# Phenomenology of jet angularities at the LHC

Daniel Reichelt111Email<EMAIL_ADDRESS>Institute for Particle
Physics Phenomenology, Department of Physics, Durham University,
South Road, Durham DH1 3LE, United Kingdom Institut für Theoretische Physik,
Georg-August-Universität Göttingen,
Friedrich-Hund-Platz 1, 37077 Göttingen, Germany Simone Caletti222Email:
<EMAIL_ADDRESS>Dipartimento di Fisica, Università di Genova and
INFN, Sezione di Genova,
Via Dodecaneso 33, 16146, Genoa, Italy Oleh Fedkevych333Email:
<EMAIL_ADDRESS>Dipartimento di Fisica, Università di Genova and
INFN, Sezione di Genova,
Via Dodecaneso 33, 16146, Genoa, Italy Simone Marzani444Email:
<EMAIL_ADDRESS>Dipartimento di Fisica, Università di Genova and
INFN, Sezione di Genova,
Via Dodecaneso 33, 16146, Genoa, Italy Steffen Schumann555Email:
<EMAIL_ADDRESS>Institut für Theoretische Physik,
Georg-August-Universität Göttingen,
Friedrich-Hund-Platz 1, 37077 Göttingen, Germany Gregory Soyez666Email:
<EMAIL_ADDRESS>Institut de Physique Théorique, Paris Saclay University,
CNRS, CEA,
Orme des Merisiers, Bât 774, F-91191 Gif-sur-Yvette, France

###### Abstract

We compute resummed and matched predictions for jet angularities in hadronic
dijet and $Z$+jet events with and without grooming the candidate jets using
the SoftDrop technique. Our theoretical predictions also account for non-
perturbative corrections from the underlying event and hadronisation through
parton-to-hadron level transfer matrices extracted from dedicated Monte Carlo
simulations with S  HERPA . Thanks to this approach we can account for non-
perturbative migration effects in both the angularities and the jet transverse
momentum. We compare our predictions against recent measurements from the CMS
experiment. This allows us to test the description of quark- and gluon-jet
enriched phase-space regions separately. We supplement our study with S  HERPA
results based on the matching of NLO QCD matrix elements with the parton
shower. Both theoretical predictions offer a good description of the data,
within the experimental and theoretical uncertainties. The latter are however
sizeable, motivating higher-accuracy calculations.

###### Contents

  1. 1 Introduction
  2. 2 Phase space and observable definition
  3. 3 Theoretical framework for jet-angularity predictions
    1. 3.1 Monte Carlo simulations with S  HERPA
    2. 3.2 $\text{NLO}+\text{NLL}^{\prime}$ resummation for dijet final states
    3. 3.3 Non-perturbative corrections in the transfer-matrix approach
      1. 3.3.1 The transfer-matrix approach
      2. 3.3.2 Transfer matrices for jet-angularity observables
  4. 4 Results for jet angularities in dijet and $Z$+jet production
    1. 4.1 Selected parton-level results
    2. 4.2 Hadron-level results
  5. 5 Conclusions
  6. A Results for jet width and jet thrust

##  1 Introduction

The LHC data-taking campaigns during run I and II recorded huge amounts of
high-quality data which can be explored to perform high-precision tests of the
Standard Model. Recent studies have demonstrated the particular potential of
jet cross-section measurements. For example, jet data can be used in
determinations of the strong coupling constant, see e.g. [1, 2, 3, 4, 5], and
form an important input for constraining parton distribution functions (PDFs)
[6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Information on the physical processes
underlying jet formation can be gained by studying so-called jet substructure
observables (for recent reviews see e.g. [16, 17, 18]). In this context, a
popular jet substructure technique used to remove radiation in phase-space
regions dominated by non-perturbative physics is SoftDrop grooming [19],
including the modified Mass Drop Tagger (mMDT) as a special case [20, 21].
Physics observables measured on SoftDrop jets have been shown to be less
affected by non-perturbative corrections and to be amenable to precision
calculations in QCD [22, 23, 24, 25, 26, 27]. This has triggered the attention
of the jet substructure community, both theorists and experimentalists [28,
29, 30], over the past few years.

One of the central goals of jet substructure analyses is to determine
differences between jets originating from quarks versus gluons. This
information can be exploited, for instance, to enhance sensitivity in searches
for new physics. Consequently, the consistent modelling of quark and gluon
jets has been an active area of research over the past years, see for example
[31, 32, 33, 34, 35, 36, 37, 38, 39]. An often-studied benchmark, widely used
in many of the above references, is the so-called generalised jet angularities
[40, 41]. A sound theoretical understanding of these angularities is thus
critical to the task of quark–gluon discrimination. These observables, and
closely related ones, have hence been studied in a variety of theoretical
frameworks [40, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]. Recent
measurements of jet angularities at the LHC have for example been presented in
[53, 54, 55, 30].

In this paper we focus on the wealth of data presented by the CMS experiment
in [30], which measured angularities, with and without SoftDrop grooming, on
jets produced either in association with a $Z$ boson or in dijet events. In an
earlier article, [51], we already presented $\text{NLO}+\text{NLL}^{\prime}$
accurate predictions for the $Z$+jet case, derived using the resummation
plugin to the S  HERPA event generator framework [56, 57]. Here, we aim to
extend this study to the case of dijet production. In this context, we
introduce an improved prescription to apply non-perturbative corrections,
extracted from Monte Carlo (MC) simulations, to our resummed analytic
predictions. Since $Z$+jet and dijet events receive different contributions
from quark- and gluon-initiated jets, a combined study of (groomed) jet shapes
such as the angularities in both channels is expected to shed light on our
fundamental understanding of jet formation from hard quarks and gluons. To
that aim, besides providing results for the angularity differential
distributions, we also consider the angularities mean values in quark- and
gluon-dominated phase-space regions of $Z$+jet and dijet final states.

The article is organised as follows: in Sec. 2 we introduce the considered
angularity observables and define the fiducial phase space used in our study
of jets in $Z$+jet and dijet production. In Sec. 3 we introduce the
theoretical methods used for making predictions at
$\text{NLO}+\text{NLL}^{\prime}$ accuracy, based on the implementation of the
C  AESAR formalism in S  HERPA . Here, we also present our new approach to
account for non-perturbative corrections based on transfer matrices extracted
from MC simulations, see Sec. 3.3. We present our final
$\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ results in Sec. 4, alongside with
MC predictions from S  HERPA , and compare those against data from the CMS
experiment. We compile our conclusions and give an outlook in Sec. 5.

##  2 Phase space and observable definition

To be able to directly compare to the CMS measurement presented in [30], based
on proton–proton collisions at a centre-of-mass energy of
$\sqrt{s}=$13\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$, the exact definition
of angularities and the event-selection criteria have to be carefully matched
to the experimental setup. For completeness, this section summarises the final
definitions and cuts and we refer the reader to Ref. [30] for additional
details and motivations.

The events considered are conceptually triggered either by an approximately
on-shell $Z$ boson, decaying into a pair of muons, accompanied by a hard jet
($Z$+jet case), or by two high-$p_{\text{T}}$ jets (dijet case). The precise
phase space for the two cases is defined by the selection criteria summarised
in table 1. Substructure observables are then calculated for the jet with the
highest $p_{\text{T}}$ in the $Z$+jet case. For dijet events, the two jets
with the largest transverse momenta are considered and the measurement is
performed separately for the more central (smaller rapidity $|y|$) and more
forward (larger $|y|$) of those jets. For the $Z$+jet process
$p_{T,\text{jet}}$ bins in the interval $[50,1500]$ GeV were considered, while
in the dijet case this range was extended to $[50,4000]$ GeV.

The substructure observables we are interested in belong to the family of jet
angularities. These probe both the angular and the transverse momentum
distribution of particles within a given jet. They are defined from the
momenta of jet constituents as follows:

$\lambda_{\alpha}^{\kappa}=\sum_{i\in\text{jet}}\left(\frac{p_{T,i}}{\sum_{j\in\text{jet}}p_{T,j}}\right)^{\kappa}\left(\frac{\Delta_{i}}{R}\right)^{\alpha}\,.$
(2.1)

Here

$\Delta_{i}=\sqrt{(y_{i}-y_{\text{jet}})^{2}+(\phi_{i}-\phi_{\text{jet}})^{2}}\,,$
(2.2)

is the azimuth-rapidity distance of particle $i$ from the jet axis, which is
determined using the Winner-Take-All (WTA) recombination scheme [46] for
$\alpha\leq 1$, and using the default $E$-scheme axis otherwise. $R$ denotes
the radius parameter used for the anti-$k_{t}$ [58] jet clustering. We
restrict our analysis to the infrared-and-collinear-safe case $\kappa=1$ with
$\alpha=0.5,1,2$, while Ref. [30] in addition considered the infrared-unsafe
combinations $\kappa=0,\,\alpha=0$ and $\kappa=2,\,\alpha=0$. For the
different choices of $\alpha$ we follow the common naming convention [31, 59,
30], and refer to $\lambda^{1}_{0.5}$ as Les Houches Angularity (LHA),
$\lambda^{1}_{1}$ as jet width, and $\lambda^{1}_{2}$ as jet thrust.

The CMS collaboration also measured the same jet angularities based on the jet
constituents obtained after SoftDrop grooming, as implemented in the F  AST J
ET [60] `contrib` package. The SoftDrop parameters are chosen as $\beta=0$ and
$z_{\text{cut}}=0.1$. All results are provided either using all the particles
in the jet (both charged and neutral), or using only the charged jet
constituents. Note that the $p_{T,\text{jet}}$ bin is always determined from
the full jet, without any grooming or restriction on the charge of particles.

Process | Selection cuts
---|---
$Z$+jet |  Require at least two muons with $p_{T,\mu}>26~{}\mathrm{GeV},\;|\eta_{\mu}|<2.4,$ oppositely charged with invariant mass $m_{\mu^{+}\mu^{-}}\in[70,110]$ GeV and $p_{T,\mu^{+}\mu^{-}}>30$ GeV. The leading (highest $p_{\text{T}}$) $R=0.4$ (or $R=0.8$) anti-$k_{t}$ jet has to satisfy $|y_{\rm jet}|<1.7$, $\left|\phi_{\mu^{+}\mu^{-}}-\phi_{\rm jet}\right|>2$ and $\left|p_{T,\rm jet}-p_{T,\mu^{+}\mu^{-}}\right|/\left|p_{T,\rm jet}+p_{T,\mu^{+}\mu^{-}}\right|<0.3$ .
Dijet |  Require at least two $R=0.4$ (or $R=0.8$) anti-$k_{t}$ jets with $p_{\text{T}}>30~{}\text{GeV}$. The two leading (highest $p_{\text{T}}$) jets have to satisfy $|y_{\rm jet}|<1.7$, $p_{T,\rm jet}>30$ GeV, $\left|\phi_{j_{1}}-\phi_{j_{2}}\right|>2$ and $\left|p_{T,j_{1}}-p_{T,j_{2}}\right|/\left|p_{T,j_{1}}+p_{T,j_{2}}\right|<0.3$ .
Table 1: Event-selection cuts used by the CMS collaboration in [30]. The muons
used to reconstruct the $Z$ boson are considered as “bare”, _i.e._ without QED
emissions, and are excluded from the input to the jet clustering.

