11institutetext: Laboratoire de Physique des Rayonnements et de leurs
Intéractions avec la Matière,
Département de Physique, Faculté des Sciences de la Matière,
Université de Batna–1, Batna, Algeria22institutetext: Department of Physics,
Faculty of Science, Islamic University of Madinah,
Madinah 42351, Saudi Arabia

# Jet mass distribution in Higgs/vector boson + jet events at hadron colliders
with ${\boldsymbol{k}}_{\boldsymbol{t}}$ clustering

Naïma Ziani 11<EMAIL_ADDRESS>Kamel Khelifa-Kerfa 22
<EMAIL_ADDRESS>Yazid Delenda  (corresponding author)11
<EMAIL_ADDRESS>
###### Abstract

We address the issues of clustering and non-global logarithms for jet shapes
in the process of production of a Higgs/vector boson associated with a single
hard jet at hadron colliders. We perform an analytical fixed-order calculation
up to second order in the coupling as well as an all-orders estimation for the
specific invariant mass distribution of the highest-$p_{t}$ jet, for various
jet algorithms. Our results are derived in the eikonal (soft) limit and are
valid up to next-to-leading logarithmic accuracy. We perform a matching of the
resummed distribution to next-to-leading order results from MCFM and compare
our findings with the outputs of the Monte Carlo event generators Pythia 8 and
Herwig 7. After accounting for non-perturbative effects we compare our results
with available experimental data from the CMS collaboration for the $Z$ \+ jet
production. We find good agreement over a wide range of the observable.

###### Keywords:

QCD, Jets, Hadron collisions, Resummation

###### pacs:

12.38.Bx and 12.38.Cy and 13.87.-Ce and 13.85.Fb

##  1 Introduction

The invariant mass of a jet is a typical example of a jet shape that plays an
important role in the study of the substructure of jets, testing QCD, and
identifying new-physics signals. Being sensitive to soft and/or collinear
emissions from the parton initiating the jet and from the other incoming and
outgoing partons, this observable provides an indispensable mean for probing
various aspects that are relevant to achieving better accuracy in QCD
calculations. Examples of such aspects include, on the non-perturbative side,
hadronisation corrections, underlying event, pile-up interactions, and on the
perturbative side, initial and final-state radiation, colour flow, resummation
of large logarithms, etc. Analytical calculations for these aspects pave the
way for a deeper insight into QCD processes, a better control of theoretical
uncertainties, and a precise quantification of missing higher-order
contributions and their significance, all of which are issues not very clear
in Monte Carlo event generators.

In this paper we shed light on the resummation of large logarithms that arise
due to a miscancellation of soft and collinear singularities between real
emissions and their corresponding virtual corrections. The convergence of the
perturbative series, in the invariant jet mass ($m_{j}$) distribution, is
spoiled by the presence of large logarithms in the ratio of the jet mass and
its transverse momentum $p_{t}$, $L=\ln(m_{j}/p_{t})$, at each order in
perturbation theory. In the exponent of the integrated distribution, these
logarithms take the form $\alpha_{s}^{n}L^{m}$, with $\alpha_{s}$ being the
strong coupling constant and $m\leq(n+1)$, and thus they require an all-orders
resummation. A next-to-leading logarithmic (NLL) resummation ensures that all
single logarithms of the form $\alpha_{s}^{n}L^{n}$ are resummed, in addition
to the leading (double) logarithms (LL) $\alpha_{s}^{n}L^{n+1}$.

The jet mass is a non-global observable, i.e., an exclusive observable that is
sensitive only to gluon emissions which end up inside the jet. To ensure a
proper NLL resummation then its distribution must carefully be treated for a
class of large single logarithms known as non-global logarithms (NGLs), which
are related to secondary non-Abelian emissions of soft gluons Dasgupta:2001sh
; Dasgupta:2002bw . Furthermore, another type of large single logarithms known
as clustering logarithms (CLs) Banfi:2005gj ; Delenda:2006nf , related to
primary gluon emissions off the hard Born configuration, needs to be resummed
when the jets are reconstructed using jet algorithms such as $k_{t}$
Catani:1993hr ; Ellis:1993tq  and Cambridge-Aachen (C-A) Dokshitzer:1997in ;
Wobisch:1998wt . The anti-$k_{t}$ clustering algorithm Cacciari:2008gp  is
known to cause no CLs (see for instance refs. Banfi:2010pa ; Dasgupta:2012hg
). The full resummation of both NGLs and CLs has thus far proven to be a
formidable challenge. The resummation of NGLs is usually achieved numerically
via a Monte Carlo approach Dasgupta:2001sh ; Dasgupta:2002bw  in the
large-$\mathrm{N_{c}}$ limit ($\mathrm{N_{c}}$ being the number of quark
colours), though full-colour numerical resummation has been provided in refs.
Hatta:2013iba ; Hagiwara:2015bia  based on an analogy between small-$x$ BFKL
resummation in Regge scattering and the Weigert equation Weigert:2003mm .
Additionally, NGLs may also be resummed via an evolution equation known as the
Banfi-Marchesini-Smye (BMS) equation Banfi:2002hw  valid at large
$\mathrm{N_{c}}$.

In this work we study the jet mass distribution in the process of production
of a single jet associated with a vector boson ($\gamma$, $Z$ or $W$) or a
Higgs boson $H$ at the Large Hadron Collider (LHC). In ref. Dasgupta:2012hg ,
the jet mass distribution was calculated at NLL accuracy combined with next-
to-leading order (NLO) results in $Z$ \+ jet and di-jet processes at hadron
colliders, 111Note that our convention for LO, NLO, etc, is different from
that in ref. Dasgupta:2012hg . In our convention, the LO differential
distribution is proportional a delta function. for jets defined with the
anti-$k_{t}$ clustering algorithm. The NGLs were computed therein analytically
at fixed order (at $\mathcal{O}(\alpha_{s}^{2})$) and numerically to all
orders in the large-$\mathrm{N_{c}}$ approximation. In the context of soft-
collinear effective theory the jet mass distribution was also studied in ref.
Liu:2014oog  for di-jet events, in ref. Chien:2012ur  for $\gamma$ +jet
events, and in ref. Jouttenus:2013hs  for $H$ \+ jet events. We elaborate
herein on the work of ref. Dasgupta:2012hg  by considering the jet mass
distribution when jets are reconstructed using $k_{t}$ or C-A clustering
algorithms. We additionally consider other vector bosons, namely $\gamma$ and
$W$, as well as Higgs boson + jet production processes. On the experimental
side, the jet mass distribution in $W/Z$ \+ jet events at the LHC was studied
by the CMS collaboration Chatrchyan:2013rla , where the jets were
reconstructed using various jet algorithms. Additional jet substructure
techniques such as trimming, filtering, and pruning, were also addressed in
the same work Chatrchyan:2013rla . We do not address these techniques in the
present paper.

We compute NGLs and CLs at fixed order, specifically at
$\mathcal{O}(\alpha_{s}^{2})$ where they first appear, for the invariant mass
distribution of the highest-$p_{t}$ jet. We provide results for the following
three jet algorithms: $k_{t}$, C-A and anti-$k_{t}$, where we note that for
the latter algorithm NGLs were first computed in ref. Dasgupta:2012hg  and
that CLs are absent. Moreover, we approximate the all-orders resummed CLs and
NGLs by an exponential of the $\mathcal{O}(\alpha_{s}^{2})$ result in the case
of $k_{t}$ and C-A algorithms. This is justified by the fact that for the
anti-$k_{t}$ algorithm the said exponential approximates the all-orders
numerical result very well as we shall demonstrate. 222While the all-orders
numerical resummation of NGLs for the anti-$k_{t}$ algorithm may be computed
using the Monte Carlo code of ref. Dasgupta:2001sh , as was done in ref.
Dasgupta:2012hg , we found that this code produces unreliable results for some
dipoles in the case of $k_{t}$ clustering. Note that the C-A algorithm is not
implemented in the code of ref. Dasgupta:2001sh . We then compare the NLL-
resummed and NLO-matched result for the jet mass distribution, which includes
the resummed global and non-global (NGLs and CLs) form factors convoluted with
the Born cross-section and corrected for NLO effects for each of the four
$V/H$ \+ jet processes, with results from Pythia 8 Sjostrand:2014zea  and
Herwig 7 Bahr:2008pv ; Bellm:2015jjp  parton showers. Finally, we estimate the
non-perturbative corrections to this distribution and compare our predictions
with experimental data from the CMS collaboration Chatrchyan:2013rla  for the
jet mass distribution in $Z$ \+ jet events at the LHC.

This paper is organised as follows. In section 2 we discuss kinematics of the
processes under consideration and define our observable. We calculate, in
section 3, the distribution of the jet mass at leading order and construct the
resummed global form factor up to NLL accuracy in the exponent. In section 4
we compute the leading CLs at $\mathcal{O}(\alpha_{s}^{2})$ for both $k_{t}$
and C-A clustering algorithms, which happen to give identical results at this
particular order. We also calculate NGLs at $\mathcal{O}(\alpha_{s}^{2})$ for
the aforementioned jet algorithms in addition to the anti-$k_{t}$. We are then
able to assess the impact of the various clustering algorithms on NGLs. In
section 5 we discuss the all-orders resummation of NGLs and CLs. In section 6
we compare our NLL-resummed result including NLO corrections for the jet mass
distribution with the outputs of Pythia 8 and Herwig 7 parton showers. In
section 7 we estimate the non-perturbative corrections, which include
hadronisation corrections and the underlying event, on the distribution, and
compare our results with the experimental data. Finally, in section 8, we draw
our conclusions.

##  2 Setup

###  2.1 Processes and kinematics

In this paper we are interested in the calculation of both CLs and NGLs at
single logarithmic accuracy, for the jet mass distribution in the process of
production of a single jet associated with a vector ($W/Z/\gamma$) or Higgs
boson at hadron colliders. For this purpose, it suffices to consider the
eikonal (soft) approximation in the squared matrix elements for the emission
of gluons. The emitted gluons are assumed to be strongly ordered in transverse
momenta, i.e., $k_{tn}\ll\cdots\ll k_{t2}\ll k_{t1}\ll p_{t}$, where $k_{ti}$
is the transverse momentum of the $i^{\mathrm{th}}$ emission and $p_{t}$ is
that of the outgoing hard jet. The latter ordering simplifies the calculations
of the emission amplitudes while being sufficient for capturing the single
logarithmic CLs and NGLs.

For a vector boson + one jet production in hadron collisions, there are three
partonic channels that contribute to the Born process, namely: $q\bar{q}\to
g$, $qg\to q$, and $\bar{q}g\to\bar{q}$. For $W^{\pm}$ production, flavour
changing needs to be taken into account at the Born level, but this does not
affect the QCD structure of initial and final-state radiation. As for the
Higgs + one jet process there are four partonic channels to be considered.
These are: $qg\to qH$, $\bar{q}g\to\bar{q}H$, $q\bar{q}\to Hg$, and $gg\to
Hg$. As far as QCD calculations are concerned all mentioned channels, whether
for Higgs or vector boson production, are in fact identical as they all
involve three hard coloured (QCD) partons and a colour-neutral boson. This
means that the resummation of the jet mass distribution is essentially
identical in all of the said channels, with differences pertaining to just the
Born cross-section and the associated colour factors for the various channels.
We note that the relevant total cross-sections have been calculated up to
next-to-next-to-leading order (NNLO): Higgs + jet in refs. Boughezal:2015dra ;
Boughezal:2015aha ; Caola:2015wna ; Chen:2016zka , $Z$ \+ jet in refs.
Ridder:2015dxa ; Boughezal:2015ded , $W$ \+ jet in ref. Boughezal:2015dva ,
and $\gamma$ \+ jet in ref. Campbell:2016lzl .

In the current work, we henceforth consider the three partonic channels shown
in figure 1: $(\delta_{1}):q\bar{q}\to g+X$, $(\delta_{2}):qg\to q+X$, and
$(\delta_{3}):gg\to g+X$, 333For our QCD calculations, the channel
$\bar{q}g\to\bar{q}$ is in fact identical to $qg\to q$. where $X$ refers to
the colour-neutral boson ($\gamma$, $Z$, $W^{\pm}$ or $H$).

Figure 1: Partonic processes considered in this paper.

