Buccioni:2019sur  and Recola Actis:2016mpe ; Biedermann:2017yoi  for
the fixed order calculation. The NP corrections are based on S  HERPA parton
shower simulations at MC@NLO accuracy Frixione:2002ik ; Hoeche:2011fd
hadronised with S  HERPA ’s cluster fragmentation model Winter:2003tt .

Our results are reported in Fig. 3, where we show the $p_{t\,Z}$ distribution
for events where the highest-$p_{t}$ jet is quark initiated, for the taggers
selected for this study. We show the $p_{t\,Z}$ distribution with no-tagging,
with tagging on standard jets and with tagging on SoftDrop jets. We note that
NP corrections are rather sizeable at low $p_{t\,Z}$, making this observable
most reliable in the high transverse momentum region. The latter is actually
_per se_ interesting because it allows us to probe the proton dynamics
described by the PDFs at large values of the momentum fraction $x$, _i.e._ in
a kinematic region where they are less constrained. To qualitatively assess
the values of $x$ which would be accessible, we show, on the upper horizontal
axis, the Born-level momentum fraction computed at central (zero) rapidities:
$\bar{x}=x_{1,2}=\frac{p_{t}e^{\pm y_{J}}+\sqrt{p_{t}^{2}+m_{Z}^{2}}e^{\pm
y_{Z}}}{\sqrt{S}}\Big{|}_{y_{J}=y_{Z}=0}.$

We have also studied $p_{t\,J}$ distributions. However, as anticipated, these
distributions turn out to be less robust, essentially because jet dynamics can
be significantly altered by the cut on the angularity, as well as by the
grooming procedure 555Note that if $\beta=0$, the groomed $p_{t\,J}$
distribution is not even IRC safe.. In contrast, the $p_{t\,Z}$ spectrum is
inclusive with respect to the jet activity and thus, for a given working point
of the tagger, less dependent on the details of the tagging procedure itself.
We can therefore take modifications in such distribution as more directly
related to the bias that the tagger induces on the composition of the partonic
initial state, which is what we want to achieve.

## Conclusions and future developments

We have shown how _q/g_ tagging can be successfully applied to $Z$+jets events
in order to significantly enhance the gluon-initiated contributions. In
particular, our tagger is realised through a simple cut on a jet angularity,
which is an IRC safe observable and therefore can be studied in perturbation
theory. Exploiting MC simulations, we have performed a study of their
efficiencies and their dependence on NP effects, exploring different
angularities and different levels of grooming. We have explicitly studied the
transverse momentum of the $Z$ boson, providing theoretical predictions that
included both all-order resummation and matching to NLO. We have shown that we
can achieve initial-state gluon purities close to 95%.

We see several possible directions for future work in this context. First, we
would like to assess the impact of this type of observables on PDF fits. In
particular, while our study does show an increase in the gluon purity, this
comes at the cost of a noticeable reduction of the available dataset. Thus,
despite the relative large cross-section of the process we are considering, we
may need to optimise the tagger’s working point, taking this further aspect
into consideration. For instance, as mentioned above, we could choose
$\lambda_{\text{cut}}$ so that the signal-to-background ratio is maximised.
Second, the results presented in this study are based on a calculation of the
angularity spectra, and hence of closely related efficiencies, at NLO+NLL′.
The resummed calculation can be promoted to higher accuracy (Frye:2016okc, ;
Frye:2016aiz, ; Kardos:2020gty, ). The fixed-order can also be improved by
including the two-loop correction to $Z$+1 jet (see (Gehrmann-
DeRidder:2015wbt, ) and references therein). An improvement in the description
of the angularity distribution away from $\lambda_{\alpha}=0$ is much more
challenging because it requires $Z$+2 partons at NNLO, which may become
available in the near future. Finally, with the aim of enhancing performance
while maintaining calculability, it would be interesting to consider more
powerful, albeit more sophisticated, taggers. In this context, the Les Houches
multiplicity (Amoroso:2020lgh, ) is very promising and, although its
theoretical understanding is only in its infancy, we believe that achieving
NLL accuracy is within reach.

## Acknowledgments

We thank S. Schumann and G. Soyez for collaboration on related topics. We also
thank our ATLAS and CMS colleagues: R. Aggleton, J. Ferrando, A. Hinzmann, M.
LeBlanc, B. Nachman, and F. Sforza, for useful discussions. This work is
supported by Università di Genova under the curiosity-driven grant “Using jets
to challenge the Standard Model of particle physics” and by the Italian
Ministry of Research (MUR) under grant PRIN 20172LNEEZ. DR further
acknowledges funding from the European Union Horizon 2020 research and
innovation programme as part of the Marie Sklodowska Curie Innovative Training
Network MCnet ITN3 (grant agreement no. 722104), from BMBF (contract
05H18MGCA1) and by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) - project number 456104544. Figures were created with the
Matplotlib (Hunter:2007ouj, ) and NumPy (NumPy, ) libraries.

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