##  3 Theoretical framework for jet-angularity predictions

We here consider two sets of theoretical predictions for jet angularities in
$Z$+jet and dijet production to confront with the CMS data. The first are
simulations with the S  HERPA multi-purpose event generator. The jet-
angularity distributions presented in [30] were compared against several
event-generator predictions. The highest accuracy was reached in two
simulations based on LO merging [61, 62] of tree level matrix elements with
additional jets dressed with parton showers, one using M  ADGRAPH 5_aMC@NLO
[63] in conjunction with P  YTHIA [64], the other one using S  HERPA [65, 57].
In addition, results obtained from the parton showers of P  YTHIA , H  ERWIG
[66, 67] and S  HERPA , based on Born matrix elements, were presented. We here
provide S  HERPA predictions at NLO accuracy for the $Z$+jet and $jj$
processes, which were not covered in [30].

The second set of predictions is obtained from
$\text{NLO}+\text{NLL}^{\prime}$ semi-analytic calculations, based on the
implementation of the C  AESAR resummation formalism [43] in the S  HERPA
framework. In [30] the CMS results for $Z$+jet production were already
compared against our $\text{NLO}+\text{NLL}^{\prime}$ predictions from [51].
We here extend this comparison to the case of dijet production. In addition,
we introduce a more physical way to apply non-perturbative corrections to our
calculations, that we refer to as _transfer-matrix approach_ which we also
apply to both the $Z$+jet and dijet cases.

In the following, we describe our setup for the MC simulations with S  HERPA ,
then introduce the necessary additions to our resummation framework to carry
out the $\text{NLO}+\text{NLL}^{\prime}$ calculation for the dijet case, and
finally introduce the transfer-matrix approach.

###  3.1 Monte Carlo simulations with S  HERPA

We compile hadron-level predictions for jet angularities with the S  HERPA
[65, 57] event generator, version 2.2.11, using the NNPDF-3.0 NNLO PDF set
[68]. We consider inclusive dijet and $\mu^{+}\mu^{-}$+jet production based on
the S  HERPA implementation of the MC@NLO formalism [69], dubbed SH-MC@NLO in
what follows, matching the NLO QCD matrix elements for two-jet and
$\mu^{+}\mu^{-}$+jet production with the S  HERPA Catani–Seymour dipole shower
[70]. The involved QCD one-loop amplitudes we obtain from O  PEN L  OOPS [71],
using the C  OLLIER library [72] for the evaluation of tensor and scalar
integrals. The central values for the perturbative scales entering the
calculation are set to

$\displaystyle\mu_{\text{F}}=\mu_{\text{R}}=\mu_{\text{Q}}=\sqrt{m^{2}_{\mu^{+}\mu^{-}}+p^{2}_{T,\mu^{+}\mu^{-}}}\,\quad\quad$
$\displaystyle:$ $\displaystyle\quad Z+\text{jet process}\,,$ (3.1)
$\displaystyle\mu_{\text{F}}=\mu_{\text{R}}=H_{T}/2\,,\quad\mu_{\text{Q}}=p_{T,\text{jet}}/2\quad$
$\displaystyle:$ $\displaystyle\quad\text{dijet process}\,.$ (3.2)

To estimate the perturbative uncertainties of our predictions, we perform on-
the-fly [73] $7$-point variations [74] of the factorisation ($\mu_{\text{F}}$)
and renormalisation ($\mu_{\text{R}}$) scales in the matrix elements and the
parton shower, while we keep the parton-shower starting scale
($\mu_{\text{Q}}$) fixed as it is related to the dipole kinematics of the
parton shower. As error estimate we then consider the envelope of the results
for

$\\{(\tfrac{1}{2}\mu_{\text{R}},\tfrac{1}{2}\mu_{\text{F}}),(\tfrac{1}{2}\mu_{\text{R}},\mu_{\text{F}}),(\mu_{\text{R}},\tfrac{1}{2}\mu_{\text{F}}),(\mu_{\text{R}},\mu_{\text{F}}),(\mu_{\text{R}},2\mu_{\text{F}}),(2\mu_{\text{R}},\mu_{\text{F}}),(2\mu_{\text{R}},2\mu_{\text{F}})\\}\,.$
(3.3)

The S  HERPA Underlying Event (UE) simulation is based on its implementation
of the Sjöstrand–Zijl multiple-parton interaction model [75]. To account for
the parton-to-hadron transition we use the S  HERPA cluster fragmentation
[76].

Later, when we will use S  HERPA to account for non-perturbative corrections
to the analytic calculation, we include an estimate of the uncertainty related
to the hadronisation model. This uses the interface to P  YTHIA 6.4 [77],
provided with S  HERPA , with the Lund string fragmentation model [78, 79] as
an alternative. In both models the default set of tuning parameters is used,
see [57] for details. However, we additionally vary the default for the
rescale-exponent parameter $\alpha^{\text{MPI}}_{\text{min}}$ used to model
the dependence of the cutoff regulator $p_{T,\text{min}}$ for the $2\to 2$
cross-section integration in the UE model on the hadronic centre-of-mass
energy, _i.e._

$p_{T,\text{min}}(E_{\text{cms}})=p_{T,\text{min}}(E_{\text{ref}})\left(\frac{E_{\text{cms}}}{E_{\text{ref}}}\right)^{\alpha^{\text{MPI}}_{\text{min}}}\,,$
(3.4)

where $E_{\text{ref}}=1.8\,\text{TeV}$ and
$p_{T,\text{min}}(E_{\text{ref}})=2.895\,\text{GeV}$. As new central value we
here consider

$\alpha^{\text{MPI}}_{\text{min,central}}=0.16\,,$ (3.5)

corresponding to a somewhat enhanced UE activity with respect to the default
in public S  HERPA -2.2.11, where
$\alpha^{\text{MPI}}_{\text{min,def}}=0.244$. This is motivated by a
preliminary study comparing S  HERPA predictions for the jet angularities in
particular for the low-$p_{T,\text{jet}}$ region against the data from [30].
As systematic up- and down-variations of the MPI activity we consider
$\alpha^{\text{MPI}}_{\text{min,up}}=0.08$ and
$\alpha^{\text{MPI}}_{\text{min,down}}=0.24$.

###  3.2 $\text{NLO}+\text{NLL}^{\prime}$ resummation for dijet final states

Throughout this paper, we target a description of the angularity distribution
at the $\text{NLO}+\text{NLL}^{\prime}$ accuracy, whereby the exact
distribution at order $\alpha_{\text{s}}^{2}$ (NLO)***Note, in this counting
the powers of $\alpha_{\text{s}}$ from the underlying Born process, _i.e._
$\alpha_{\text{s}}^{2}$ for dijet and $\alpha_{\text{s}}^{1}$ for $Z$+jet, are
not included. is matched to the all-order resummation, which is relevant for
small values of the observable at hand, carried out at the
$\text{NLL}^{\prime}$ accuracy.

The NLL resummation is performed using the S  HERPA implementation of the C
AESAR formalism [43]. This plugin to S  HERPA was first presented in [56] and
further developed in subsequent work, _e.g._ to introduce new matching schemes
[80], to include the SoftDrop phase-space constraints [81], or to support some
non-global configurations in Ref. [51]. The master formula for the all-order
resummation can be written as the following sum over partonic channels
$\delta$:

$\displaystyle\Sigma_{\text{res}}(v)$
$\displaystyle=\sum_{\delta}\Sigma^{\delta}_{\text{res}}(v)\,,$ (3.6)
$\displaystyle\Sigma^{\delta}_{\text{res}}(v)$
$\displaystyle=\int\mathop{d\mathcal{B}^{\delta}}\frac{\mathop{d\sigma^{\delta}}}{\mathop{d\mathcal{B}}}\exp\left[-\sum_{l\in\mathcal{B}^{\delta}}R^{\mathcal{B}^{\delta}}_{l}(L)\right]\mathcal{S}^{\mathcal{B}^{\delta}}(L)\,\mathcal{P}^{\mathcal{B}^{\delta}}(L)\,\mathcal{F^{\mathcal{B}^{\delta}}}(L)\,\Theta_{\text{hard}}^{\mathcal{B}^{\delta}}\,.$
(3.7)

In this expression, $\mathop{d\sigma^{\delta}}/\mathop{d\mathcal{B}}$ is the
Born-level cross section, obtained through the C  OMIX matrix element
generator [82]; the collinear radiators $R_{l}$ are directly obtained from the
C  AESAR paper [43] or extended to include the effect of SoftDrop (see _e.g._
Refs. [19, 81]); $\mathcal{S}$ is the soft function including global and non-
global colour evolution; $\mathcal{P}$ includes corrections associated with
PDF evolution for initial-state radiation (which are absent in our case,
_i.e._ $\mathcal{P}=1$); $\mathcal{F}$ is the multiple-emission function,
which for our application is simply given by
$e^{-\gamma_{E}R^{\prime}}/\Gamma(1+R^{\prime})$ with
$R^{\prime}=\partial_{L}R$ since angularities are additive observables.
Finally, $\Theta_{\text{hard}}$ implements the cuts on the Born-level phase
space, _cf._ table 1. The phase-space integration as well as the separation
over different partonic channels is greatly facilitated by the integration of
the plugin into the S  HERPA framework. With this approach, one reaches NLL
accuracy including an estimate of the resummation scale uncertainty
(introducing a rescaling parameter $x_{L}$), as well as a treatment of the
endpoint corrections and associated uncertainties. Modulo an enhanced
treatment of non-global logarithms to account for the more complicated colour
structure of dijet events that we discuss below, our usage of the resummation
plugin to S  HERPA is the same as in our study of angularities in $Z$+jet
events and we therefore refer the reader to Ref. [51] for details. In
particular, through the iterative flavour-$k_{t}$ clustering procedure [83]
(referred to as BSZ algorithm in what follows), our automated resummation
plugin allows us to keep track of the jet flavour (see again [51] for a
detailed description of the algorithm), yielding the so-called
$\text{NLL}^{\prime}$ resummation accuracy.