We label the incoming partons with $(a)$ and $(b)$ and the outgoing parton
initiating the hard jet with $(j)$. The four-momenta of the three hard Born
partons and the emitted soft gluons are given by

$\displaystyle p_{a}$
$\displaystyle=x_{a}\,\frac{\sqrt{s}}{2}\left(1,0,0,1\right),$ (1a)
$\displaystyle p_{b}$
$\displaystyle=x_{b}\,\frac{\sqrt{s}}{2}\left(1,0,0,-1\right),$ (1b)
$\displaystyle p_{j}$ $\displaystyle=p_{t}\left(\cosh
y,\cos\varphi,\sin\varphi,\sinh y\right),$ (1c) $\displaystyle k_{i}$
$\displaystyle=k_{ti}\left(\cosh\eta_{i},\cos\phi_{i},\sin\phi_{i},\sinh\eta_{i}\right),$
(1d)

where $\eta_{i}$ and $\phi_{i}$ are the rapidity and azimuth of the
$i^{\mathrm{th}}$ emission and $y$ and $\varphi$ are those of the outgoing
hard jet, measured with respect to the beam axis. The incoming partons $a$ and
$b$ carry momentum fractions $x_{a}$ and $x_{b}$ of the incoming protons, and
$\sqrt{s}$ is the collision centre-of-mass energy. We shall be ignoring recoil
against soft emissions throughout, as it is beyond single logarithmic
accuracy.

###  2.2 Jet mass observable and jet algorithms

We study the normalised (squared) invariant mass of the outgoing hard jet $j$
defined by

$\begin{split}\varrho&\equiv\frac{1}{p_{t}^{2}}\left(p_{j}+\sum_{i\in\mathrm{jet}}k_{i}\right)^{2}\approx\frac{2}{p_{t}^{2}}\sum_{i\in\mathrm{jet}}k_{i}\cdot
p_{j}=\sum_{i\in\mathrm{jet}}\varrho_{i}\,,\\\
\varrho_{i}&\equiv\,\frac{2\,k_{i}\cdot
p_{j}}{p_{t}^{2}}=2\,\frac{k_{ti}}{p_{t}}\left[\cosh(\eta_{i}-y)-\cos(\phi_{i}-\varphi)\right],\end{split}$
(2)

where the sum is over all emitted soft gluons which end up inside the hard jet
after the application of a jet algorithm on the final state partons. Notice
that we are considering massless quarks and that the soft approximation has
been assumed in the above equation whereby $p_{j}\cdot k_{i}\gg k_{\ell}\cdot
k_{m}$.

The $k_{t}$, C-A and anti-$k_{t}$ jet algorithms work as follows. For each
pair $(im)$ of hadrons in the final state one defines a distance

$\displaystyle
d_{im}=\min(k_{ti}^{2p},k_{tm}^{2p})\left[(\eta_{i}-\eta_{m})^{2}+(\phi_{i}-\phi_{m})^{2}\right],$
(3a) and for each single hadron a beam distance $\displaystyle
d_{i}=k_{ti}^{2p}\,R^{2}\,,$ (3b)

for some fixed jet radius parameter $R$. Here the parameter $p=1,0,-1$ for
$k_{t}$, C-A, and anti-$k_{t}$ clustering, respectively. If the smallest of
all of these distances is $d_{im}$, then particles $i$ and $m$ are combined
into a single particle with four-momentum $p_{i}+p_{m}$, whereas if the
smallest is $d_{i}$ then particle $i$ is considered as a jet and is removed
from the list of particles. This procedure is iterated until one is left only
with jets in the final state.

For the $k_{t}$ algorithm, and in the regime of strongly-ordered emissions,
the clustering of particles starts with the softest real gluon. Then, in a
given event this softest gluon is dragged towards the next-to-softest real
parton within a circle of radius $R$ in the $(\eta,\phi)$ plane. If no such
harder parton exists then this softest gluon is considered as a jet and is
removed from the list of partons. The process is then repeated until no
particles are left. When clustering two partons together, the resulting
pseudo-jet is essentially aligned along the direction of the harder, and its
four-momentum is just that of the harder parton.

For the anti-$k_{t}$ algorithm, on the other hand, clustering starts with the
hardest particle, and hence it works in an apposite way to $k_{t}$ clustering.
For the C-A algorithm, only geometric distances between partons in the
$(\eta,\phi)$ plane decide how clustering happens. Particles which are closest
to each other get clustered first.

###  2.3 Jet mass distribution

In what follows we calculate at NLL accuracy the jet mass distribution for a
given channel $\delta$, defined by (following the notation of refs.
Dasgupta:2012hg ; Banfi:2004yd )

$\frac{\mathrm{d}\Sigma_{\delta}(\rho)}{\mathrm{d}\mathcal{B}_{\delta}}=\int_{0}^{\rho}\frac{\mathrm{d}^{2}\sigma_{\delta}}{\mathrm{d}\mathcal{B}_{\delta}\,\mathrm{d}\varrho}\,\mathrm{d}\varrho\,,$
(4)

where
$\mathrm{d}^{2}\sigma_{\delta}/\mathrm{d}\mathcal{B}_{\delta}\,\mathrm{d}\varrho$
is the differential cross-section with respect to both the Born configuration
$\mathcal{B}_{\delta}$ and the jet mass observable $\varrho$. Details of the
differential Born configuration $\mathrm{d}\mathcal{B}_{\delta}$ are discussed
further in appendix A. The integrated jet mass distribution is obtained by
integrating $\mathrm{d}\Sigma_{\delta}(\rho)/\mathrm{d}\mathcal{B}_{\delta}$
over $\mathcal{B}_{\delta}$ with some chosen kinematical cuts (which we denote
by $\Xi_{\mathcal{B}}$), and summing over all Born channels. That is

$\displaystyle\Sigma(\rho)=\sum_{\delta}\int\mathrm{d}\mathcal{B}_{\delta}\,\frac{\mathrm{d}\Sigma_{\delta}(\rho)}{\mathrm{d}\mathcal{B}_{\delta}}\,\Xi_{\mathcal{B}}\,.$
(5)

Following ref. Dasgupta:2012hg , we write eq. (4) in the region $\rho\ll 1$ in
the factorised form

$\displaystyle\frac{\mathrm{d}\Sigma_{\delta}(\rho)}{\mathrm{d}\mathcal{B}_{\delta}}=\frac{\mathrm{d}\sigma_{0,\delta}}{\mathrm{d}\mathcal{B}_{\delta}}\,f_{\mathcal{B},\delta}(\rho)\,C_{\mathcal{B},\delta}(\rho)\,,$
(6)

where $\mathrm{d}\sigma_{0,\delta}/\mathrm{d}\mathcal{B}_{\delta}$ is the
differential partonic Born cross-section for channel $\delta$ (see appendix A)
and the factor $C_{\mathcal{B},\delta}(\rho)$ depends on the Born kinematics
and has the perturbative expansion

$C_{\mathcal{B},\delta}(\rho)=1+\alpha_{s}\,C_{\mathcal{B},\delta}^{(1)}(\rho)+\alpha_{s}^{2}\,C_{\mathcal{B},\delta}^{(2)}(\rho)+\cdots\,,$
(7)

where $C_{\mathcal{B},\delta}^{(n)}(\rho)$ are channel-dependent terms that
correct the resummation for non-logarithmically-enhanced terms. The
$\rho$-dependent function $f_{\mathcal{B},\delta}(\rho)$ resums all the large
logarithms. It has the form Banfi:2004yd

$f_{\mathcal{B},\delta}(\rho)=\exp\left[L\,\tilde{g}_{1}(\alpha_{s}L)+\tilde{g}_{2}(\alpha_{s}L)+\alpha_{s}\,\tilde{g}_{3}(\alpha_{s}L)+\cdots\right],$
(8)

where the function $L\,\tilde{g}_{1}$ resums the leading (double) logarithms
(LL) of the form $\alpha_{s}^{n}L^{n+1}$, $\tilde{g}_{2}$ resums next-to-
leading (single) logarithms (NLL) of the form $\alpha_{s}^{n}L^{n}$, and
$\alpha_{s}\,\tilde{g}_{3}$ resums next-to-next-to-leading logarithms (NNLL)
of the form $\alpha_{s}^{n}L^{n-1}$, and so on, with $L=\ln(R^{2}/\rho)$. The
LL function $\tilde{g}_{1}$ receives contributions from soft-collinear
emissions from the parton initiating the jet and depends on its colour Casimir
scalar. The NLL function $\tilde{g}_{2}$ receives contributions from various
sources:

  1. (a)

hard-collinear emissions from the outgoing hard parton,

  2. (b)

soft wide-angle emissions from all hard partons,

  3. (c)

starting at $\mathcal{O}(\alpha_{s}^{2})$, NGLs from soft wide-angle
correlated secondary emissions, and

  4. (d)

CLs, when jet algorithms other than anti-$k_{t}$ are implemented for jet
reconstruction, from soft wide-angle primary emissions off the hard partons.
These again appear starting from $\mathcal{O}(\alpha_{s}^{2})$.

The whole function $\tilde{g}_{1}$ and parts of $\tilde{g}_{2}$, namely
contributions (a) and (b) stated above, have been determined in ref.
Dasgupta:2012hg  for the anti-$k_{t}$ algorithm. The exact same result also
applies for the case of $k_{t}$ and C-A clustering as the effect of jet
algorithms first appears at $\mathcal{O}(\alpha_{s}^{2})$. Our task is to
determine the other two contributions to $\tilde{g}_{2}$, namely (c) NGLs and
(d) CLs, for $k_{t}$ and C-A algorithms. Before doing so, we review in the
next section the basic calculations that lead to the determination of
$\tilde{g}_{1}$ and contributions (a) and (b) of $\tilde{g}_{2}$.

##  3 One-gluon emission

###  3.1 Fixed-order calculation

In this section we compute the jet mass distribution at leading order in QCD
and present the all-orders resummed result. Our calculations are valid in the
eikonal approximation and accurate up to NLL accuracy. First, we define the
following antenna functions relevant for the squared matrix elements for the
emission of soft gluons

$\displaystyle w_{\alpha\beta}^{i}$
$\displaystyle=\frac{k_{ti}^{2}}{2}\,\frac{p_{\alpha}\cdot
p_{\beta}}{(p_{\alpha}\cdot k_{i})\,(k_{i}\cdot p_{\beta})}\,,$ (9a)
$\displaystyle\mathcal{A}_{\alpha\beta}^{ik}$
$\displaystyle=w^{i}_{\alpha\beta}\left(w^{k}_{\alpha
i}+w^{k}_{i\beta}-w^{k}_{\alpha\beta}\right).$ (9b)

It is worth noting that these antennae are purely angular functions, i.e.,
they involve no energy or momentum dependence.

Consider the process of emission of a single soft gluon off the three-hard-
legs Born configuration $(abj)$, i.e., the process $a+b\to j+k_{1}$ shown in
figure 2.

Figure 2: Feynman diagrams for the emission of one gluon off the three-hard-
partons configuration $(abj)$.

The corresponding factorised eikonal amplitude squared is given by

$\mathcal{W}_{1,\delta}^{\scriptscriptstyle\mathrm{R}}=\sum_{(i\ell)\in\Delta_{\delta}}\mathcal{C}_{i\ell}\,w_{i\ell}^{1}\,,$
(10)

with $\Delta_{\delta}=\\{(ab),(aj),(bj)\\}$ denoting the three dipoles formed
from the partons in channel $\delta$. The colour factor $\mathcal{C}_{i\ell}$
is defined as

$\mathcal{C}_{i\ell}=-2\,\mathbf{T}_{i}\cdot\mathbf{T}_{\ell}\,,$ (11)

where $\mathbf{T}_{i}$ are the generators of the $\mathrm{SU}(\mathrm{N_{c}})$
group with Casimir scalar given by
$\mathbf{T}_{i}^{2}=\mathrm{C_{F}}=(\mathrm{N_{c}^{2}}-1)/(2\,\mathrm{N_{c}})$
for quarks (and anti-quarks) and
$\mathbf{T}_{i}^{2}=\mathrm{C_{A}}=\mathrm{N_{c}}$ for gluons. Conservation of
colour implies that for our leading-order process $a+b\to j+k_{1}$ we have
Delenda:2015tbo : $\mathbf{T}_{a}+\mathbf{T}_{b}+\mathbf{T}_{j}=0$, where the
generators are taken as if all partons were incoming. Explicitly written, the
colour factors relevant to our dipoles are:
$\mathcal{C}_{q\tilde{q}}=2\,\mathrm{C_{F}}-\mathrm{C_{A}}=-1/\mathrm{N_{c}}$
and $\mathcal{C}_{qg}=\mathcal{C}_{gg}=\mathrm{C_{A}}=\mathrm{N_{c}}$.