Let us now discuss our treatment of soft logarithms and, in particular, of
non-global logarithms. In practice, the soft function $\mathcal{S}$ can be
factorised in a global contribution $\mathcal{S}_{\text{global}}$ and a non-
global [84, 85] part $\mathcal{S}_{\text{non-global}}$. Although we perform
the resummation of non-global logarithms in the large-$N_{C}$ limit, we can
account, to all orders, for the full set of colour correlations induced by the
first emission. In order to achieve this we note that, for a given Born-level
partonic channel $\mathcal{B}_{\delta}$, the overall contribution from soft
logarithms can be written as

$\mathcal{S}^{\mathcal{B}_{\delta}}(t(L))=\operatorname{Tr}\left[He^{-t\left(\mathbf{\Gamma}^{\mathcal{B}_{\delta}}\right)^{\dagger}}ce^{-t\mathbf{\Gamma}^{\mathcal{B}_{\delta}}}\right]/\operatorname{Tr}\left[cH\right]\,,$
(3.8)

with

$t\mathbf{\Gamma}^{\mathcal{B}_{\delta}}=\sum_{i>j\in\mathcal{B}_{\delta}}\mathbf{T}_{i}\cdot\mathbf{T}_{j}(I_{ij}t+f_{ij}(t))\,,\qquad\text{
and }\qquad
t(L)=\int^{\mu_{Q}}_{\mu_{Q}e^{-L}}\frac{dk_{t}}{k_{t}}\frac{\alpha_{\text{s}}\left(k_{t}\right)}{\pi}=\frac{-\ln(1-2\alpha_{\text{s}}\beta_{0}L)}{2\pi\beta_{0}}\,,$
(3.9)

where $\alpha_{\text{s}}=\alpha_{\text{s}}(\mu_{\text{Q}})$. Note that these
contributions arise only at NLL accuracy and so the actual scale choice for
$\alpha_{\text{s}}$ is subleading. In practice, we evaluate
$\alpha_{\text{s}}$ at $\mu_{\text{R}}$. Here, $H$ and $c$ are the hard
function and colour metric for this channel. The $I_{ij}t$ coefficients in
$t\mathbf{\Gamma}^{\mathcal{B}_{\delta}}$ correspond to global contributions,
stemming from soft-wide-angle radiation. These depend only on the jet radius
$R$ and are computed in the small-$R$ limit. They are taken from [86] and we
note that, compared to the $Z$+jet case, the dijet case also has to include
dipoles involving the recoiling jet.

The non-global contributions to $\mathcal{S}$ are encoded in the functions
$f_{ij}(t)$ in $t\mathbf{\Gamma}^{\mathcal{B}_{\delta}}$. They are computed
independently for each dipole using a Monte Carlo approach as described in
[84].†††This procedure requires a cutoff $\theta_{\text{min}}$ on collinear
radiation. In practice, we compute $f(t)$ for different cuts and extrapolate
numerically to $\theta_{\text{min}}\to 0$. The main difference compared to our
previous work on $Z$+jet events is the fact that one also has to include the
$f_{ij}(t)$ corresponding to dipole configurations involving the recoiling
jet. On top of the standard dependence on $t(L)$ and on the jet radius, these
dipoles involving the recoiling jet also depend on the rapidity separation
between the measured jet and the recoiling one (see _e.g._ [86]). This
approach allows us to compute the non-global soft function for ungroomed
angularities. For SoftDrop groomed distributions, the non-global factor
remains the same for $v\geq z_{\text{cut}}$ and saturates at that value,
_i.e._ $\mathcal{S}_{\text{non-
global}}^{\text{(groomed)}}(v)=\mathcal{S}_{\text{non-
global}}^{\text{(ungroomed)}}(\text{max}(v,z_{\text{cut}}))$ . Finally, we
note that while for $Z$+jet processes the matrix structure of Eq. (3.8)
disappears, the situation for dijet events is more complex and the full matrix
structure needs to be taken into account.

The above-described resummed observable calculation is matched to exact
distributions at NLO. These are obtained using S  HERPA with R  ECOLA [87, 88]
and O  PEN L  OOPS [71] as one-loop generators in conjunction with the C
OLLIER library [72], and using Catani–Seymour dipole subtraction [89, 90]. All
tree-level contributions are obtained from the built-in C  OMIX [82]
generator. For a generic angularity $\lambda$, the matching is done at the
level of cumulative distributions $\Sigma^{\delta}(\lambda)$ defined for each
flavour channel $\delta$. For this, we expand the (fixed-order) jet cross
section $\sigma_{\text{fo}}(p_{\text{T}})$ as well as the fixed-order and
resummed cumulative distributions, $\Sigma_{\text{fo}}(\lambda)$ and
$\Sigma_{\text{res}}(\lambda)$ in series of $\alpha_{\text{s}}$:

$\displaystyle\sigma^{\delta}_{\text{fo}}(p_{\text{T}})$
$\displaystyle=\sigma^{\delta(0)}+\sigma^{\delta(1)}_{\text{fo}}+\sigma^{\delta(2)}_{\text{fo}}+\dots\,,$
(3.10) $\displaystyle\Sigma^{\delta}_{\text{fo}}(\lambda)$
$\displaystyle=\sigma^{\delta(0)}+\Sigma^{\delta(1)}_{\text{fo}}+\Sigma^{\delta(2)}_{\text{fo}}+\dots\,,$
(3.11) $\displaystyle\Sigma^{\delta}_{\text{res}}(\lambda)$
$\displaystyle=\sigma^{\delta(0)}+\Sigma^{\delta(1)}_{\text{res}}+\Sigma^{\delta(2)}_{\text{res}}+\dots\,,$
(3.12)

where the $p_{\text{T}}$ dependence is left implicit, the number in
parentheses indicates the respective order in $\alpha_{\text{s}}$ (without
including the Born-level factors of $\alpha_{\text{s}}$ and, possibly,
$\alpha_{\text{EW}}$), and $\sigma^{\delta(0)}$ is the Born-level jet cross
section in the specified flavour channel. Our final matched expression for the
cumulative distribution reads:

$\Sigma^{\delta}_{\text{matched}}(\lambda)=\Sigma^{\delta}_{\text{res}}(\lambda)\left(1+\frac{\Sigma^{\delta(1)}_{\text{fo}}-\Sigma^{\delta(1)}_{\text{res}}}{\sigma^{\delta(0)}}-\frac{\bar{\Sigma}^{\delta(2)}_{\text{fo}}+\Sigma^{\delta(2)}_{\text{res}}}{\sigma^{\delta(0)}}-\frac{\Sigma^{\delta(1)}_{\text{res}}}{\sigma^{\delta(0)}}\frac{\Sigma^{\delta(1)}_{\text{fo}}-\Sigma^{\delta(1)}_{\text{res}}}{\sigma^{\delta(0)}}\right)\,,$
(3.13)

where
$\bar{\Sigma}^{(2)}_{\text{fo}}=\sigma_{\text{fo}}^{(2)}-\Sigma^{(2)}_{\text{fo}}=\int_{\lambda}^{1}d\sigma^{(2)}$
can be calculated without the need for the two-loop virtual corrections to the
jet cross section.

To estimate the perturbative uncertainties of our predictions, we again
consider $7$-point variations of the factorisation and renormalisation scales
in the matrix elements and the resummation, _cf._ Eq. (3.3). The central
values in the $Z$+jet case are taken from [51], _i.e._ the transverse momentum
of the $Z$ boson, multiplied by the jet radius in the case of
$\mu_{\text{Q}}$. Similarly, in the dijet case we use
$\mu_{\text{R}}=\mu_{\text{F}}=\mu_{\text{Q}}/R=H_{T}/2$, closely matching the
scale choice in the SH-MC@NLO simulations see Eq. (3.2). When implementing the
matching between our fixed-order and resummed results, we treat the endpoint
of the resummed expression and of its expansion by setting

$L=\frac{1}{p}\ln\left[1+\left(\frac{x_{L}}{v}\right)^{p}-\left(\frac{x_{L}}{v_{\text{max}}}\right)^{p}\right]\,,$
(3.14)

with $v_{\text{max}}$ the kinematic endpoint of the fixed-order distribution.
Varying $x_{L}$ between $\tfrac{1}{2}$ and $2$ then allows us to quantify the
resummation uncertainty (we keep $p=1$ fixed). As total perturbative
uncertainty estimate we consider the envelope of all $(\mu_{R},\mu_{F})$ and
$x_{L}$ variations.

###  3.3 Non-perturbative corrections in the transfer-matrix approach

So far we have explained our analytic description of jet angularities in
perturbative QCD. Clearly, these observables probe QCD dynamics in the
infrared regime and are therefore also sensitive to non-perturbative (NP)
corrections. At a hadron collider such as the LHC, this includes the
Underlying Event — _i.e._ remnant–remnant interactions — on top of the parton-
to-hadron transition. Because jet angularities are IRC-safe observables, NP
effects can be considered as power corrections and are therefore expected to
become less important as one increases the transverse momentum of the jet they
are measured on. However, at moderate $p_{T,\text{jet}}$ their effect can be
sizeable and it is crucial to include them in any realistic phenomenological
prediction. Additionally, the size of the jet-radius parameter impacts the
susceptibility to NP corrections.

In previous works [22, 23, 91, 51], we have adopted a rather simple procedure
to account for NP effects. We have computed predictions for a given jet-
angularity distribution in the considered fiducial phase space, _i.e._
applying the phase-space constraints on jets and restricting to a given
$p_{T,\text{jet}}$ bin, both at parton and hadron level, using a general-
purpose MC generator. The ratio between these two distributions was then used
to correct our resummed and matched predictions for that particular observable
and phase-space region. While this model captures the dominant NP effects of
redistributing the partonic jet-constituent momenta onto hadrons, it is still
rather crude, as it does not fully account for alterations of the underlying
parton-level event kinematics due to NP effects. In particular, both
hadronisation and UE affect the transverse-momentum distribution of jets,
potentially leading to migration between the considered
$p_{T,\text{jet}}$-bins. By neglecting this important feature of NP
corrections, this approach turns out to be rather sensitive to cutoff-effects
in the employed parton shower. In particular, NP corrections can become
pathological for observable values corresponding to scales below the shower
cutoff. This generator dependence can be partially overcome by considering
several MC programs and using the spread of the extracted NP ratios as an
uncertainty on the NP corrections, as was done in Ref. [51], and consequently
used in the comparison to CMS data in [30].