The term $\mathcal{W}_{1,\delta}^{\scriptscriptstyle\mathrm{R}}$ is the
eikonal amplitude squared for the emission of a real soft gluon in the
partonic sub-process $\delta$. The corresponding virtual correction in the
eikonal limit is simply
$\mathcal{W}_{1,\delta}^{\scriptscriptstyle\mathrm{V}}=-\mathcal{W}_{1,\delta}^{\scriptscriptstyle\mathrm{R}}$.
Notice that we are adopting the notation used in our previous work on eikonal
amplitudes for $e^{+}e^{-}\to$ di-jet process Delenda:2015tbo . In our recent
paper Khelifa-Kerfa:2020nlc  we have generalised the latter to the case of
hadron collisions, specifically considering three-hard-legs Born processes.
The corresponding phase-space factor is given by

$\mathrm{d}\Phi_{1}=\bar{\alpha}_{s}\,\frac{\mathrm{d}\xi_{1}}{\xi_{1}}\,\mathrm{d}\eta_{1}\,\frac{\mathrm{d}\phi_{1}}{2\pi}\,,$
(12)

where $\bar{\alpha}_{s}=\alpha_{s}/\pi=g_{s}^{2}/4\pi^{2}$, $g_{s}$ is the
strong coupling, and $\xi_{1}=k_{t1}/p_{t}$. The running of the coupling is
irrelevant at one loop and only becomes important at higher orders. The full
resummation that we present later will include running-coupling effects.

Following the procedure of measurement operators (see for instance ref.
Khelifa-Kerfa:2015mma ), we write the jet mass distribution at one loop as

$f_{\mathcal{B},\delta}^{(1)}(\rho)=-\int\mathrm{d}\Phi_{1}\,\mathcal{W}_{1,\delta}^{\scriptscriptstyle\mathrm{R}}\,\Theta(\varrho_{1}-\rho)\,\Xi_{\mathrm{in}}(k_{1})\,,$
(13)

where the function $\Xi_{\mathrm{in}}(k_{1})$ ensures that the angular
integration region for gluon $k_{1}$ is such that it gets clustered to the
hard jet when the jet algorithm is applied. At this order all jet algorithms
essentially work in the same manner, and $\Xi_{\mathrm{in}}(k_{1})$ is then a
simple Heaviside step function;
$\Xi_{\mathrm{in}}(k_{1})=\Theta_{\mathrm{in}}(k_{1})=\Theta\left[R^{2}-(\eta_{1}-y)^{2}-(\phi_{1}-\varphi)^{2}\right]$.
At higher loops, as we shall see, this is not as simple.

Substituting the expression of the eikonal amplitude squared (10) into eq.
(13) we obtain

$\displaystyle f_{\mathcal{B},\delta}^{(1)}(\rho)$
$\displaystyle=-\sum_{(i\ell)\in\Delta_{\delta}}\mathcal{C}_{i\ell}\,\bar{\alpha}_{s}\int\frac{\mathrm{d}\xi_{1}}{\xi_{1}}\,\mathrm{d}\eta_{1}\,\frac{\mathrm{d}\phi_{1}}{2\pi}\,w_{i\ell}^{1}\,\times$
$\displaystyle\times\Theta(\varrho_{1}-\rho)\,\Theta[R^{2}-(\eta_{1}-y)^{2}-(\phi_{1}-\varphi)^{2}]\,,$
(14)

with the antenna function $w_{i\ell}^{1}$ for each emitting dipole in
$\Delta_{\delta}$ given by

$\displaystyle w_{ab}^{1}$ $\displaystyle=1\,,$ (15a) $\displaystyle
w_{aj}^{1}$
$\displaystyle=\frac{1}{2}\,\frac{\exp(\eta_{1}-y)}{\cosh(\eta_{1}-y)-\cos(\phi_{1}-\varphi)}\,,$
(15b) $\displaystyle w_{bj}^{1}$
$\displaystyle=\frac{1}{2}\,\frac{\exp(y-\eta_{1})}{\cosh(\eta_{1}-y)-\cos(\phi_{1}-\varphi)}\,.$
(15c)

Note that the upper limit of the $k_{t1}$ integral is the renormalisation
scale $\mu_{\mathrm{R}}=p_{t}$, which translates into an upper limit $1$ on
$\xi_{1}$. In order to perform the angular integrations we introduce the polar
variables $(r_{1},\theta_{1})$ such that

$\eta_{1}-y=R\,r_{1}\cos\theta_{1}\,,\qquad\phi_{1}-\varphi=R\,r_{1}\sin\theta_{1}\,,$
(16)

and make a change of variables in the integration such that
$\mathrm{d}\eta_{1}\,\mathrm{d}\phi_{1}=\frac{1}{2}\,R^{2}\,\mathrm{d}r_{1}^{2}\,\mathrm{d}\theta_{1}$.
One may expand the jet mass $\varrho_{1}$ defined in eq. (2) as a series in
$R$ as follows

$\varrho_{1}=\xi_{1}\,R^{2}\,r_{1}^{2}+\frac{1}{12}\,\xi_{1}\,R^{4}\,r_{1}^{4}\cos(2\,\theta_{1})+\cdots\,.$
(17)

In fact, at single logarithmic accuracy it suffices to keep just the first
term in this expansion, and thus we write the step function in eq. (14) as
$\Theta\left(\xi_{1}\,R^{2}\,r_{1}^{2}-\rho\right)$. We now perform the
integrations for each dipole.

  * •

The dipole $(ab)$:

The contribution of the in-in dipole $(ab)$ to eq. (14) at single logarithmic
accuracy may be written as follows

$\displaystyle f_{\mathcal{B},(ab)}^{(1)}(\rho)$
$\displaystyle=-\mathcal{C}_{ab}\,\bar{\alpha}_{s}\,\frac{R^{2}}{2}\int\frac{\mathrm{d}\xi_{1}}{\xi_{1}}\,\mathrm{d}r_{1}^{2}\,\frac{\mathrm{d}\theta_{1}}{2\pi}\,\Theta\left(1-r_{1}^{2}\right)\times$
$\displaystyle\quad\times\Theta\left[\xi_{1}\,R^{2}\,r_{1}^{2}-\rho\right]$
$\displaystyle=-\mathcal{C}_{ab}\,\bar{\alpha}_{s}\,\frac{R^{2}}{2}\,L\,,$
(18)

with $L=\ln(R^{2}/\rho)$ being the large logarithm that we aim to resum. This
contribution corresponds to soft wide-angle radiation from the in-in dipole
into the interior of the measured outgoing jet, and is thus free from
collinear logarithms.

  * •

The dipole $(aj)$:

For the in-jet dipole $(aj)$ eq. (14) reads

$\displaystyle
f_{\mathcal{B},(aj)}^{(1)}=-\mathcal{C}_{aj}\,\bar{\alpha}_{s}\,\frac{R^{2}}{2}\int\frac{\mathrm{d}\xi_{1}}{\xi_{1}}\,\mathrm{d}r_{1}^{2}\,\frac{\mathrm{d}\theta_{1}}{2\pi}\,\Theta\left[\xi_{1}\,R^{2}\,r_{1}^{2}-\rho\right]\times$
$\displaystyle\times\frac{1}{2}\,\frac{\exp(R\,r_{1}\cos\theta_{1})}{\cosh(R\,r_{1}\cos\theta_{1})-\cos(R\,r_{1}\sin\theta_{1})}\,\Theta\left(1-r_{1}^{2}\right).$
(19)

Note here that the step function
$\Theta\left[\xi_{1}\,R^{2}\,r_{1}^{2}-\rho\right]$ which restricts
$\xi_{1}>\rho/(R^{2}\,r_{1}^{2})$ also implies that $R^{2}\,r_{1}^{2}>\rho$
since $\xi_{1}<1$. This serves as a collinear regulator for the integral over
$r_{1}$, which would otherwise diverge, resulting in an overall double
logarithm as well as a single logarithm. Evaluating the $\xi_{1}$ integration
yields

$\displaystyle
f_{\mathcal{B},(aj)}^{(1)}(\rho)=-\mathcal{C}_{aj}\,\bar{\alpha}_{s}\,\frac{R^{2}}{2}\int_{\rho/R^{2}}^{1}\ln\frac{R^{2}\,r_{1}^{2}}{\rho}\,\mathrm{d}r_{1}^{2}\,\times$
$\displaystyle\times\int_{0}^{2\pi}\frac{\mathrm{d}\theta_{1}}{2\pi}\,\frac{1}{2}\,\frac{\exp(R\,r_{1}\cos\theta_{1})}{\cosh(R\,r_{1}\cos\theta_{1})-\cos(R\,r_{1}\sin\theta_{1})}\,.$
(20)

We perform the integration over $\theta_{1}$ by expanding the integrand as a
series in $R$ and neglecting higher-order terms that have small coefficients.
Thus we find

$\displaystyle\frac{R^{2}}{2}\int_{0}^{2\pi}\frac{\mathrm{d}\theta_{1}}{2\pi}\,\frac{1}{2}\,\frac{\exp(R\,r_{1}\cos\theta_{1})}{\cosh(R\,r_{1}\cos\theta_{1})-\cos(R\,r_{1}\sin\theta_{1})}=$
$\displaystyle=\frac{1}{2\,r_{1}^{2}}+\frac{R^{2}}{8}+\frac{r_{1}^{2}\,R^{4}}{288}+\mathcal{O}(R^{8})\,.$
(21)

The first term in this expansion corresponds to soft and collinear emissions
from the outgoing hard leg $(j)$ into its own jet. It contributes at the
double logarithmic level, giving the result

$f_{\mathcal{B},(aj)}^{(1)\mathrm{DL}}=-\mathcal{C}_{aj}\,\bar{\alpha}_{s}\,\frac{L^{2}}{4}\,,$
(22)

which is independent of the jet radius (other than in the argument of the
logarithm). The other terms in the expansion (21) are purely soft wide-angle
contributions, hence we can set $\rho\to 0$ in the lower limit of integration
over $r_{1}^{2}$, and throw away the sub-leading $\ln r_{1}^{2}$ term in the
integrand. Performing the integration we obtain

$f_{\mathcal{B},(aj)}^{(1)\mathrm{SL}}=-\mathcal{C}_{aj}\,\bar{\alpha}_{s}\,L\left(\frac{1}{8}\,R^{2}+\frac{1}{576}\,R^{4}+\mathcal{O}(R^{8})\right).$
(23)

We note that the coefficient of $R^{8}$ in this expression is vanishingly
small ($\mathcal{O}(10^{-7})$).

  * •

The dipole $(bj)$:

For the other in-jet dipole $(bj)$ the only differences relative to the dipole
$(aj)$ are the colour factor $\mathcal{C}_{aj}\to\mathcal{C}_{bj}$ and a minus
sign to be inserted in the exponent of the exponential in the integrand of eq.
(• ‣ 3.1), i.e.,
$\exp(R\,r_{1}\cos\theta_{1})\to\exp(-R\,r_{1}\cos\theta_{1})$. This is
equivalent to a change $R\to-R$ (the rest of the integral is invariant under
this change). This actually does not produce any differences in the
integration since only even powers of $R$ appear in the results (22) and (23).

We can therefore write the assembled soft-collinear double-logarithmic result
as

$f_{\mathcal{B},\delta}^{(1)\mathrm{DL}}=-\left(\mathcal{C}_{aj}+\mathcal{C}_{bj}\right)\bar{\alpha}_{s}\,\frac{L^{2}}{4}\,,$
(24)

and the soft wide-angle single-logarithmic contribution as

$f_{\mathcal{B},\delta}^{(1)\mathrm{SL}}=-\bar{\alpha}_{s}\,L\left[\mathcal{C}_{ab}\,\frac{R^{2}}{2}+\left(\mathcal{C}_{aj}+\mathcal{C}_{bj}\right)h(R)\right],$
(25)

with

$h(R)=\frac{1}{8}\,R^{2}+\frac{1}{576}\,R^{4}+\mathcal{O}(R^{8})\,.$ (26)

This result was first derived in ref. Dasgupta:2012hg , and it actually
exponentiates to all orders. However, the running coupling, whose argument is
the invariant transverse momentum
$\kappa_{t1,(i\ell)}^{2}=k_{t1}^{2}/w_{i\ell}^{1}$ of the emission $k_{1}$
with respect to the emitting dipole $(i\ell)$ Catani:1999ss , contributes at
higher orders and modifies the single logarithmic contribution
$f_{\mathcal{B},\delta}^{(1)\mathrm{SL}}$ with a change

$-\bar{\alpha}_{s}\,L\to\frac{1}{2\pi\beta_{0}}\ln\left(1-2\,\alpha_{s}\,\beta_{0}\,L\right),$
(27)

where $\beta_{0}$ is the one-loop coefficient of the QCD $\beta$ function.
Accounting for the running coupling for the double logarithmic contribution
$f_{\mathcal{B},\delta}^{(1)\mathrm{DL}}$ is more subtle. In fact the running
coupling introduces additional single logarithmic components which depend on
the renormalisation scheme. We discuss the all-orders resummed result in the
following subsection.

###  3.2 Resummed global result

The full NLL-resummed global form factor
$f_{\mathcal{B},\delta}^{\mathrm{global}}(\rho)$ has been computed in ref.
Dasgupta:2012hg  (eqs. (3.3), (3.11) and appendix C therein). The interested
reader is referred to the latter reference, together with ref. Banfi:2004yd ,
for details. Here we only state its form, which is given by Dasgupta:2012hg

$\displaystyle
f_{\mathcal{B},\delta}^{\mathrm{global}}(\rho)=\frac{1}{\Gamma\left[1+\mathcal{R}^{\prime}_{\delta}(\rho)\right]}\exp\left[-\mathcal{R}_{\delta}(\rho)-\gamma_{E}\,\mathcal{R}^{\prime}(\rho)\right],$
(28)

with $\gamma_{E}$ the Euler-Mascheroni constant ($\gamma_{E}\approx 0.577$)
and $\Gamma$ denotes the Gamma function. The radiator $\mathcal{R}$ and its
derivative with respect to $L$, $\mathcal{R}^{\prime}$, are presented in
appendix B. We note that the global form factor is identical for all jet
algorithms. We also note that the expression of
$f_{\mathcal{B},\delta}^{\mathrm{global}}(\rho)$ may be deduced from the
general form presented in ref. Banfi:2004yd  as we show in appendix B.

In the next section we treat the case of two-gluon emission where clustering
and non-global logarithms first pop-up.

##  4 Two-gluon emission

In the eikonal approximation, the factorised squared amplitude for the
emission of two real gluons $k_{1}$ and $k_{2}$ off the three-hard-legs Born
configuration is given by Khelifa-Kerfa:2020nlc

$\mathcal{W}_{12,\delta}^{{\scriptscriptstyle\mathrm{R}}{\scriptscriptstyle\mathrm{R}}}=\mathcal{W}_{1,\delta}^{\scriptscriptstyle\mathrm{R}}\,\mathcal{W}_{2,\delta}^{\scriptscriptstyle\mathrm{R}}+\overline{\mathcal{W}}_{12,\delta}^{{\scriptscriptstyle\mathrm{R}}{\scriptscriptstyle\mathrm{R}}}\,,$
(29)

where the one-loop amplitude squared
$\mathcal{W}_{i,\delta}^{{\scriptscriptstyle\mathrm{R}}}$, which builds up the
reducible part of the above two-gluon squared amplitude (first term on the
right-hand side), is given in eq. (10), and the irreducible contribution
$\overline{\mathcal{W}}_{12,\delta}^{{\scriptscriptstyle\mathrm{R}}{\scriptscriptstyle\mathrm{R}}}$
reads

$\overline{\mathcal{W}}_{12,\delta}^{{\scriptscriptstyle\mathrm{R}}{\scriptscriptstyle\mathrm{R}}}=\mathrm{C_{A}}\sum_{(i\ell)\in\Delta_{\delta}}\mathcal{C}_{i\ell}\,\mathcal{A}_{i\ell}^{12}\,.$
(30)

The virtual corrections at this order are

$\displaystyle\mathcal{W}_{12,\delta}^{{\scriptscriptstyle\mathrm{R}}{\scriptscriptstyle\mathrm{V}}}$
$\displaystyle=-\mathcal{W}_{12,\delta}^{{\scriptscriptstyle\mathrm{R}}{\scriptscriptstyle\mathrm{R}}}\,,$
(31a)
$\displaystyle\mathcal{W}_{12,\delta}^{{\scriptscriptstyle\mathrm{V}}{\scriptscriptstyle\mathrm{R}}}$
$\displaystyle=-\mathcal{W}_{1,\delta}^{{\scriptscriptstyle\mathrm{R}}}\,\mathcal{W}_{2,\delta}^{{\scriptscriptstyle\mathrm{R}}}\,,$
(31b)
$\displaystyle\mathcal{W}_{12,\delta}^{{\scriptscriptstyle\mathrm{V}}{\scriptscriptstyle\mathrm{V}}}$
$\displaystyle=-\mathcal{W}_{12,\delta}^{{\scriptscriptstyle\mathrm{V}}{\scriptscriptstyle\mathrm{R}}}\,.$
(31c)

Following ref. Khelifa-Kerfa:2015mma , and implementing the measurement-
operator method, we write the jet mass distribution at this order as

$\displaystyle f_{\mathcal{B},\delta}^{(2)}(\rho)$
$\displaystyle=\int_{\xi_{1}>\xi_{2}}\mathrm{d}\Pi_{12}\,\Xi^{\mathrm{p}}(k_{1},k_{2})\,\mathcal{W}_{1,\delta}^{\scriptscriptstyle\mathrm{R}}\,\mathcal{W}_{2,\delta}^{\scriptscriptstyle\mathrm{R}}-$
$\displaystyle-\int_{\xi_{1}>\xi_{2}}\mathrm{d}\Pi_{12}\,\Xi^{\mathrm{NG}}(k_{1},k_{2})\,\overline{\mathcal{W}}_{12,\delta}^{{\scriptscriptstyle\mathrm{R}}{\scriptscriptstyle\mathrm{R}}}\,,$
(32)

with phase-space factor
$\mathrm{d}\Pi_{12}=\mathrm{d}\Phi_{1}\,\mathrm{d}\Phi_{2}\,\Theta(\varrho_{1}-\rho)\,\Theta(\varrho_{2}-\rho)$.
Here, the first integral produces the primary-emission contribution, which
contains CLs, and the second integral gives NGLs. The functions
$\Xi^{\mathrm{p}}$, where $\mathrm{p}$ stands for primary, and
$\Xi^{\mathrm{NG}}$, where $\mathrm{NG}$ stands for non-global, result from
the application of the jet algorithm and restrict the angular integration
regions for gluons $k_{1}$ and $k_{2}$.

###  4.1 Clustering logarithms

In this subsection we focus on the primary-emission integral in eq. (4) and
leave the treatment of the correlated-emission NGLs term to the next
subsection. The primary-emission contribution may be split into two parts. The
first is the global component which results from integrating both gluons
within the measured jet region. This has, however, been accounted for by the
all-orders resummed formula (28) discussed in the previous section, and will
thus be skipped here. The second part is related to the way jet algorithms
cluster gluons and results in large single logarithms that are referred to as
clustering logarithms Banfi:2005gj ; Delenda:2006nf ; Delenda:2012mm . These
logarithms are a result of miscancellation between real emissions and virtual
corrections. The key point is that while real gluons may be dragged into/out
of the jet by other real gluons and thus get clustered together, virtual
gluons can neither drag nor get dragged. We note that CLs are totally absent
when jets are reconstructed using the anti-$k_{t}$ algorithm. At two loops,
the C-A and $k_{t}$ algorithms produce identical CLs, but they start to differ
at higher orders as was shown in ref. Delenda:2012mm .

To perform the first integral in eq. (4) we begin by simplifying the
clustering function $\Xi^{\mathrm{p}}(k_{1},k_{2})$. To this end, we introduce
the same change of variables as in eq. (16) such that
$(\eta_{1},\phi_{1})\to(r_{1},\theta_{1})$ and
$(\eta_{2},\phi_{2})\to(r_{2},\theta_{2})$. Note that the upper limit of
$r_{i}$ is $\pi/\left(R\,|\sin\theta_{i}|\right)$ since we have
$-\pi<\phi_{i}-\varphi<\pi$. We then have for the $k_{t}$ clustering algorithm
Delenda:2006nf ; Delenda:2012mm

$\displaystyle\Xi^{\mathrm{p}}(k_{1},k_{2})=\Theta\left(R^{2}-d_{1j}\right)\Theta\left(R^{2}-d_{2j}\right)+$
$\displaystyle+\Theta\left(d_{1j}-R^{2}\right)\Theta\left(R^{2}-d_{2j}\right)\Theta\left(d_{2j}-d_{12}\right)$
$\displaystyle=\Theta\left(1-r_{1}^{2}\right)\Theta\left(1-r_{2}^{2}\right)+$
$\displaystyle+\Theta\left(r_{1}^{2}-1\right)\Theta\left(1-r_{2}^{2}\right)\Theta\left(2\,r_{2}\cos(\theta_{1}-\theta_{2})-r_{1}\right),$
(33)

where the algorithm distances $d_{im}$ have been defined in eq. (3). The first
term exactly reproduces half the square of the one-loop result (14), i.e.
$1/2!\,[f_{\mathcal{B},\delta}^{(1)}]^{2}$, and persists at higher orders as
$1/n!\,[f_{\mathcal{B},\delta}^{(1)}]^{n}$. This signifies that the one-loop
result simply exponentiates into the global form factor discussed before. It
is the second term, the CLs term, that we shall focus on in the remainder of
this subsection.

We write the CLs contribution at this order as follows

$\displaystyle\mathcal{C}_{2,\delta}(\rho)=\bar{\alpha}_{s}^{2}\,R^{4}\int_{\xi_{1}>\xi_{2}}\frac{\mathrm{d}\xi_{1}}{\xi_{1}}\,r_{1}\,\mathrm{d}r_{1}\,\frac{\mathrm{d}\theta_{1}}{2\pi}\,\frac{\mathrm{d}\xi_{2}}{\xi_{2}}\,r_{2}\,\mathrm{d}r_{2}\,\frac{\mathrm{d}\theta_{2}}{2\pi}\,\times$
$\displaystyle\times\Theta\left(r_{1}^{2}-1\right)\Theta\left(1-r_{2}^{2}\right)\Theta\left(2\,r_{2}\cos(\theta_{1}-\theta_{2})-r_{1}\right)\times$
$\displaystyle\Theta(\varrho_{1}-\rho)\Theta(\varrho_{2}-\rho)\left[\sum_{(ik)\in\Delta_{\delta}}\mathcal{C}_{ik}\,w_{ik}^{1}\right]\left[\sum_{(\ell
m)\in\Delta_{\delta}}\mathcal{C}_{\ell m}w_{\ell m}^{2}\right].$ (34)

The expressions inside the square brackets are the one-loop eikonal amplitudes
squared (10) for gluons $k_{1}$ and $k_{2}$, respectively. To single
logarithmic accuracy the $\xi$ integrations factor out from the rest of the
integrals yielding the result $L^{2}/2!$, and we are left with

$\displaystyle\mathcal{C}_{2,\delta}(\rho)=\frac{1}{2!}\,\bar{\alpha}_{s}^{2}\,L^{2}\,\mathcal{F}_{2}^{\delta}(R)\,,$
(35)

where

$\displaystyle\mathcal{F}_{2}^{\delta}(R)=\sum_{(ik)\in\Delta_{\delta}}\sum_{(\ell
m)\in\Delta_{\delta}}\mathcal{C}_{ik}\,\mathcal{C}_{\ell
m}\,R^{4}\int_{0}^{1}r_{2}\,\mathrm{d}r_{2}\,\frac{\mathrm{d}\theta_{2}}{2\pi}\,\times$
$\displaystyle\times\int_{1}^{2}r_{1}\,\mathrm{d}r_{1}\,\frac{\mathrm{d}\theta_{1}}{2\pi}\,\Theta\left(2\,r_{2}\cos(\theta_{1}-\theta_{2})-r_{1}\right)\,w_{ik}^{1}\,w_{\ell\,m}^{2}$
$\displaystyle=\sum_{(ik)\in\Delta_{\delta}}\mathcal{C}_{ik}^{2}\,\mathcal{F}_{2,\mathrm{dip}}^{(ik)}+\sum_{(ik)\neq(\ell
m)\in\Delta_{\delta}}\mathcal{C}_{ik}\,\mathcal{C}_{\ell
m}\,\mathcal{F}_{2,\mathrm{int}}^{(ik,\ell m)}\,,$ (36)

where the first term represents contributions from independent dipoles, that
is, each dipole consecutively emits softer gluons at each order independently
of the other dipoles. This situation is analogous to that in $e^{+}e^{-}$
annihilation to di-jet process (see for instance ref. Delenda:2012mm ). The
second term in eq. (36) represents contributions arising from the interference
of dipoles in channel $\delta$.