####  3.3.1 The transfer-matrix approach

In this new study, we attempt to overcome the limitations of the method for
extracting NP corrections we previously used. To this end we develop and
implement a more realistic and detailed model to include NP corrections, in
which we account for the change of parton-level event kinematics due to
hadronisation and the UE. In particular, considering double-differential
measurements in jet-transverse momentum and the angularity observables, we
derive non-perturbative transfer matrices which account for the alteration of
both $p_{T,\text{jet}}$ and the angularity. Through the migration in
transverse momentum we significantly reduce the sensitivity to phase-space
restrictions and non-perturbative parameters in the underlying MC simulations,
in particular to the parton-shower cutoff parameter. We here present our
approach in full generality, applicable to an arbitrary set of observables,
measured in a multi-differential way. We thereby aim to keep track both of the
migration in the underlying event-kinematical variables, used to define the
fiducial phase space, that get partially integrated over, as well as changes
in the actual observable of interest, _e.g._ a specific angularity variable.

Let us consider a scattering process which results in a partonic configuration
$\mathcal{P}$. Through NP effects the set of parton momenta is then mapped
onto a hadron-level configuration $\mathcal{H}\left(\mathcal{P}\right)$. The
map $\mathcal{H}$, which does not need to be fully specified at this point,
accounts for hadronisation and UE corrections. It could be derived from field-
theoretical considerations (see for example Refs. [92, 93] for recent work on
SoftDrop observables) or it can be extracted from any given parton-shower
simulation interfaced to a model of NP phenomena. For a given configuration
$\mathcal{P}$ or $\mathcal{H}\left(\mathcal{P}\right)$, we then measure a set
of $m$ observables, $\vec{V}\left(\mathcal{P}\right)$ or
$\vec{V}\left(\mathcal{H}\left(\mathcal{P}\right)\right)$.‡‡‡Here, we have
chosen the same set of observables $\vec{V}$ on the parton- and hadron-level
configurations for simplicity. It is of course trivial to extend this to
differing sets of observables for parton and hadron level, for example using
additional auxiliary observables to parameterise the parton-level phase space,
or adding selection criteria like, for instance, particle charge, at hadron
level. We define the transfer operator as the conditional probability to
measure a hadron-level set of observables $\vec{v}_{h}$, evaluated on
$\mathcal{H}\left(\mathcal{P}\right)$, given that the parton-level observables
were $\vec{v}_{p}$:

$\mathcal{T}(\vec{v}_{h}|\vec{v}_{p})=\frac{\int\mathop{d\mathcal{P}}\frac{\mathop{d\sigma}}{\mathop{d\mathcal{P}}}\delta^{(m)}\left(\vec{v}_{p}-\vec{V}\left(\mathcal{P}\right)\right)\delta^{(n)}\left(\vec{v}_{h}-\vec{V}\left(\mathcal{H}\left(\mathcal{P}\right)\right)\right)}{\int\mathop{d\mathcal{P}}\frac{\mathop{d\sigma}}{\mathop{d\mathcal{P}}}\delta^{(m)}\left(\vec{v}_{p}-\vec{V}\left(\mathcal{P}\right)\right)}\,.$
(3.15)

This way, the multi-differential distribution for the set of hadron-level
observables $\vec{v}_{h}$ can be written as

$\frac{\mathop{d^{m}\sigma^{\text{HL}}}}{\mathop{dv_{h,1}\dots
dv_{h,m}}}=\int\mathop{d^{m}\vec{v}_{p}}\,\mathcal{T}(\vec{v}_{h}|\vec{v}_{p})\,\frac{\mathop{d^{m}\sigma^{\text{PL}}}}{\mathop{dv_{p,1}\dots
dv_{p,m}}}\,.$ (3.16)

When performing numerical studies, we often work with binned distributions,
_i.e._ we consider binned cross sections obtained by integrating the multi-
differential distribution over hypercubes in the observables’ space. If we
consider, for instance, the parton-level case, the cross section in any given
hyper-bin $p$ is written as

$\displaystyle\Delta\sigma^{\text{PL}}_{p}$
$\displaystyle=\int\mathop{d\mathcal{P}}\frac{\mathop{d\sigma}}{\mathop{d\mathcal{P}}}\Theta_{p}\left(\mathcal{P}\right)\,,$
(3.17)

where

$\displaystyle\Theta_{p}\left(\mathcal{P}\right)=\prod_{i=1}^{m}\theta(V_{i}(\mathcal{P})-v^{\text{min}}_{p,i})\theta(v^{\text{max}}_{p,i}-V_{i}(\mathcal{P}))\,.$
(3.18)

If we now consider a binned distribution at hadron level, the transfer
operator from parton-level bin $p$ to a given hadron-level bin $h$ becomes a
matrix of the form

$\mathcal{T}_{hp}=\frac{\int\mathop{d\mathcal{P}}\frac{\mathop{d\sigma}}{\mathop{d\mathcal{P}}}\Theta_{p}\left(\mathcal{P}\right)\Theta_{h}\left(\mathcal{H}\left(\mathcal{P}\right)\right)}{\int\mathop{d\mathcal{P}}\frac{\mathop{d\sigma}}{\mathop{d\mathcal{P}}}\Theta_{p}\left(\mathcal{P}\right)}\,,$
(3.19)

with

$\displaystyle\Theta_{h}\left(\mathcal{\mathcal{H}(\mathcal{P})}\right)=\prod_{i=1}^{m}\theta\left(V_{i}\left(\mathcal{\mathcal{H}(\mathcal{P})}\right)-v^{\text{min}}_{h,i}\right)\theta\left(v^{\text{max}}_{h,i}-V_{i}\left(\mathcal{\mathcal{H}(\mathcal{P})}\right)\right)\,.$
(3.20)

Consequently, the final hadron-level distribution in the hyper-bin $h$ is
obtained by the weighted sum of all parton-level contributions

$\mathop{\Delta\sigma_{h}^{\text{HL}}}=\sum_{p}\mathcal{T}_{hp}\mathop{\Delta\sigma_{p}^{\text{PL}}}\,.$
(3.21)

We note that the $\Theta$-functions introduced in Eqs. (3.18) and (3.20) can
also be used to represent event-selection cuts, such as the ones reported in
Tab. 1, that define the fiducial region of a measurement, at parton- and
hadron level, respectively. We further note that the parton- and hadron-level
bins do not necessarily have to be the same. For example, one would typically
define underflow and overflow bins at parton level so as to include their
contribution to the final hadron-level predictions.

The elements of the transfer matrices appearing in Eq. (3.21) can easily be
extracted from a multi-purpose generator in a single run, given that events
are accessible at different stages of the simulation process. Note however
that, while the parton shower and hadronisation are treated in a factorised
form in all multi-purpose event generators [94], this is not necessarily the
case for the UE. In particular P  YTHIA [77, 64] makes use of an interleaved
evolution of the initial-state shower and the secondary interactions [95, 96].
Accordingly, in a full event simulation within such model there is no notion
of an intermediate parton-level final state that is directly comparable to a
resummed calculation. However, in S  HERPA the parton showers off the hard
process and the simulation of multiple-parton interactions are fully
separated, _i.e._ the UE is simulated only after the shower evolution of the
hard interaction is completed. The secondary scatterings then get showered and
ultimately the partonic final state consisting of the showered hard process
and multiple-parton interactions gets hadronised. In what follows we therefore
determine the transfer matrices using the S  HERPA generator.

Of course, the matrix $\mathcal{T}_{hp}$ depends on the model or the specific
generator used to derive them. However, we argue that a milder dependence than
with the naive parton-to-hadron level ratios we previously adopted can be
expected. As the matrices are based on the conditional probabilities for
transitions between bins, _e.g._ in $p_{T,\text{jet}}$, they only depend on
the parton shower insofar as it determines what exact phase space those
probabilities are averaged over.

It is interesting to note that, the task of correcting PL predictions for NP
effects closely resembles the situation of accounting for detector effects in
an experimental analysis. For this purpose proposals have recently been made
(see for example Refs. [97, 98, 99, 100, 101]) to use machine-learning
techniques to accomplish such a mapping for either binned or unbinned
distributions. One can envisage that similar methods can be applied for the
problem of non-perturbative corrections, possibly even in an invertible way,
relating the observed hadronic final state to an underlying parton-level
prediction.§§§In practice, treating non-perturbative corrections in an
invertible way may be complicated by the fact that the mapping between partons
and hadrons is not one-to-one. There exist however approaches allowing for
maps which are many-to-many, such as the one using conditional invertible
networks (see e.g. Ref. [99]).

####  3.3.2 Transfer matrices for jet-angularity observables

For our study of jet angularities we follow the above strategy to determine
$\mathcal{T}_{hp}$ from MC simulations with S  HERPA , and use them to correct
our binned perturbative $\text{NLO}+\text{NLL}^{\prime}$ predictions. To this
end we employ the R  IVET analysis tool [102, 103], which allows us to
directly access the HepMC [104, 105] event record and extract the intermediate
truth-level information on the event evolution after parton showering, prior
to UE and hadronisation. This enables us to consistently derive the desired
migration matrices for all angularity observables (with and without grooming,
both jet radii, and based on all/charged hadrons) for a particular UE
parameter set and hadronisation model in a single generator run for $Z$+jet
and dijet production, respectively. To assess the systematic uncertainties
related to the UE activity and hadronisation, we derive alternative transfer
matrices for the up and down variation of the
$\alpha^{\text{MPI}}_{\text{min}}$ parameter and using the string
fragmentation model, _cf._ Sec. 3.1. In total, we consider the envelope of

$(\text{model},\alpha^{\text{MPI}}_{\text{min}})=\\{(\text{cluster},0.16),(\text{string},0.16),(\text{cluster},0.08),(\text{cluster},0.24)\\}~{},$
(3.22)

using cluster hadronisation with $\alpha^{\text{MPI}}_{\text{min}}=0.16$ as
our default.

Ideally, the uncertainties on the parton-to-hadron transfer matrix should
include a contribution associated with the choice of the parton-shower cutoff
scale, especially if one then wants to use the transfer matrix with analytic
parton-level results which do not have an explicit cutoff. Varying the parton-
shower cutoff would however require a full re-tuning of the non-perturbative
parameters of the Monte Carlo, a task which is clearly beyond the scope of the
present study. In practice, our use of two hadronisation models, cluster and
string fragmentation, should at least partially include the effect of varying
the parton-shower stopping scale, at a much lower cost.

In order to illustrate our transfer-matrix method, we discuss some concrete
examples. First, we consider the change in transverse momentum for the leading
jet in $Z$+jet production. We apply the transfer matrix extracted from the SH-
MC@NLO simulation for the $Z$+jet event selection for ungroomed jets, based on
all hadrons, to our parton level (PL) $\text{NLO}+\text{NLL}^{\prime}$
resummed calculation. Thus, we consider $\vec{v}_{p}$ and $\vec{v}_{h}$ in Eq.
(3.15) as two-component arrays, with elements given by the jet transverse
momentum and the jet thrust, at parton level,
$\vec{v}_{p}=\\{p^{\text{PL}}_{T,\text{jet}},\lambda^{1,\text{PL}}_{2}\\}$,
and at hadron level,
$\vec{v}_{h}=\\{p^{\text{HL}}_{T,\text{jet}},\lambda^{1,\text{HL}}_{2}\\}$.

We first focus on the jet-$p_{T}$ distribution. Integrating out the angularity
at hadron level, our transfer-matrix approach yields the following result:

$\displaystyle\frac{\mathop{d\sigma^{\text{HL}}}}{\mathop{dp^{\text{HL}}_{T,\text{jet}}}}$
$\displaystyle=\int\mathop{d\lambda^{1,\text{HL}}_{2}}\frac{\mathop{d^{2}\sigma^{\text{HL}}}}{\mathop{dp^{\text{HL}}_{T,\text{jet}}d\lambda^{1,\text{HL}}_{2}}}$
$\displaystyle=\int\mathop{d\lambda^{1,\text{HL}}_{2}}\int\mathop{dp_{T,\text{jet}}^{\text{PL}}}\mathop{d\lambda_{2}^{\text{1,PL}}}\mathcal{T}\left(\\{p^{\text{HL}}_{T,\text{jet}},\lambda^{1,\text{HL}}_{2}\\}|\\{p_{T,\text{jet}}^{\text{PL}},\lambda_{2}^{\text{1,PL}}\\}\right)\,\frac{\mathop{d^{2}\sigma^{\text{PL}}}}{\mathop{dp_{T,\text{jet}}^{\text{PL}}d\lambda_{2}^{\text{1,PL}}}}$
$\displaystyle=\int\mathop{dp_{T,\text{jet}}^{\text{PL}}}\mathop{d\lambda_{2}^{\text{1,PL}}}\mathcal{T}\left(\\{p^{\text{HL}}_{T,\text{jet}}\\}|\\{p_{T,\text{jet}}^{\text{PL}},\lambda_{2}^{\text{1,PL}}\\}\right)\,\frac{\mathop{d^{2}\sigma^{\text{PL}}}}{\mathop{dp_{T,\text{jet}}^{\text{PL}}d\lambda_{2}^{\text{1,PL}}}}\,,$
(3.23)

where we consider binned distributions, and multiply our result for
$\lambda^{1}_{2}$ at $\text{NLO}+\text{NLL}^{\prime}$ accuracy by the matrix
form of
$\mathcal{T}(\\{p^{\text{HL}}_{T,\text{jet}}\\}|\\{p^{\text{PL}}_{T,\text{jet}},\lambda^{1,\text{PL}}_{2}\\})$.
We note that Eq. (3.23) still non-trivially depends on the assumed _double_
-differential distribution of $\sigma^{\text{PL}}$, even though in itself it
is only differential in one observable. This happens because the above
expression uses different parton-level distributions for the determination of
the transfer matrix $\mathcal{T}$, here SH-MC@NLO simulations, and for the
parton-level cross section $\sigma^{\text{PL}}$, given by the
$\text{NLO}+\text{NLL}^{\prime}$ prediction.

Figure 1: Illustration of cross-section migration between parton level and
hadron level with respect to the jet transverse momentum for the double-
differential $\text{NLO}+\text{NLL}^{\prime}$ resummed distribution of
$\lambda^{1}_{2}$ in $Z$+jet production. The used jet-radius parameter is
$R=0.4$, the employed transfer matrix corresponds to the case of _all_ hadrons
contributing at hadron level.

It should be noted that there is an additional contribution from events where
the parton level does not pass some of the event-selection cuts, like the
lowest transverse momentum bin edge but also any other cut in principle,
whereas the corresponding hadron-level event actually does pass the cuts. We
have found that this contribution is basically negligible starting from the
$p_{T,\text{jet}}\in[88-120]$ GeV bin. For the Z+jet case, we added an
underflow bin, dependent on a technically necessary cut $p_{T,\text{jet}}>30$
GeV on matrix-element level, and confirmed that this contribution is indeed
dominated by lower $p_{T,\text{jet}}$ regions. In the following, we will
consider distributions starting from $p_{T,\text{jet}}>88$ GeV and neglect
this additional contribution.

The result is illustrated in Fig. 1, where we show the set of transverse-
momentum bins measured by CMS and show the effect of bin migration from
parton- to hadron level on the respective $\text{NLO}+\text{NLL}^{\prime}$
cross section for anti-$k_{t}$ jets with $R=0.4$. For example, the hadron-
level cross section for $p_{T,\text{jet}}\in[65,88]\,\text{GeV}$ receives non-
negligible contributions from parton-level events with smaller transverse
momentum, _e.g._ through an additive contribution from the UE, as well as
higher slices, _e.g._ through the widening of jets due to hadronisation. It is
evident that these effects, which were completely ignored in our previous
study, can be rather sizeable, especially at low and moderate
$p_{T,\text{jet}}$, and are therefore important to consider in the derivation
of NP corrections. Note that we do not illustrate the flow in or out of the
underflow bin at $p_{T,\text{jet}}<50$ GeV for simplicity. The net flow over
this border appears to be negligible. In principle, a similar study could be
performed for all the other variables that are used to characterise the event
kinematics and to define the fiducial region, for example the jet rapidity.
However, we expect that their impact is much less pronounced and, therefore,
in order to keep the dimensionality of the transfer matrices as low as
possible, we have ignored them.

As second step in our assessment of NP corrections, we study their effect on
angularity distributions. While at parton level the angularities are measured
on all the partons in the jet, at hadron level they can either be measured on
all the particles or using charged particles only. To illustrate the effect of
NP corrections, we consider in Figs. 2 and 3 the LHA, $\lambda^{1}_{0.5}$,
measured on ungroomed jets, using different approximations for the
$p_{T,\text{jet}}$ migration from parton to hadron level: including only the
contribution of the transfer matrix for the same $p_{T,\text{jet}}$ bin
(dashed orange curve), including as well migration from the two neighbouring
bins (dashed red curve), and including the full transfer matrix (solid black
curve). Each time, the same approximation is used for both the numerator and
the denominator of the normalised distribution. For reference, the parton-
level result is shown by the solid blue line. Fig. 2 concentrates on a
medium-$p_{T,\text{jet}}$ bin, $p_{T,\text{jet}}\in[120,150]\,\text{GeV}$, for
both the $Z$+jet and the central dijet selection and Fig. 3 shows the
corresponding results for the highest $p_{T,\text{jet}}$ interval,
$p_{T,\text{jet}}\in[1000,4000]\,\text{GeV}$.

Figure 2: Effect of NP corrections on the $\lambda^{1}_{0.5}$ observable with
$p_{T,\text{jet}}\in[120,150]\,\text{GeV}$ for the central dijet (left) and
$Z$+jet (right) selection. Shown are the purely perturbative result (blue),
the distribution after transferring events between different observable bins
within that $p_{T,\text{jet}}$ bin (orange), after including migration from
the neighbouring $p_{T,\text{jet}}$ bins (red) and our final prediction at
hadron level (black). The uncertainty band corresponds to variations in the UE
model.

Figure 3: Same as Fig. 2 but for jets with
$p_{T,\text{jet}}\in[1000,4000]\,\text{GeV}$ in the central (left) and forward
(right) dijet selection.

In Fig. 2, we see that including only the same $p_{T,\text{jet}}$ bin already
accounts for the bulk of the non-perturbative correction, and that NP effects
cause sizeable migrations between the different angularity bins. However, we
see a non-negligible contribution of events from the neighbouring
$p_{T,\text{jet}}$ bins. The effect of $p_{T,\text{jet}}$ migration beyond the
two closest bins differs from the full correction only at large values of the
angularity. For the final prediction, using the full transfer matrix, we also
derive an uncertainty estimate, shown as a grey band, corresponding to the
variation of the UE $\alpha^{\text{MPI}}_{\text{min}}$ parameter as described
above. We observe that, despite the large overall effect of the NP
corrections, they are fairly stable under the considered variations. The
observed uncertainty bands do not fully cover up the case where we discard the
effect of $p_{T,\text{jet}}$ migration from the neighbouring bins, in
particular for large $\lambda^{1}_{0.5}$ values. However, as one might expect,
NP effects decrease when considering the higher $p_{T,\text{jet}}$ bin in Fig.
3. Despite being also much smaller at high $p_{T,\text{jet}}$, the migration
between different observable bins is still by no means negligible. The effect
of migration of events from lower $p_{T,\text{jet}}$ regions does however
appear to be very small in this case, and is mostly covered by the UE
variations.

Figure 4: Transfer matrix
$\mathcal{T}(\\{\lambda_{1}^{1,\text{HL}}\\}|\\{\lambda_{1}^{1,\text{PL}}\\})$
for the jet width, $\lambda^{1}_{1}$, for central dijet events with $R=0.8$
and $p_{T,\text{jet}}\in[120,150]$ GeV. The angularities at hadron level are
measured on charged particles only. For simplicity of the presentation, we
here assumed no $p_{T,\text{jet}}$ migration. The left (right) plot
corresponds to ungroomed (groomed) jet width.

To better illustrate the effect of hadronisation in our transfer-matrix
approach, we directly plot the transfer matrix itself in Fig. 4. We select
jets with $R=0.8$ and $p_{T,\text{jet}}\in[120,150]$ GeV in central dijet
events and measure the jet width, $\lambda_{1}^{1}$, either on ungroomed jets
(left plot) or on groomed jets (right plot). At hadron level, the jet width is
measured on charged particles only. The transfer matrix for ungroomed jets
shows clear signs of bin migration, especially at low-to-mid angularities
where the parton-level values predominantly get pushed to larger values of the
jet width at hadron level. If we instead apply grooming, the transfer matrix
appears to be much more diagonal, albeit with longer tails away from the
diagonal.¶¶¶At larger $p_{T,\text{jet}}$, not shown here, the transfer matrix
for groomed jets shows clear signs of migration from relatively large partonic
values of the angularity to very small hadron-level values. These are likely
related to subjets passing the SoftDrop condition at parton level being pushed
below the SoftDrop cut after hadronisation, an effect which is typically
included in analytic treatments of non-perturbative corrections of SoftDrop
observables [21, 23, 92].