To carry out the integrations we expand the integrand as a power series in $R$
and use the change of variable $\theta_{1}-\theta_{2}\to\theta_{1}$ for the
angular integrations. We obtain the following result

$\displaystyle\mathcal{F}_{2,\mathrm{dip}}^{(ab)}$
$\displaystyle=0.052\,R^{4}\,,$ (37a)
$\displaystyle\mathcal{F}_{2,\mathrm{dip}}^{(aj)}=\mathcal{F}_{2,\mathrm{dip}}^{(bj)}$
$\displaystyle=0.046+0.047\,R^{2}+0.009\,R^{4}+$
$\displaystyle+0.0004\,R^{6}+\mathcal{O}(R^{8})\,,$ (37b) for the independent-
dipoles part, and
$\displaystyle\mathcal{F}_{2,\mathrm{int}}^{(aj,bj)}=\mathcal{F}_{2,\mathrm{int}}^{(bj,aj)}$
$\displaystyle=0.046+0.004\,R^{2}+0.0004\,R^{4}+$
$\displaystyle+0.00004\,R^{6}+\mathcal{O}(R^{8})\,,$ (37c)
$\displaystyle\mathcal{F}_{2,\mathrm{int}}^{(aj,ab)}=\mathcal{F}_{2,\mathrm{int}}^{(bj,ab)}$
$\displaystyle=0.032\,R^{2}+0.013\,R^{4}+$
$\displaystyle+0.0006\,R^{6}+\mathcal{O}(R^{8})\,,$ (37d)
$\displaystyle\mathcal{F}_{2,\mathrm{int}}^{(ab,aj)}=\mathcal{F}_{2,\mathrm{int}}^{(ab,bj)}$
$\displaystyle=0.071\,R^{2}+0.013\,R^{4}+$
$\displaystyle+0.0003\,R^{6}+\mathcal{O}(R^{8})\,,$ (37e)

for the dipole-interference part. Notice that the interference term
$\mathcal{F}_{2,\mathrm{int}}$ is not symmetric under the interchange of the
dipoles $(aj)$ and $(ab)$, or $(bj)$ and $(ab)$, as apposed to the dipoles
$(aj)$ and $(bj)$. This stems from the fact that integrands such as
$w_{aj}^{1}\,w_{ab}^{2}$ and $w_{aj}^{2}\,w_{ab}^{1}$ are not identical,
though symmetric under $(r_{1},\theta_{1})\leftrightarrow(r_{2},\theta_{2})$.
Since the angular restrictions on $k_{1}$ and $k_{2}$ are not identical then
the results one obtains for the two mentioned terms are different. This boils
down to the effect of the $k_{t}$ algorithm which does not treat the two
gluons symmetrically. Furthermore, independent and interference terms
involving the in-in $(ab)$ dipole vanish in the limit $R\to 0$.

Substituting the results (37) back into eq. (36) we obtain the corresponding
CLs coefficients for each channel. They read

$\displaystyle\mathcal{F}_{2}^{\delta_{1}}(R)=\mathrm{C_{F}^{2}}\,0.207\,R^{4}+$
$\displaystyle+\mathrm{C_{F}}\,\mathrm{C_{A}}\left[0.414\,R^{2}-0.103\,R^{4}+0.004\,R^{6}+\mathcal{O}(R^{10})\right]+$
$\displaystyle+\mathrm{C_{A}^{2}}\left[0.183-0.103\,R^{2}+0.019\,R^{4}-0.001\,R^{6}+\mathcal{O}(R^{8})\right],$
(38a) for channel $q\bar{q}\to g+X$,
$\displaystyle\mathcal{F}_{2}^{\delta_{2}}(R)=$
$\displaystyle=\mathrm{C_{A}^{2}}\left[0.087\,R^{2}+0.069\,R^{4}+0.001\,R^{6}+\mathcal{O}(R^{10})\right]+$
$\displaystyle+\mathrm{C_{F}}\,\mathrm{C_{A}}\left[0.034\,R^{2}+0.017\,R^{4}+0.0003\,R^{6}+\mathcal{O}(R^{10})\right]+$
$\displaystyle+\mathrm{C_{F}^{2}}\left[0.183+0.190\,R^{2}+0.037\,R^{4}+0.002\,R^{6}+\mathcal{O}(R^{8})\right],$
(38b) for channel $qg\to q+X$, and
$\displaystyle\mathcal{F}_{2}^{\delta_{3}}(R)=$
$\displaystyle\mathrm{C_{A}^{2}}\left[0.183+0.310\,R^{2}+0.122\,R^{4}+0.003\,R^{6}+\mathcal{O}(R^{8})\right],$
(38c)

for $gg\to g+H$. We show in figure 3 a plot of the CLs coefficient
$\frac{1}{2!}\,\mathcal{F}_{2}^{\delta}$ as a function of $R$ for the various
channels $\delta$.

Figure 3: Two-loops CLs coefficient as a function of jet radius $R$ in the
$k_{t}$ and C-A algorithms for the three channels.

We notice that gluon-initiated jets have larger CLs coefficient than quark-
initiated jets, mainly due to the corresponding colour factors
($\mathrm{C_{A}}=3$ and $\mathrm{C_{F}}=4/3$, respectively). These series
expansions in $R$ converge, and at small values of $R$ it suffices to keep
only the leading terms. At very small values of $R$ we observe that

$\lim_{R\to
0}\mathcal{F}_{2}^{\delta_{1,3}}=\mathrm{C_{A}^{2}}\,0.183\,,\qquad\lim_{R\to
0}\mathcal{F}_{2}^{\delta_{2}}=\mathrm{C_{F}^{2}}\,0.183\,.$ (39)

This result for CLs obtained here for $H/W/Z/\gamma$ \+ jet events at hadron
colliders coincides with that found in refs. Delenda:2012mm ; Banfi:2010pa ;
KhelifaKerfa:2011zu  for jet mass distribution in $e^{+}e^{-}\to$ di-jet
events. It does, however, deviate from it as $R$ increases due to initial-
state radiation from the incoming partons. Inline with the findings of refs.
Delenda:2006nf ; Delenda:2012mm  we expect the term (35) to simply
exponentiate to all orders. Nonetheless, there will be new CLs terms at each
order that are not captured by the latter exponential and that are highly non-
trivial to deduce (see ref. Delenda:2012mm ). Moreover, we expect that at
higher orders the small-$R$ limit of the CLs coefficient in $H/V$ \+ jet
events at the LHC will coincide with that in $e^{+}e^{-}\to$ di-jet events
found in ref. Delenda:2012mm .

In the next subsection we compute NGLs at two loops.

###  4.2 Non-global logarithms

####  4.2.1 k${}_{\text{t}}$ and C-A clustering algorithms

We now turn to the evaluation of the correlated secondary-emission
contribution in eq. (4) for the $k_{t}$ and C-A clustering algorithms. To this
end we write

$\displaystyle\mathcal{S}_{2,\delta}(\rho)$
$\displaystyle=-\int_{\xi_{1}>\xi_{2}}\mathrm{d}\Phi_{1}\,\mathrm{d}\Phi_{2}\,\Theta\left(\varrho_{1}-\rho\right)\Theta\left(\varrho_{2}-\rho\right)\times$
$\displaystyle\times\Xi^{\mathrm{NG}}(k_{1},k_{2})\left[\sum_{(i\ell)\in\Delta_{\delta}}\mathrm{C_{A}}\,\mathcal{C}_{i\ell}\,\mathcal{A}_{i\ell}^{12}\right],$
(40)

where the clustering function reads

$\displaystyle\Xi^{\mathrm{NG}}(k_{1},k_{2})=\Theta\left(d_{1j}-R^{2}\right)\Theta\left(R^{2}-d_{2j}\right)\Theta\left(d_{12}-d_{2j}\right)$
$\displaystyle=\Theta\left(r_{1}^{2}-1\right)\Theta\left(1-r_{2}^{2}\right)\Theta\left(r_{1}-2\,r_{2}\cos(\theta_{1}-\theta_{2})\right).$
(41)

As before, the integration over $\xi_{1}$ and $\xi_{2}$ yields $L^{2}/2!$, and
we may write

$\displaystyle\mathcal{S}_{2,\delta}(\rho)$
$\displaystyle=-\frac{1}{2!}\,\bar{\alpha}_{s}^{2}\,L^{2}\,\mathcal{G}_{2}^{\delta}(R)\,,$
(42a) with NGLs coefficient
$\displaystyle\mathcal{G}_{2}^{\delta}(R)=\mathrm{C_{A}}\sum_{(i\ell)\in\Delta_{\delta}}\mathcal{C}_{i\ell}\,R^{4}\int_{0}^{1}r_{2}\,\mathrm{d}r_{2}\,\frac{\mathrm{d}\theta_{2}}{2\pi}\,\times$
$\displaystyle\times\int_{1}^{\pi/\left(R\,|\sin\theta_{1}|\right)}r_{1}\,\mathrm{d}r_{1}\,\frac{\mathrm{d}\theta_{1}}{2\pi}\,\Theta\left(r_{1}-2\,r_{2}\cos(\theta_{1}-\theta_{2})\right)\mathcal{A}_{i\ell}^{12}$
$\displaystyle=\mathrm{C_{A}}\sum_{(i\ell)\in\Delta_{\delta}}\mathcal{C}_{i\ell}\,\mathcal{G}_{2}^{(i\ell)}(R)\,.$
(42b)

Performing the integration, as in the previous subsection, we obtain the
results for $\mathcal{G}_{2}^{(i\ell)}$ for each dipole as a series in $R$

$\displaystyle\mathcal{G}^{(ab)}_{2}(R)$ $\displaystyle=-R^{2}\ln
R+0.015\,R^{2}+0.151\,R^{4}-$
$\displaystyle-0.004\,R^{6}+\mathcal{O}(R^{8})\,,$ (43a)
$\displaystyle\mathcal{G}^{(aj)}_{2}(R)$
$\displaystyle=\mathcal{G}^{(bj)}_{2}(R)=0.366-0.103\,R^{2}+0.004\,R^{4}+$
$\displaystyle+0.0002\,R^{6}+\mathcal{O}(R^{8})\,.$ (43b)

In terms of channels we have

$\displaystyle\mathcal{G}^{\delta_{1}}_{2}=\mathrm{C_{F}}\,\mathrm{C_{A}}\left[-2\,R^{2}\ln
R+0.031\,R^{2}+0.302\,R^{4}-\right.$
$\displaystyle\left.-0.008\,R^{6}+\mathcal{O}(R^{8})\right]+\mathrm{C_{A}^{2}}\left[0.731+R^{2}\ln
R-0.222\,R^{2}-\right.$
$\displaystyle\left.-0.143\,R^{4}+0.004\,R^{6}+\mathcal{O}(R^{8})\right],$
(44a) for channel $q\bar{q}\to g+X$,
$\displaystyle\mathcal{G}^{\delta_{2}}_{2}=\mathrm{C_{F}}\,\mathrm{C_{A}}\left[0.731-0.207\,R^{2}+0.008\,R^{4}+0.0004\,R^{6}+\right.$
$\displaystyle\left.\mathcal{O}(R^{8})\right]+\mathrm{C_{A}^{2}}\left[-R^{2}\ln
R+0.015\,R^{2}+0.151\,R^{4}-\right.$
$\displaystyle\left.-0.004\,R^{6}+\mathcal{O}(R^{8})\right],$ (44b) for
channel $qg\to q+X$, and
$\displaystyle\mathcal{G}^{\delta_{3}}_{2}=\mathrm{C_{A}^{2}}\left[0.731-R^{2}\ln
R-0.191\,R^{2}+0.159\,R^{4}-\right.$
$\displaystyle\left.-0.003\,R^{6}+\mathcal{O}(R^{8})\right],$ (44c)

for $gg\to g+H$. Moreover, in the small-$R$ limit we observe that

$\displaystyle\lim_{R\to
0}\mathcal{G}_{2}^{\delta_{1,3}}=\mathrm{C_{A}^{2}}\,0.731\,,\qquad\lim_{R\to
0}\mathcal{G}_{2}^{\delta_{2}}=\mathrm{C_{F}}\,\mathrm{C_{A}}\,0.731\,.$ (45)

The result for channel $\delta_{2}$ is exactly the same small-$R$ limit found
in the case of jet shapes in $e^{+}e^{-}\to$ di-jet events (see for instance
ref. KhelifaKerfa:2011zu ). Results for channels $\delta_{1}$ and $\delta_{3}$
in the limit $R\to 0$ are also the same and differ from those for channel
$\delta_{2}$ only in the colour factor. In figure 4 we plot the NGLs
coefficient $\frac{1}{2!}\,\mathcal{G}_{2}^{\delta}$ at this order as a
function of jet radius $R$.

Figure 4: Two-loops NGLs coefficient as a function of jet radius for C-A and
$k_{t}$ clustering.

Once again we notice that gluon-initiated jets have larger NGLs coefficient
due to their large gluon-emission colour factor ($\mathrm{C_{A}}$). We observe
from the plots in figures 3 and 4 that the CLs coefficient for the $gg\to g$
channel grows larger with $R$ while that for NGLs does not change much.

Moreover, in order to assess the overall impact of CLs and NGLs at this order,
we plot in figure 5 the combined coefficient of the single logarithm
$\bar{\alpha}_{s}^{2}\,L^{2}$ resulting from the non-global nature of our
observable.

Figure 5: The coefficient of the single logarithm
$\bar{\alpha}_{s}^{2}\,L^{2}$ resulting from both CLs and NGLs with C-A and
$k_{t}$ clustering.

We note that at large jet radii ($R\gtrsim 1.0$) and for all partonic channels
the CLs and NGLs tend to balance each other out, but not entirely though. For
small jet radii the said single logarithmic CLs + NGLs coefficient is quite
large in magnitude especially for gluon-initiated jets.

####  4.2.2 Anti-k${}_{\text{t}}$ clustering algorithm

For the sake of assessing the effect of clustering on NGLs, we report the
results for the NGLs coefficient in the anti-$k_{t}$ algorithm. Note that
there are no CLs in this case. The corresponding integral is identical to that
in eq. (40) except for the clustering function. It reads

$\displaystyle\mathcal{S}^{\mathrm{ak_{t}}}_{2,\delta}(\rho)$
$\displaystyle=-\int_{\xi_{1}>\xi_{2}}\mathrm{d}\Phi_{1}\,\mathrm{d}\Phi_{2}\,\Theta\left(\varrho_{1}-\rho\right)\Theta\left(\varrho_{2}-\rho\right)\times$
$\displaystyle\times\Theta\left(r_{1}^{2}-1\right)\Theta\left(1-r_{2}^{2}\right)\left[\sum_{(i\ell)\in\Delta_{\delta}}\mathrm{C_{A}}\,\mathcal{C}_{i\ell}\,\mathcal{A}_{i\ell}^{12}\right]$
$\displaystyle=-\frac{1}{2!}\,\bar{\alpha}_{s}^{2}\,L^{2}\,\mathcal{G}_{2}^{\delta,\mathrm{ak_{t}}}(R)\,.$
(46)

The results we obtain for each dipole are

$\displaystyle\mathcal{G}^{(ab),\mathrm{ak_{t}}}_{2}$ $\displaystyle=-R^{2}\ln
R+0.500\,R^{2}+0.125\,R^{4}-$
$\displaystyle-0.003\,R^{6}+\mathcal{O}(R^{8})\,,$ (47a)
$\displaystyle\mathcal{G}^{(aj),\mathrm{ak_{t}}}_{2}$
$\displaystyle=\mathcal{G}^{(bj),\mathrm{ak_{t}}}_{2}=0.822+0.003\,R^{4}+\mathcal{O}(R^{8})\,.$
(47b)

In terms of channels we have

$\displaystyle\mathcal{G}^{\delta_{1},\mathrm{ak_{t}}}_{2}=\mathrm{C_{F}}\,\mathrm{C_{A}}\left[-2\,R^{2}\ln
R+R^{2}+0.250\,R^{4}-\right.$
$\displaystyle\left.-0.007\,R^{6}+\mathcal{O}(R^{8})\right]+\mathrm{C_{A}^{2}}\left[1.645+R^{2}\ln
R-0.500\,R^{2}-\right.$
$\displaystyle\left.-0.118\,R^{4}+0.003\,R^{6}+\mathcal{O}(R^{8})\right],$
(48a)
$\displaystyle\mathcal{G}^{\delta_{2},\mathrm{ak_{t}}}_{2}=\mathrm{C_{F}}\,\mathrm{C_{A}}\left[1.645+0.007\,R^{4}+\mathcal{O}(R^{8})\right]+$
$\displaystyle+\mathrm{C_{A}^{2}}\left[-R^{2}\ln
R+0.500\,R^{2}+0.125\,R^{4}-0.003\,R^{6}+\right.$
$\displaystyle\left.+\mathcal{O}(R^{8})\right],$ (48b)
$\displaystyle\mathcal{G}^{\delta_{3},\mathrm{ak_{t}}}_{2}=\mathrm{C_{A}^{2}}\left[1.645-R^{2}\ln
R+0.500\,R^{2}+0.132\,R^{4}-\right.$
$\displaystyle\left.-0.003\,R^{6}+\mathcal{O}(R^{8})\right].$ (48c)

These results are in agreement with those reported in ref. Dasgupta:2012hg .
Notice again that the $R\to 0$ limit of the above expressions produces a
result (which is proportional to $1.645=\zeta_{2}$) that is identical to that
reported in ref. KhelifaKerfa:2011zu  for $e^{+}e^{-}\to$ di-jet process.

We plot in figure 6 the NGLs coefficient
$\frac{1}{2!}\,\mathcal{G}_{2}^{\delta,\mathrm{ak_{t}}}$ with
anti-$k_{t}$–clustered jets as a function of the jet radius $R$ for the
various partonic channels.

Figure 6: NGLs coefficient with anti-$k_{t}$–clustered jets at two loops.

As is clearly evident from the plots, NGLs in the anti-$k_{t}$ algorithm are
much larger compared to those in the C-A or $k_{t}$ clustering case. This is
made clearer in figure 7 where NGLs coefficients for each dipole are plotted
for both $k_{t}$ and anti-$k_{t}$ algorithms.

Figure 7: NGLs coefficients at two loops with anti-$k_{t}$ and $k_{t}$
clustering.

This observation was also made in previous studies of NGLs with $k_{t}$
clustering Delenda:2006nf ; Delenda:2012mm ; Appleby:2002ke . While $k_{t}$
clustering induces another tower of large single logarithms, namely CLs, it
actually diminishes the impact of NGLs. Additionally, as we observed in the
previous subsection 4.2.1, the induced CLs play a role of further reducing
NGLs since their coefficients have opposite signs. This may hint at a
(phenomenological) favour for the $k_{t}$ (or C-A) clustering algorithm over
the anti-$k_{t}$ algorithm.

##  5 All-orders treatment of CLs and NGLs

Including the resummation of NGLs and CLs together with the global form factor
(28) then the all-orders NLL-resummed jet mass distribution may be cast into

$\frac{\mathrm{d}\Sigma_{\delta}(\rho)}{\mathrm{d}\mathcal{B}_{\delta}}=\frac{\mathrm{d}\sigma_{0,\delta}}{\mathrm{d}\mathcal{B}_{\delta}}\,\mathcal{S}_{\delta}(\rho)\,\mathcal{C}_{\delta}(\rho)\,f_{\mathcal{B},\delta}^{\mathrm{global}}(\rho)\,C_{\mathcal{B},\delta}(\rho)\,,$
(49)

where $\mathcal{S}_{\delta}(\rho)$ and $\mathcal{C}_{\delta}(\rho)$ account
for the resummation of NGLs and CLs, respectively. We note that, unlike the
global form factor, the factors $\mathcal{S}_{\delta}$ and
$\mathcal{C}_{\delta}$ are algorithm-dependent.

In the anti-$k_{t}$ algorithm, the NGLs form factor results from multiple
correlated gluons outside the jet that coherently emit the softest gluon into
the jet. For the $k_{t}$ and C-A clustering algorithms, gluons can be moved
into and out of the jet by the clustering, thus NGLs can be induced when more
than one gluon is emitted within the jet region from an ensemble of harder
gluons. The NGLs factor $\mathcal{S}_{\delta}$ can be computed numerically and
in general only in the large-$\mathrm{N_{c}}$ limit Dasgupta:2001sh ;
Banfi:2002hw . For the $e^{+}e^{-}\to$ di-jet process, finite-$\mathrm{N_{c}}$
results do exist though Hatta:2013iba ; Hagiwara:2015bia . Moreover, the CLs
form factor results from multiple independent (primary) emissions that are
clustered by the $k_{t}$ or C-A algorithm. Just like NGLs, the latter CLs can
also be resummed numerically.

For the anti-$k_{t}$ algorithm, the all-orders numerical resummation of NGLs
may be obtained from the dipole-evolution Monte Carlo code of ref.
Dasgupta:2001sh  as reported in ref. Dasgupta:2012hg  for the various dipoles.
We see from figure 8

Figure 8: The full resummed differential jet mass distribution in the
anti-$k_{t}$ jet algorithm with NGLs factor as an exponential of the two-loops
result and as an all-orders numerical result obtained from ref.
Dasgupta:2012hg . We explain the in the next section how these plots are
obtained.

that the exponential of the two-loops result (46) approximates very well the
all-orders numerical result for the NGLs factor in the $Z$ \+ jet process. The
same is observed for the other processes. Hence we shall confine ourselves to
simply using the exponential of the two-loops result for the $k_{t}$ and C-A
algorithms. To this end we write, for a given channel $\delta$,

$\mathcal{S}_{\delta}(\rho)\approx\exp\left[-\frac{1}{2!}\,\mathcal{G}_{2}^{\delta}(R)\,t^{2}\right],$
(50)

where $\mathcal{G}_{2}^{\delta}$, for the $k_{t}$ and C-A algorithms, is given
in eq. (44), and the evolution parameter $t$ is defined by

$\displaystyle
t=-\frac{1}{2\pi\beta_{0}}\,\ln\left(1-2\,\alpha_{s}\,\beta_{0}\,L\right).$
(51)

Note that at fixed order $t$ reduces to just $\bar{\alpha}_{s}\,L$.

As for CLs, it was shown in refs. Delenda:2006nf ; Delenda:2012mm  that the
perturbative CLs series exhibits a pattern of exponentiation, and that the
exponential of the two-loops result is a very good approximation to the
numerically-resummed CLs factor obtained from the code of ref. Dasgupta:2001sh
. Therefore, and just as we did with NGLs, we shall be using the exponential
of the two-loops result for the CLs resummed factor
$\mathcal{C}_{\delta}(\rho)$. Thence

$\displaystyle\mathcal{C}_{\delta}(t)\approx\exp\left[\frac{1}{2!}\,\mathcal{F}_{2}^{\delta}(R)\,t^{2}\right],$
(52)

where $\mathcal{F}_{2}^{\delta}$ is given for $k_{t}$ and C-A algorithms in
eq. (38).