Finally, it is interesting to compare our new method with the simpler approach
we have used in earlier studies, which consisted in accounting for NP
contributions via a bin-by-bin HL/PL ratio. In Fig. 5 we perform such
comparison for an observable known to be rather sensitive to NP contributions,
namely the groomed LHA $\lambda_{0.5}^{1}$ measured on charged tracks on the
hardest jet in $Z$+jet events [51]. We again consider the moderate
$p_{T,\text{jet}}$ region, here both for $R=0.4$ (left), and $R=0.8$ (right).
The old approach, used in [51], is shown in blue,∥∥∥Note that, for a
meaningful comparison between the two methods, both the approach based on the
HL/PL ratio and the one based on transfer matrices are derived using the S
HERPA generator with variations of the UE parameter
$\alpha^{\text{MPI}}_{\text{min}}$. while the results obtained with the new
transfer-matrix approach are shown in red. For both predictions we estimate
theoretical uncertainties, illustrated by the bands, corresponding to the
envelope of the $7$-point scale variations, the alternative $x_{L}$-parameter
settings, and variations of the $\alpha^{\text{MPI}}_{\text{min}}$ parameter
of the UE model. We see that the difference in the nominal predictions is
rather substantial, and that the new treatment of NP corrections yields a
significantly better agreement with the CMS data, shown in black, although
visible differences remain for the $R=0.8$ case. One notices that the
uncertainty estimates for the results based on the transfer-matrix approach
are somewhat larger than for the old ratio method. A source of this increase
is the larger range of kinematics probed when allowing for migration from
lower and higher $p_{T,\text{jet}}$ slices.

Figure 5: Hadron-level predictions for the groomed Les Houches Angularity
$\lambda^{1}_{0.5}$ in $Z$+jet production, measured on the charged hadrons in
$R=0.4$ (left) and $R=0.8$ (right) jets with
$p_{T,\text{jet}}\in[120,150]\,\text{GeV}$. Results are obtained based on the
$\text{NLO}+\text{NLL}^{\prime}$ perturbative predictions including NP
correction using the HL/PL ratio ($\times\text{HL}/\text{PL}$) and the new
transfer-matrix approach ($\otimes T(\text{HL}|\text{PL})$).

We conclude this section by noting that a similar procedure to account for NP
effects was employed by the ALICE collaboration in Ref. [54], where a
measurement of various angularity observables in inclusive jet production in
$pp$ collisions at $\sqrt{s}=$5.02\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$$
was reported. These data, based on SoftDrop-groomed charged-tracks jets, was
compared to analytical predictions at $\text{NLL}^{\prime}$ accuracy obtained
from SCET [48, 49]. Two techniques for correcting the analytic predictions for
NP effects and the selection of charged-particle jets were considered. The
first being based on “folding” the $\text{NLL}^{\prime}$ result with a
response matrix extracted from MC simulations that maps parton-level jets with
$(p^{\text{PL}}_{T},\lambda^{\text{PL}})$ to hadron-level jets with
$(p^{\text{HL}}_{T},\lambda^{\text{HL}})$, thereby accounting for
hadronisation corrections only. To incorporate UE effects an additional bin-
wise correction has been applied. As an alternative a NP shape-function
approach [49] to simultaneously correct for hadronisation and the UE has been
employed. We refer to [54] for additional details.

##  4 Results for jet angularities in dijet and $Z$+jet production

In this section, we present the results obtained from the calculation detailed
in section 3, _i.e_ $\text{NLO}+\text{NLL}^{\prime}$ accurate predictions
accounting for NP corrections through the transfer-matrix approach. We start
in section 4.1 with a few considerations at the purely perturbative level. In
section 4.2, we then discuss our results with NP corrections included, and
present full hadron-level predictions at NLO QCD accuracy from S  HERPA .

###  4.1 Selected parton-level results

(a) central dijets

(b) forward dijets

(c) $Z$+jet

Figure 6: Fraction of gluon jets as a function of the jet transverse momentum
$p_{T,\text{jet}}$ in dijet (left and middle) and $Z$+jet (right) processes.
For the dijet production we present results for the leading central and
forward jet, respectively. We show predictions at Born (red bands) and at one-
loop level (blue bands). Flavour tagging for the considered anti-$k_{t}$ jets
relies on the algorithm described in [51], that employs the BSZ
flavour-$k_{t}$ algorithm, and in turn defines the partonic channels $\delta$
in our implementation of NLL C  AESAR resummation in S  HERPA .

As discussed in section 3.2, the implementation of the C  AESAR resummation in
the S  HERPA plugin [56] relies on a separation into different flavour
channels $\delta$. Since we obtain fully exclusive parton-level events from C
OMIX , we can easily apply the (BSZ) flavour-clustering algorithm from Ref.
[83] so as to identify the flavour assignment of a given phase-space point. We
use this algorithm iteratively, following the procedure introduced in [51] for
the leading jet in the $Z$+jet case. This can trivially be extended to the
central/forward jet in dijet production. It allows us to talk about the
flavour of a particular anti-$k_{t}$ jet of radius $R$ in an IR-safe way. Note
that in practice we run the _bland_ variant of the algorithm whereby each jet
gets identified as either quark- or gluon-like. While flavour identification
is a necessity in the context of our matching scheme (at least at LO in the
angularity distribution), these results also provide a well defined way to
analyse the flavour decomposition of jet cross sections. Such truth-level
flavour assignment could for example be used as well-defined input, _e.g._ for
machine-learning based methods, or as benchmark for flavour-tagging algorithms
[106, 107, 108, 109, 110, 111, 112].

We start our discussion by considering the fractions of gluon and quark
anti-$k_{t}$ jets in the $Z$+jet and dijet final states. These are compiled in
Fig. 6 for the central- and forward jet in dijet events, as well as in $Z$+jet
production, as a function of the jet transverse momentum, $p_{T,\text{jet}}$.
On each plot, two curves are shown together with their respective scale
uncertainties from the $7$-point scale variation: gluon fractions at Born
level (red bands), _i.e._ $\mathcal{O}(\alpha_{\text{s}}^{2})$ for dijets and
$\mathcal{O}(\alpha_{\text{EW}}^{2}\alpha_{s})$ for $Z$+jet******Note that, as
described in Sec. 2, we always include the decay of the $Z$ boson into muons
in the calculation., and for the NLO QCD matrix element (blue bands), at
$\mathcal{O}(\alpha_{\text{s}}^{3})$ and
$\mathcal{O}(\alpha_{\text{EW}}^{2}\alpha_{\text{s}}^{2})$, respectively. Note
that the (NLO) one-loop result corresponds to the LO accuracy for jet-
angularity distributions. One sees that for low-$p_{\text{T}}$ dijet events,
both the central and forward jets have a large fraction of gluon jets ($\sim
70\%$), and that high-$p_{\text{T}}$ dijet events and $Z$+jet events at any
$p_{T,\text{jet}}$, with gluon fractions of about $20{-}30\%$, are dominated
by quarks.

A striking feature of Fig. 6 is the size of the NLO QCD corrections. For dijet
production, we see an (absolute) decrease of about 15% of the gluon fraction,
regardless of the transverse momentum and rapidity of the jet, marginally
larger than the estimated one-loop perturbative uncertainty. For $Z$+jet
events, NLO corrections cause a $2{-}10$% (absolute) increase with a clearly-
visible dependence on the jet $p_{T}$. This is in line with earlier
observations that the NLO corrections to the gluon fraction in $Z$+jet events
are both larger than in dijet events and show a stronger jet-$p_{T}$
dependence (see _e.g._ [113, 24]). It is also worth commenting on the relative
size of the theoretical uncertainty, which appears to increase when going from
Born-level to NLO QCD. The reason for this counterintuitive behaviour can be
traced back to the fact that the considered gluon fractions are determined by
cross-section ratios. At Born-level the dependence on $\alpha_{\text{s}}$, and
hence on the renormalisation scale, exactly cancels between numerator and
denominator, leaving only a rather weak dependence on the factorisation scale.

Our results can be directly compared to the corresponding gluon-jet fractions
reported by CMS in [30] (see Fig. 2 therein and the corresponding discussion),
although that study was done using a different definition of gluon jets based
on M  ADGRAPH 5_aMC@NLO +P  YTHIA simulations as well as a series of other
generators. Qualitatively, the gluon fractions presented in [30] show the same
pattern as the ones obtained here. However, with the BSZ-based approach we
find gluon fractions that are generally smaller than the ones reported by CMS.
This is especially the case in Fig. 6(c) for the $Z$+jet process where CMS
found gluon fractions reaching almost 40% at large $p_{T,\text{jet}}$. This is
in agreement with their comment that other generators predict up to 25%
smaller gluon fractions in the $Z$+jet sample, which is also where the largest
NLO corrections are observed.

Figure 7: $\text{NLO}+\text{NLL}^{\prime}$ accurate predictions for the quark-
and gluon-jet contribution to the Les Houches Angularity $\lambda^{1}_{0.5}$
for the central (left) and forward (right) dijet event selection, with jets in
the range $p_{T,\text{jet}}\in[1000,4000]~{}\text{GeV}$. The lower panels show
the gluon fractions in the respective angularity bins.

Figure 8: $\text{NLO}+\text{NLL}^{\prime}$ accurate predictions for the quark-
and gluon-jet contribution to the Les Houches Angularity $\lambda^{1}_{0.5}$
for the central dijet (left) and the $Z$+jet (right) event selections, with
jets in the range $p_{T,\text{jet}}\in[120,150]~{}\text{GeV}$. The lower
panels show the gluon fractions in the respective angularity bins.

Next, Figs. 7 and 8 show examples of our matched
$\text{NLO}+\text{NLL}^{\prime}$ results for the LHA $\lambda^{1}_{0.5}$
distribution, for different $p_{T,\text{jet}}$ and event selections. In each
case, we have separated the total cross section into contributions from quark
and gluon jets. The total result is given by the sum of both components,
indicated by the solid (black) line. The fraction of gluon jets in each bin is
shown in the lower panels. The selected $p_{T,\text{jet}}$ ranges are chosen
such that they coincide with the gluon- and quark-enriched samples studied by
CMS in [30] (see also table 2 in the following section). The results confirm
the findings from Fig. 6: we indeed see that the high-$p_{\text{T}}$ dijets
and the $Z$+jet sample for $p_{T,\text{jet}}\in[120,150]~{}\text{GeV}$ are
dominated by quark jets, and the low-$p_{\text{T}}$ dijet events are instead
dominated by gluon jets. It is also clearly visible in these figures that the
gluon distributions contribute at larger values of the angularity variable
than the quark distributions, indicative of their potential as a quark–gluon
discriminator.

###  4.2 Hadron-level results

Figure 9: $\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ and hadron-level SH-
MC@NLO predictions for the differential cross section in the Les Houches
Angularity $\lambda^{1}_{0.5}$ for ungroomed (top row) and groomed (bottom
row) $R=0.4$ anti-$k_{t}$ jets with
$p_{T,\text{jet}}\in[120,150]~{}\text{GeV}$, compared to data from CMS [30].
The left and middle panel correspond to the central and forward jet in dijet
events, respectively, the right one to the leading jet in $Z$+jet production.
The NP corrections to the perturbative $\text{NLO}+\text{NLL}^{\prime}$
prediction have been obtained with the transfer-matrix approach.

In this section we present our final
$\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ predictions, with NP corrections
implemented through the transfer-matrix approach, for jet angularities in both
dijet and $Z$+jet production and compare them to the results obtained by the
CMS collaboration. To avoid the proliferation of plots, we concentrate mostly
on the LHA $\lambda^{1}_{0.5}$ measured on (groomed or ungroomed) $R=0.4$
anti-$k_{t}$ jets at moderate transverse momentum, namely
$p_{T,\text{jet}}\in[120,150]~{}\text{GeV}$. As before, we consider separately
the central and forward jet in dijet events, as well as the leading jet in
$Z$+jet events, we thereby restrict to the data for _all_ hadrons.
Corresponding results for the jet width $\lambda^{1}_{1}$ and jet thrust
$\lambda^{1}_{2}$ we compile in App. A. The full set of predictions, _i.e._
for all the $p_{T,\text{jet}}$ slices considered by CMS in [30], for jet
radius $R=0.8$, as well as based on charged tracks, and their comparison to
the data from CMS are also publicly available [114]††††††Our results are also
attached to this paper as supplementary material.. To estimate the theoretical
uncertainty of our $\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ predictions we
consider $7$-point variations of the factorisation and renormalisation scales,
as well as variations of the $x_{L}$ parameter, see section 3.2 for details.
Besides the NP corrections corresponding to the default parameter set in S
HERPA , we also consider up- and down variations of the UE activity and use
Lund string fragmentation as alternative hadronisation model, _cf._ section
3.1. This way we derive four alternative hadron-level distributions for _each_
combination of scale- and $x_{L}$-variations. The final uncertainty is then
obtained by taking the envelope of all those variations.

Alongside the $\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ results we present
hadron-level simulations with S  HERPA at NLO QCD accuracy, as described in
section 3.1. The perturbative uncertainty of the SH-MC@NLO predictions are
taken as the envelope of $7$-point variations of $\mu_{\text{F}}$ and
$\mu_{\text{R}}$ in the matrix elements and the parton shower.

In Fig. 9 we present $\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ and SH-MC@NLO
predictions for normalised differential cross sections in $\lambda^{1}_{0.5}$
and compare them with the CMS experimental data. The top row of plots thereby
corresponds to ungroomed jets, while the bottom row ones are obtained with
SoftDrop ($\beta=0,\,z_{\text{cut}}=0.1$) applied to the jets prior to the
observable evaluation. The left-hand plots correspond to LHA measurements on
the central jet, the middle ones on the forward jet in dijet events,
respectively, and the right-hand ones on the leading jet in $Z$+jet
production.

Overall, our resummed and matched predictions when corrected of NP effects
(shown in red) provide a good description of the hadron-level data, with the
exception of the last (and in some cases the second to last) bin at large
values of the angularity. The corresponding region of phase space is outside
the jurisdiction of the all-orders calculation and one might have hoped that
it would be well-described by the NLO contribution. However, the last bin
contains the kinematic endpoint of the fixed-order calculation, accordingly,
this part of the distribution is very sensitive to the effect of multiple
emissions. Indeed the SH-MC@NLO predictions (in blue) are able to populate
this region of phase space through additional parton-shower real radiation,
resulting in a better description of the data. In the future, it would be
interesting to see if higher-order, _e.g._ NNLO, corrections will yield an
improved description in this large-angularity region. For the groomed
distribution, we also see that the MC simulation provides a better description
of the peak region than the analytic calculation. One should however bear in
mind that all the $\lambda_{0.5}^{1}$ bins, except the lowest one, are in a
region with $\lambda_{0.5}^{1}\geq z_{\text{cut}}$, _i.e._ not directly
affected by SoftDrop at NLL accuracy. In this region, non-trivial subleading
effects can have a sizeable impact (see also Refs. [91, 115] for discussions
and potential improvements).

A crucial difference between the resummed and the SH-MC@NLO results is the
size of the theoretical uncertainty, which is much larger for the
$\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ calculation. We have investigated
this feature by decomposing the total uncertainty into its various
perturbative and non-perturbative contributions. This clearly led us to the
conclusion that this effect is dominated by the resummation-scale variation,
_i.e._ the variation of $x_{L}$ in Eq. (3.14). A systematic reduction of this
uncertainty would require to improve the accuracy of the resummation, _i.e._
to include NNLL contributions. We will comment on the feasibility of this
calculation in the conclusions. We furthermore note that in the here shown MC
simulations no corresponding variation is performed (see section 3.1). It
would be interesting to study systematic variations when using a different
shower model, _e.g._ the DIRE cascade [116] available in the S  HERPA
framework.

Figure 10: Results for the mean value of the Les Houches Angularity
$\langle\lambda^{1}_{0.5}\rangle$ for ungroomed (top row) and groomed (bottom
row) $R=0.4$ anti-$k_{t}$ jets as a function of $p_{T,\text{jet}}$. The left
and middle panel correspond to the central and forward jet in dijet events,
respectively, the right one to the leading jet in $Z$+jet production.

Following the analysis performed by the CMS collaboration in Ref. [30] we also
consider the mean value of the angularity distributions, as a function of
$p_{T,\text{jet}}$. Corresponding results for the case of the
$\lambda^{1}_{0.5}$ angularity are shown in Fig. 10. The layout is organised
as in Fig. 9 with ungroomed jets in the top row and SoftDrop-groomed jets in
the bottom row and with, from left to right, the central and forward jet in
dijet events and the leading jet in $Z$+jet events. Analogous results for the
jet-width and jet-thrust angularities are collected in App. A.

The comparison between the resummed and the MC predictions highlights the
feature previously discussed, namely that the estimated uncertainties of the
$\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ results are significantly larger
than the ones obtained for the SH-MC@NLO simulations, with the former being
dominated by the $x_{L}$ variation. We observe that both theoretical
predictions consistently underestimate the experimentally measured mean
values. From the ratio plots in the bottom panels we can read off that the
theory-to-data ratios for the central values are almost constant, exhibiting
only a very mild dependence on the transverse momentum. This holds in
particular for the SH-MC@NLO predictions that undershoot the data by about
$5-10\%$. Larger deviations, reaching up to 18% at the highest
$p_{T,\text{jet}}$ values, are seen for the
$\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ results in the ungroomed case. In
the groomed case, although our $\text{NLO}+\text{NLL}^{\prime}+\text{NP}$
predictions systematically undershoot the data, we observe agreement, within
the theoretical uncertainties. In Ref. [30], similar results have been found
for the LO MC generators considered there. Part of this effect can be
explained by the observed underestimation of large angularity values in the
theoretical predictions when comparing to data. However, in this region the
cross section is quite low, thus it contributes rather little to the average
value $\langle\lambda^{1}_{0.5}\rangle$. As we have seen in section 3.3, the
angularity distribution is significantly affected by NP corrections.
Accordingly, it would clearly be interesting to include the new data on the
jet angularities in tunes of the NP models, thereby further investigating
which parameters affect their description. In turn, improved NP transfer
matrices could be derived and applied to the $\text{NLO}+\text{NLL}^{\prime}$
predictions.

Detailed studies of jet angularities allow us to quantitatively assess how
well MC simulations, and more generally theoretical calculations, describe the
particle distribution inside jets. In turn, they offer potential to analyse
the description of QCD radiation off quarks and gluons, separately. This issue
was, for instance, studied in Refs. [32, 33]. The measurement of CMS presented
in [30] now allows us to test theoretical predictions on jets from gluon-
enriched and quark-enriched samples. For this purpose, the analysis identified
five interesting phase-space regions and parameter choices, detailed in table
2, that can be classified as having an enhanced gluon- or quark-jet
contribution in the dijet or $Z$+jet process, respectively. We refer to
section 4.1 and Fig. 6 in particular for a discussion on flavour fractions in
these specific channels.

configuration | type of jet |  $p_{T,\text{jet}}$ [GeV] |  $g$-enriched |  $q$-enriched
---|---|---|---|---
(1) | ungroomed $R=0.4$ | [120,150] | dijet central |  $Z$+jet
(2) | ungroomed $R=0.4$ | [1000,4000] | dijet central | dijet forward
(3) | ungroomed $R=0.8$ | [120,150] | dijet central |  $Z$+jet
(4) | ungroomed $R=0.4$ (tracks only) | [120,150] | dijet central |  $Z$+jet
(5) |  SoftDrop ($\beta=0$, $z_{\text{cut}}=0.1$) $R=0.4$ | [120,150] | dijet central |  $Z$+jet
Table 2: Configurations selected in Ref. [30] to test theory predictions for
gluon-enriched and quark-enriched samples. Figure 11: Jet-angularity mean
values for the phase-space regions enriched with gluon- and quark jets in
dijet and $Z$+jet production, respectively. See Tab. 2 for details on the
configurations and the respective process considered as $g$\- and
$q$-enriched.

For these five configurations we consider the three IRC safe angularities
$\lambda^{1}_{\alpha}$ with $\alpha\in\\{0.5,1,2\\}$, _i.e._ the LHA, jet
width, and jet thrust. We present in Fig. 11 our
$\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ and SH-MC@NLO predictions for the
mean values $\langle\lambda^{1}_{\alpha}\rangle$ of the five configurations,
separately for the gluon- (upper row) and quark enriched (bottom row) samples
and compare them with the corresponding experimental results. Both theoretical
approaches yield results that are in fair agreement with the data. The
deviations of the central values from the measured ones are in fact rather
similar for all three observables, and for all the configurations under
consideration. Also, we do not observe a qualitative difference in the
description of the $g$\- or $q$-enriched samples. Note that, both in the data
and in the predictions, we observe smaller jet angularity mean values for the
$q$-enriched samples compared to the $g$-enriched case. This is theoretically
anticipated given that gluons carry more colour charge and accordingly radiate
more. For the high-$p_{T}$ configuration (2) this effect is, as expected,
rather marginal, since the fractions of gluon jets are relatively similar in
the central and forward cases. Our theoretical calculations nicely capture
this effect even quantitatively. As seen for the LHA already, the theoretical
uncertainties on the $\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ predictions
appear to be much larger than the SH-MC@NLO ones, motivating to consider the
evaluation of the NNLL, and eventually NNLO, corrections. From these
considerations we can conclude that for the calculations considered here,
_i.e._ $\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ and SH-MC@NLO, QCD radiation
off both hard quarks and hard gluons is well-modelled. This is interesting as
it was noted before [32, 33] that general-purpose MC event generators do not
always agree in their description of QCD radiation off gluons, while they
largely do for radiation off quarks, heavily constrained by tunes on LEP data.
However, it has to be noted that, although the considered samples are
certainly $q$\- and $g$-enriched, they still contain significant contributions
from the respective other flavour channel, _cf._ Fig. 6.