##  6 Comparison to Pythia 8 and Herwig 7 parton showers

In this section we present comparisons of our results for the jet mass
distribution with those obtained from Pythia 8 Sjostrand:2014zea  and Herwig 7
Bahr:2008pv ; Bellm:2015jjp  parton showers (PS), where the jets are clustered
with FastJet Cacciari:2011ma . The resummed result is obtained by convoluting
$\mathrm{d}\Sigma_{\delta}/\mathrm{d}\mathcal{B}_{\delta}$ given in eq. (6)
with parton distribution functions (we use MSTW 2008 (NLO) PDFs Martin:2009iq
and $\mu_{\mathrm{F}}=\mu_{\mathrm{R}}=200\,\mathrm{GeV}$). For double-
checking we perform the convolution using two different methods. In one method
we simply use a Monte Carlo code to integrate over the momentum fractions of
the partons $x_{a}$ and $x_{b}$ and over the transverse momentum $p_{t}$ and
rapidity $y$ of the jet, as explained in detail in appendix A. In the other
approach we generate a set of unweighted parton-level Born events using
MadEvent from MadGraph Maltoni:2002qb ; Alwall:2014hca  in the “Les Houches
Event File” format Alwall:2006yp , with the cuts $\Xi_{\mathcal{B}}$ being
applied. We then weigh each event by the resummed form factor
$\mathcal{S}_{\delta}(\rho)\,\mathcal{C}_{\delta}(\rho)\,f_{\mathcal{B},\delta}^{\mathrm{global}}(\rho)$,
sum over all events, and divide by the effective luminosity
$\mathcal{L}=N_{\mathrm{tot}}/\sigma_{0}$, with $N_{\mathrm{tot}}$ the total
number of events and $\sigma_{0}$ the Born cross-section calculated with
MadGraph. This results in the integrated distribution given in eq. (5) from
which the differential distribution can straightforwardly be obtained. To
avoid low-$p_{t}$ resummation we impose a cut on $p_{t}$ of the final-state
jet, e.g., $p_{t}>200\,\mathrm{GeV}$, i.e., we only consider high-$p_{t}$
jets, at a centre-of-mass energy $\sqrt{s}=7\,\mathrm{TeV}$.

In our resummed result we also include an approximation to the NLO effects on
the distribution through the NLO factor $C^{(1)}_{\mathcal{B},\delta}(\rho)$.
The full NLO distribution may ideally be analytically calculated using the
full squared amplitude with two partons in the final state as well as virtual
corrections to the Born cross-section. Though possible this is a delicate
task. The alternative numerical approach would be to exploit fixed-order
programs and obtain the factor $C^{(1)}_{\mathcal{B},\delta}(\rho)$ as a fully
differential distribution in the Born configuration, and then perform the
integration including the resummed form factor over the Born kinematics.
Practically this is not feasible. Instead, one could obtain an NLO factor
$C^{(1)}_{\mathcal{B},\delta}(\rho)$ that is averaged over the Born
configuration Banfi:2010xy  and insert it in eq. (6) as if it were
unintegrated over $\mathrm{d}\mathcal{B}_{\delta}$. In this paper we employ
this method and estimate the Born-configuration–averaged factor
$C^{(1)}_{\delta}(\rho)$ as was done in refs. Dasgupta:2012hg ; Banfi:2010xy ,
using the NLO jet mass distribution obtained from the fixed-order program MCFM
Campbell:2015qma ; Campbell:2019dru .

In refs. Dasgupta:2012hg ; Banfi:2010xy , the NLO factor
$C^{(1)}_{\delta}(\rho)$ was calculated in the small-$\rho$ limit as a
constant, and then the $\rho$-dependence of the NLO contribution to the jet
mass distribution was included at the stage of matching. This is equivalent to
using the full $\rho$-dependence of $C^{(1)}_{\delta}(\rho)$, which we do in
the present work. The factor $\alpha_{s}\,C^{(1)}_{\delta}(\rho)$ for channel
$(\delta)$ is simply given by the NLO integrated jet mass distribution
$\Sigma_{\mathrm{NLO}}^{(\delta)}(\rho)$ (obtained from MCFM) minus the
expansion of the integrated pure-resummed distribution
$\Sigma^{(\delta)}_{\mathrm{NLL},\alpha_{s}}(\rho)$, and then the result is
divided by the Born cross-section $\sigma_{0,\delta}$ Banfi:2010xy

$\alpha_{s}\,C^{(1)}_{\delta}(\rho)=\frac{1}{\sigma_{0,\delta}}\left(\Sigma_{\mathrm{NLO}}^{(\delta)}(\rho)-\Sigma^{(\delta)}_{\mathrm{NLL},\alpha_{s}}(\rho)\right).$
(53)

At NLO there are new channels that open up, specifically processes with
incoming $qq^{\prime}$ or two gluons, that are not present at the Born level.
These channels are not logarithmically enhanced and only contribute a small
correction to the distribution. 444In the current version of MCFM, the only
possibility is to separate the channels with incoming $qq^{\prime}$, $qg$, and
$gg$. The former mixes channels with incoming $q\bar{q}$ and the remaining
$qq^{\prime}$ processes, so it is not possible to obtain a clean $q\bar{q}$
channel contribution from this program.

In figure 9 we show plots for the differential jet mass distribution
$1/\sigma\,\mathrm{d}\Sigma/\mathrm{d}\sqrt{\rho}$, where $\Sigma(\rho)$ is
defined in eq. (5), in $Z$ \+ jet events at the LHC with $k_{t}$ clustering.
We choose two values for the jet radius, one for which the size of NGLs + CLs
is expected to be small, $R=1.0$, and another where NGLs + CLs are expected to
be important, e.g., $R=0.6$. The global and pure-resummed distributions are
normalised to the Born cross-section, while the resummed + $C^{(1)}$, Pythia
8, and Herwig 7 distributions are normalised to the total cross-section.

Figure 9: Resummed differential jet mass distribution in $Z$ \+ jet events
with $k_{t}$ clustering compared to Pythia 8 and Herwig 7 PS.

We observe from the $R=1.0$ plot in figure 9 that the global and full-resummed
distributions are quite close to the Pythia 8 PS result, indicating the
smallness of the effect of NGLs and CLs factors in this case. For the $R=0.6$
plot, there is a clear difference between the global and Pythia 8 PS curves,
and our full resummation, which is based on the exponential of the two-loops
NGLs and CLs result, seems to do better. We also note that the NLO term
$C^{(1)}$ slightly modifies the peak and tail of the distribution especially
for $R=0.6$, bringing it even closer to the Pythia 8 PS result.

As is clear from the plots, the Pythia 8 PS result seems to be in better
agreement with our resummed distribution near the peak than Herwig 7. This
observation was also made in ref. Dasgupta:2012hg . It should be noted,
however, that a more comprehensive comparison is feasible only when one
includes non-perturbative effects, where different event generators are then
expected to be in agreement. We do this in the next section.

In figure 10 we plot the same distribution employing the C-A algorithm. We
recall that up to two loops both $k_{t}$ and C-A algorithms produce identical
results. This means that the resummed formula that includes the exponential of
the two-loops NGLs and CLs as well as the $C^{(1)}_{\delta}$ terms for all
channels are the same in both algorithms. We expect, however, differences
between the two cases when one performs an all-orders NGLs and CLs
resummation, and also when one includes the higher-order $C^{(n)}$ terms. We
compare, in figure 10, the resummed + $C^{(1)}$ result with the Pythia 8 PS
result employing both $k_{t}$ and C-A algorithms, where we notice that the
peak of the distribution is slightly higher in the latter algorithm.

Figure 10: Resummed differential jet mass distribution in $Z$ \+ jet events
with C-A algorithm compared to Pythia 8 PS.

We additionally show in figure 11 the differential jet mass distribution in
the process $gg\to Hg$ for jet radii $R=1.0$ and $R=0.6$.

Figure 11: Resummed differential jet mass spectrum with $k_{t}$ clustering
compared to Pythia 8 and Herwig 7 parton showers for $R=1.0$ and $R=0.6$ in
the process $gg\to gH$.

Notice from figures 5 and 6 that, for this channel, the combined effect of
NGLs and CLs at two loops in the $k_{t}$ algorithm is small compared to that
of NGLs with anti-$k_{t}$ clustering, and so we expect our resummed
distribution to fit well with the PS result in the case of $k_{t}$ clustering.
This is indeed the case as is clear from figure 11, particularly for Pythia 8
PS.

Finally, in figure 12, we plot the resummed differential jet mass distribution
in the processes $W$ \+ jet and $\gamma$ \+ jet at the LHC with $k_{t}$
clustering and $R=1.0$.

Figure 12: Resummed differential jet mass distribution with $k_{t}$ clustering
compared to Pythia 8 and Herwig 7 PS for $R=1.0$ in the processes $W$ \+ jet
and $\gamma$ \+ jet at the LHC.

Our results are in general in good agreement with Pythia 8 results
particularly near the peak of the differential distribution. The discrepancy
between the results of Pythia 8 and Herwig 7 may be lifted when non-
perturbative effects are included, as we do in the next section.

###  6.1 Matching to fixed order

Before we end this section we discuss the matching of the resummed result with
the NLO fixed-order distribution. In fact, including the constant term
$C^{(1)}_{\delta}(\rho)$ in eq. (6), the expansion of the resummed
distribution now agrees with the fixed-order result over the entire range of
$\rho$, except for the small correction due to the missing channels at the
Born level (specifically the channel with incoming $gg$). Additionally, as was
shown in ref. Dasgupta:2012hg , the NLO distribution has a kinematical end
point of $\rho_{\mathrm{max}}=\tan^{2}(R/2)$, which the resummed distribution
does not have. In order to match the resummed distribution to the NLO result,
specifically at the end point, we introduce the following change of the large
logarithm Banfi:2010pa

$L=\ln\frac{R^{2}}{\rho}\to
L^{\prime}=\ln\left[\frac{R^{2}}{\rho}-\frac{R^{2}}{\rho_{\mathrm{max}}}+1\right],$
(54)

such that the large logarithm $L^{\prime}$ vanishes when
$\rho\to\rho_{\mathrm{max}}$, and $L^{\prime}\to L$ when $\rho\to 0$. We then
use the simple matching formula

$\Sigma(\rho)=\Sigma_{\mathrm{NLL}}(\rho)+\Sigma_{\mathrm{NLO}}(\rho)-\Sigma_{\mathrm{NLL},\alpha_{s}}(\rho)\,,$
(55)

where now both $\Sigma_{\mathrm{NLL}}$ and $\Sigma_{\mathrm{NLL},\alpha_{s}}$
include the $C^{(1)}$ term. The subtracted term
$\Sigma_{\mathrm{NLL},\alpha_{s}}$ cancels both the large logarithms and the
$C^{(1)}$ terms in $\Sigma_{\mathrm{NLO}}(\rho)$, leaving only corrections due
to the channels missing at the Born level. We show in figure 13 a plot of the
matched differential jet mass distribution compared to the fixed-order result
from MCFM for the $Z$ \+ jet process at the LHC. In this plot the resummed
curve is plotted with the standard definition of the large logarithm
($L=\ln(R^{2}/\rho)$), and thus does not posses the end-point character, while
the matched curve does have an end point exactly as in the MCFM curve. We note
from this figure that the matched curve coincides with the resummed curve at
small $\rho$, indicating a perfect cancellation of the large logarithms
between the expanded result $\Sigma_{\mathrm{NLL},\alpha_{s}}$ and the fixed-
order MCFM result $\Sigma_{\mathrm{NLO}}$.

Figure 13: Matched differential jet mass distribution with $k_{t}$ clustering
compared to MCFM NLO fixed-order result.