Figure 12: Ratio of jet-angularity mean values for the gluon- and quark-
enriched samples in the five phase-space regions detailed in Tab. 2. The
results obtained from the $\text{NLO}+\text{NLL}^{\prime}+\text{NP}$
calculation (left) and the SH-MC@NLO simulation (right) here get divided by
the respective value reported by CMS [30].

To further study the quality in the modelling of gluon and quark jets, we next
consider ratios of the angularity mean values in the $g$\- and $q$-enriched
samples for the five phase-space selections. In Fig. 12 we present our
corresponding theoretical predictions, normalised to the result reported by
CMS. While the left-hand plot contains the results for the
$\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ calculation, the right-hand plot
shows corresponding predictions obtained from SH-MC@NLO. For both
calculational methods the central values agree remarkably well with the
measured ones _i.e._ have a ratio to data centred close to unity. For the SH-
MC@NLO simulation we observe the largest deviation, about $10\%$, for the jet-
thrust angularity for the case of SoftDrop-groomed jets of moderate-$p_{T}$,
see also Figs. 15 and 16 in App. A.

This good description reflects the fact that, despite of the notorious
underestimation of the angularity mean values seen in Fig. 10, both our
calculational approaches seem to treat quark and gluon jets similarly well,
and do not introduce an artificially-enhanced discriminative power between
both hypotheses. This is to some degree in contrast to the generator
predictions studied by CMS in [30]. There, in particular for the
low-$p_{T,\text{jet}}$ regions, predictions based on the Born matrix elements
for $Z$+jet and dijet production overestimated the difference between gluon
and quark jets by up to $20-30\%$. This might to some extent originate from
the different gluon- and quark-jet decomposition they predict for the various
phase-space regions (see Fig. 6 and the discussion of flavour fractions at
Born and NLO QCD accuracy). We finally point out that, as noted before, the
uncertainty estimate for the $\text{NLO}+\text{NLL}^{\prime}+\text{NP}$
predictions is quite large, in particular for jet thrust, $\lambda^{1}_{2}$.

##  5 Conclusions

Following a recent measurement accomplished by the CMS collaboration [30], we
have performed a detailed phenomenological analysis of IRC safe jet
angularities, considering both ungroomed and SoftDrop jets. Our theoretical
predictions account for all-order effects at next-to-leading logarithmic
accuracy, including non-global logarithms, and matching to next-to-leading
order QCD calculations. Thanks to a flavour-sensitive matching procedure,
based on the BSZ flavour-$k_{t}$ algorithm, we are able to reach
$\text{NLO}+\text{NLL}^{\prime}$ accuracy. Furthermore, our predictions are
supplemented by non-perturbative corrections, that account for hadronisation
effects and the Underlying Event.

With respect to our previous work, see _e.g._ [51], our theoretical
predictions feature two improvements. First of all, on the perturbative side,
we now include resummed and matched calculations for dijet events, in addition
to the ones for $Z$+jet production recently presented in [51] and confronted
with experimental data in [30] already. The resummation for dijet final states
features more complicated colour structures, for both the global and the non-
global part, which is dealt with by our largely-automated implementation of
colour matrices in the S  HERPA resummation framework [56, 80].

Secondly, we employ a more sophisticated approach for dealing with non-
perturbative corrections that we extract from fully exclusive hadron-level
simulations at MC@NLO accuracy with the general-purpose event generator S
HERPA . Rather than relying on a simple bin-by-bin rescaling, we have
developed and implemented a rather general transfer-matrix approach that can
account for migrations across different kinematical variable and observable
bins. In the present case, we account for migrations both in the jet
transverse momentum and in the angularity observable when going from parton-
to hadron level. We thereby largely reduce the sensitivity to differences in
the perturbative predictions from the analytic resummation and the employed MC
simulations. This cures deficiencies of our previous approach, that showed
pathological features for observable values corresponding to scales below the
parton-shower cutoff.

We compare our final $\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ results to a
plethora of distributions measured by CMS, both in dijet and $Z$+jet
production. We thereby consider differential cross sections for the LHA, jet
width and thrust, as well as their respective mean values measured as a
function of the jet transverse momentum. For comparison, we also include
results from S  HERPA at MC@NLO accuracy. Overall, we find a fair agreement
between the data and the theoretical predictions. Our results however tend to
systematically underestimate the mean values measured by CMS. With respect to
the comparisons presented in Ref. [30] for the $Z$+jet selection, we clearly
achieve an improved description of the data, thanks to the new transfer-matrix
method used to include non-perturbative corrections. Finally, still following
the CMS analysis, we compare our theoretical predictions to the measurements
performed on quark- and gluon-enriched samples. We find that both the resummed
calculation and the SH-MC@NLO prediction are able to describe the data well.
This is interesting because jet angularities have been proposed as IRC-safe
quark/gluon taggers [38, 39] that can be applied not only in the context of
searches for new physics but also to Standard Model measurements that aim for
extractions of fundamental parameters, such as the strong coupling, or parton
distribution functions. The better description of the data that we find
originates to some degree from a more accurate modelling of the flavour
fractions, _i.e._ the relative contributions of partonic channels, that
receive sizeable NLO QCD corrections, included in both our theoretical
predictions.

Throughout this paper, we have noted that our resummed calculations suffer
from rather large theoretical uncertainties, dominated by variations of the
resummation scale. An obvious way to systematically improve on this is to
promote the resummed calculation to NNLL accuracy. Many of the contributions
that are relevant for NNLL resummation of jet shapes have already been
computed [44], predominately using SCET techniques, and have reached a
considerable degree of automation, see _e.g._ [117, 118, 119]. Furthermore,
first calculations for groomed observables exist even at N3LL [25] and with an
improved description of the transition between groomed and ungroomed regimes
[115]. The resummation framework we are employing in our plugin also has
already been extended, for global observables, to NNLL [120]. The bottleneck
in this enterprise is the inclusion of subleading non-global logarithms. The
first resummation of these effects has been achieved very recently [121, 122]
and we look forward to investigating whether this method can be easily
interfaced with our framework, in order to perform higher-precision
phenomenology of jet angularities.

Another obvious improvement of our calculation would be to upgrade the fixed-
order component of the angularity distributions to NNLO accuracy. This
requires computing the hadronic production of three QCD partons, respectively
$Z$+2 partons, at two-loop accuracy. In this context, we note that the first
NNLO evaluation of three-jet observables has recently been presented [123].
Finally, the two-loop corrections to the dijet and $Z$+jet processes (see
[124, 125] and references therein) are needed to achieve NNLL′ accuracy.

In the context of assessing the impact of NP corrections, it would be
interesting to compare the transfer-matrix approach developed here to results
obtained using first-principle field-theoretical arguments (see for instance
Refs. [92, 93] for recent work on hadronisation corrections for SoftDrop
jets). In a more generic context, it would be interesting to see the impact
that the measurement of jet angularities has on tuning NP parameters of
general-purpose Monte Carlo generators. One could then also provide
independent tunes for different parton-shower cutoffs. This could be, in turn,
added to our transfer-matrix approach of NP corrections to parton-level
analytic calculations. We close by noting that it would clearly be interesting
to apply our calculational methods to jets arising from different production
modes, _e.g._ involving top-quarks [55, 53], or at different centre-of-mass
energies [54]. We leave this for future work.

## Acknowledgements

We would like to thank Andreas Hinzmann for fruitful discussions and comments
on the manuscript. We are grateful to Vincent Theeuwes for collaboration at an
early stage of the project.

This work is supported by funding from the European Union’s Horizon 2020
research and innovation programme as part of the Marie Skłodowska-Curie
Innovative Training Network MCnetITN3 (grant agreement no. 722104). SS and DR
acknowledge support from BMBF (contracts 05H18MGCA1 and 05H21MGCAB). SS
acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) - project number 456104544. The work of SC, OF and SM 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. We also acknowledge
support from the Royal Society through the project 580986 “Resum(e) The Path
To Discovery”. SC would like to thank the Institute for Particle Physics
Phenomenology (Durham University) for hospitality during the course of this
work.

##  Appendix A Results for jet width and jet thrust

Figure 13: Results for the jet width $\lambda^{1}_{1}$ for ungroomed (top row)
and groomed (bottom row) $R=0.4$ anti-$k_{t}$ jets with
$p_{T,\text{jet}}\in[120,150]~{}\text{GeV}$. The left and middle panel
correspond to the central and forward jet in dijet events, respectively, the
right one to the leading jet in $Z$+jet production.

Figure 14: Mean values of jet width, _i.e._ $\langle\lambda^{1}_{1}\rangle$,
for ungroomed (top row) and groomed (bottom row) $R=0.4$ anti-$k_{t}$ jets in
dependence of $p_{T,\text{jet}}$. The left and middle panel correspond to the
central and forward jet in dijet events, respectively, the right one to the
leading jet in $Z$+jet production.

In this appendix, we collect additional results obtained with our
$\text{NLO}+\text{NLL}^{\prime}+\text{NP}$ calculation and SH-MC@NLO for the
jet-width ($\lambda_{1}^{1}$) and jet-thrust ($\lambda_{2}^{1}$) angularities,
for anti-$k_{t}$, $R=0.4$ jets. In Fig. 13 we show the jet-width distributions
for $p_{T,\text{jet}}\in[120,150]~{}\text{GeV}$, for the central and forward
jet in dijet events, as well as the leading jet in $Z$+jet production, for
ungroomed and SoftDrop jets. In Fig. 14 we compile results for the mean value
of the jet-width distributions as a function of the jet transverse momentum.
Figs. 15 and 16 show analogous results for the jet-thrust observable.

Figure 15: Same as Fig. 13 but for jet thrust $\lambda^{1}_{2}$.

Figure 16: Same as Fig. 14 but for the mean values of jet thrust, _i.e._
$\langle\lambda^{1}_{2}\rangle$.

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