##  7 Comparison to CMS data

In order to compare our results with the experimental data we first need to
account for non-perturbative effects from hadronisation corrections and the
underlying event. One commonly used numerical approach to extract these
corrections is to compute the ratio of the results obtained from Monte Carlo
event generators with non-perturbative effects switched on and off. In this
paper we include these corrections analytically by considering the mean value
of the change in the jet mass $\delta m_{j}^{2}$ due to these non-perturbative
effects. This change was computed in ref. Dasgupta:2007wa  to be

$\langle\delta
m_{j}^{2}\rangle=\sum_{(i\ell)\in\Delta_{\delta}}\mathcal{C}_{i\ell}\,\mathcal{A}(\mu_{I})\,p_{t}\,\mathcal{N}_{i\ell}(R)\,,$
(56)

with

$\displaystyle\mathcal{N}_{ab}(R)$
$\displaystyle=\frac{1}{4}\,R^{4}+\frac{1}{4608}\,R^{8}+\cdots\,,$ (57a)
$\displaystyle\mathcal{N}_{aj}(R)$
$\displaystyle=\mathcal{N}_{bj}(R)=R+\frac{3}{16}\,R^{3}+\frac{125}{9216}\,R^{5}+\cdots\,,$
(57b)

and

$\displaystyle\mathcal{A}(\mu_{I})$
$\displaystyle=\frac{\mu_{I}}{\pi}\bigg{[}\alpha_{0}(\mu_{I})-\alpha_{s}(p_{t})$
$\displaystyle-\frac{\beta_{0}}{2\pi}\left(\ln\frac{p_{t}}{\mu_{I}}+\frac{K}{\beta_{0}}+1\right)\alpha_{s}^{2}(p_{t})\bigg{]}.$
(58)

Here $\mu_{I}$ is an arbitrary matching scale (chosen to be of order of a few
GeV) and $\alpha_{0}$ is the averaged coupling over the non-perturbative
low-$k_{t}$ region,
$\alpha_{0}=1/\mu_{I}\int_{0}^{\mu_{I}}\alpha_{s}(k_{t})\,\mathrm{d}k_{t}$.
The $\mathcal{A}(\mu_{I})$ is rescaled by the so-called Milan factor ($M=1.49$
for anti-$k_{t}$ clustering and $M=1.01$ for $k_{t}$ clustering
Dasgupta:2009tm ) to account for gluon decay. The constant $K$ is defined in
the appendix.

Non-perturbative effects are dominated by the contributions of the dipoles
involving the outgoing jet, which scale like $\mathcal{O}(R)$, and which
account for hadronisation corrections, while the smaller $\mathcal{O}(R^{4})$
contributions from the incoming legs account for the underlying event. Since
the mean value of $\delta m_{j}^{2}$ depends both on the Born channel and
kinematics, then we perform the shift on the mass of the jet on an event-by-
event basis, that is we make the change $m_{j}^{2}\to m_{j}^{2}-\delta
m_{j}^{2}$ in the resummed form factor and then perform the convolution.
Furthermore, we shift the terms $C^{(1)}(\rho)$ accordingly.

We compare, in figure 14, the NLL+NLO resummed result (with the $C^{(1)}$
term), including non-perturbative corrections, with experimental data from the
CMS collaboration Chatrchyan:2013rla ; 1224539/t32  (obtained with integrated
luminosity $\mathcal{L}=5\,\mathrm{fb}^{-5}$), in the $Z$ \+ jet process at
the LHC with anti-$k_{t}$ clustering and $R=0.7$. We also include in this
figure the Monte Carlo results obtained from interfacing MadGraph with Pythia
8 Alwall:2008qv ; Conte:2012fm  and Herwig 7 including hadronisation
corrections and the underlying event. The plots in this figure are for the un-
normalised jet mass variable $m_{j}$ rather than the normalised one
$\sqrt{\rho}=m_{j}/p_{t}$. In this figure we have
$300\,\mathrm{GeV}<p_{t}<450\,\mathrm{GeV}$. CTEQ6L parton distribution
functions Pumplin:2002vw  have been used both in the convolution and
MadGraph/Pythia 8/Herwig 7/MCFM results. For best fit we choose
$\mu_{I}=3.5\,\mathrm{GeV}$. This plot shows a good agreement between the data
and the resummed prediction over the entire range of the jet mass, as well as
with the Monte Carlo simulation.

Figure 14: Resummed differential $m_{j}$ distribution with anti-$k_{t}$
clustering and $R=0.7$ in $Z(\to\ell^{+}\ell^{-})+$ jet events, with
$\ell=e,\mu$, compared to experimental data from CMS Chatrchyan:2013rla  and
MadGraph+Pythia 8 and Herwig 7 results. The experimental data are taken from
ref. 1224539/t32 .

We note that the NLL+NLO+NP curve is cut off at around 40 GeV, which is just a
manifestation of the shift of the resummed distribution to the right, as
explained above. The value of the NLL+NLO+NP distribution at say $m_{j}=40$
GeV is related to the value of the resummed distribution at
$\sqrt{m_{j}^{2}-\langle\delta m_{j}^{2}\rangle}$, with $\sqrt{\langle\delta
m_{j}^{2}\rangle}$ (see eq. (56)) varying from 20 GeV to around 40 GeV
depending on the $p_{t}$ of the jet and partonic channel. Hence, we have no
result for the non-perturbative distribution below this value ($\sim 40$ GeV)
of the jet mass. Additionally, due to the Landau-pole singularity at small
values of the jet mass, the distribution is unreliable in the region to the
left of the Sudakov peak.

##  8 Conclusions

In this paper we have presented state-of-the-art detailed fixed-order
calculations as well as all-orders estimates of distributions of important
observables at the LHC. Specifically we have considered a typical jet-shape
observable that has been studied quite substantially in the literature, namely
the invariant jet mass. It is a member of a large class of observables known
as non-global observables, that have so far proven to be quite delicate to
treat. The subtleties in the analysis of such observables stem from the fact
that they are defined for a restricted phase-space region. This is unlike
global observables which are defined over the whole phase space. The former
non-global observables receive contributions that are totally absent for their
global counterparts. These contributions appear at each higher order in
perturbation theory and have so far shown no pattern of iteration.

We have extended the work of ref. Dasgupta:2012hg  from various angles: (a) we
have implemented two clustering algorithms, $k_{t}$ and C-A, instead of just
the anti-$k_{t}$ considered in the latter reference. Generally, computations
in $k_{t}$ and C-A algorithms are much more difficult to handle than in
anti-$k_{t}$ case; (b) we have computed CLs, which are completely absent for
anti-$k_{t}$ and thus not treated in Dasgupta:2012hg ; (c) we have
investigated the jet mass distribution in various different processes, namely
$W/Z/\gamma/H$ \+ one jet, while only $Z$ \+ jet was considered in the said
reference; and (d) we have provided analytical expressions for our results in
the form of power-series expansions in the jet radius $R$. As the experimental
data Chatrchyan:2013rla  for the jet mass distribution in $Z$ \+ jet events at
the LHC with anti-$k_{t}$ clustering were available only after the publication
of ref. Dasgupta:2012hg , we made the comparison of the resummed result with
these experimental data herein.

We have confirmed previous results that were arrived at in studies of
$e^{+}e^{-}$ annihilation processes. These include, for instance, the
observation that NGLs are decreased by the application of jet clusterings
other than anti-$k_{t}$. In other words, NGLs are more significant when
anti-$k_{t}$ is used. This may hint at the advantage of using other jet
clustering algorithms in order to bypass the difficulties posed by NGLs.
Additionally, we showed that in the limit of very small jet-radius parameter
the NGLs and CLs at hadron colliders coincide with those at $e^{+}e^{-}$
colliders. Moreover, we have been able to identify new features that are not
present in the simple $e^{+}e^{-}$ annihilation case such as the significance
of initial-state radiation and its impact on the jet mass distribution. The
jet mass provides a tool to discriminate gluon and quark-initiated jets as
their corresponding jet mass distributions were shown to be quite different.

It is worth, as a continuation to this project, investigating other crucial
hadronic processes at the LHC such as di-jet production. The latter represents
an important background for numerous potential new physics signals. Another
issue that is also worth tackling is performing calculations beyond two-gluon
emission. This will provide a deeper insight into the nature of QCD hadronic
processes that have not been fully understood so far.

## Acknowledgements

We would like to thank Amine Ahriche and Hilal Hamdellou for assistance with
MadGraph/MadAnalysis, and John M. Campbell for assistance with MCFM.

This work is supported by:

  * •

Deanship of Research at the Islamic University of Madinah (research project
No. 40/107)

  * •

PRFU: B00L02UN050120190001 (Algeria)

##  Appendix A Born cross-section

In this section we present the structure of the differential partonic Born
cross-section that is needed in the convolution with the resummed result to
obtain the jet mass distribution (eq. (6)).

The differential partonic Born cross-section (for some fixed transverse
momentum $p_{t}$ and rapidity $y$ of the outgoing hard parton) for channel
$\delta$ of the process of production of a Higgs ($H$) or a vector boson ($V$)
in association with a jet in hadron collisions may be written as follows

$\frac{\mathrm{d}\sigma_{0,\delta}}{\mathrm{d}\mathcal{B}_{\delta}}=\frac{1}{16\pi\,\hat{s}}\left|\mathcal{M}_{0,\delta}\right|^{2}\delta\left[(p_{a}+p_{b}-p_{j})^{2}-M^{2}\right],$
(59)

where $|\mathcal{M}_{0,\delta}|^{2}$ is the corresponding partonic Born
squared amplitude, summed and averaged over spins and colours, and $M$ is the
mass of the boson. For the vector-boson processes, the differential partonic
Born cross-section reduces to Ellis:1981hk ; Arnold:1988dp ; Gonsalves:1989ar

$\begin{split}&q_{i}\bar{q}_{j}\to
gV:\mathcal{K}_{V}\,\frac{2\pi\alpha_{s}\mathrm{C_{F}}}{\mathrm{N_{c}}}\,\frac{1}{\hat{s}}\,T_{0}(\hat{s},\hat{u},\hat{t})\,\delta\left(\hat{s}+\hat{t}+\hat{u}-M^{2}\right),\\\
&q_{i}g\to
q_{j}V:-\mathcal{K}_{V}\,\frac{\pi\alpha_{s}}{\mathrm{N_{c}}}\,\frac{1}{\hat{s}}\,T_{0}(\hat{t},\hat{u},\hat{s})\,\delta\left(\hat{s}+\hat{t}+\hat{u}-M^{2}\right),\end{split}$
(60)

where the couplings are

$\displaystyle\mathcal{K}_{Z}$
$\displaystyle=\delta_{ij}\,\frac{\sqrt{2}\,\mathrm{G_{F}}\,M_{Z}^{2}}{4\pi}\left(g_{iV}^{2}+g_{iA}^{2}\right),$
(61a) $\displaystyle\mathcal{K}_{W}$
$\displaystyle=\left|V_{ij}\right|^{2}\frac{\sqrt{2}\,\mathrm{G_{F}}\,M_{W}^{2}}{4\pi}\,,$
(61b) $\displaystyle\mathcal{K}_{\gamma}$
$\displaystyle=\delta_{ij}\,e_{i}^{2}\,\alpha\,,$ (61c)

with $\mathrm{G_{F}}$ the Fermi coupling, and $g_{iV}$ and $g_{iA}$ the vector
and axial-vector couplings given by

$\displaystyle g_{iV}$ $\displaystyle=T^{i}_{3}-2\,e_{i}\sin^{2}\theta_{w}\,,$
(62a) $\displaystyle g_{iA}$ $\displaystyle=T^{i}_{3}\,,$ (62b)

where $T^{i}_{3}$ is the weak isospin of the quark $q_{i}$ ($+\frac{1}{2}$ for
up-type and $-\frac{1}{2}$ for down-type quarks), $e_{i}$ is the fraction of
electric charge carried by quark $q_{i}$, and $\theta_{w}$ is the Weinberg
angle. Moreover, $V_{ij}$ are CKM matrix elements corresponding to flavours
$i$ and $j$, with $i$ and $j$ being of different magnitude of electric charge
(we exclude the top quark). In the above, $\alpha$ is the electromagnetic
coupling. The kinematical factor $T_{0}$ describing the matrix-element squared
of the underlying Born scattering is given by

$T_{0}(\hat{s},\hat{u},\hat{t})=\frac{\hat{t}^{2}+\hat{u}^{2}+2\,\hat{s}\left(\hat{s}+\hat{t}+\hat{u}\right)}{\hat{t}\,\hat{u}}\,,$
(63)

with $\hat{s}$, $\hat{t}$ and $\hat{u}$ the partonic Mandelstam variables

$\displaystyle\hat{s}$
$\displaystyle=\left(p_{a}+p_{b}\right)^{2}=x_{a}\,x_{b}\,s\,,$ (64a)
$\displaystyle\hat{t}$
$\displaystyle=\left(p_{a}-p_{j}\right)^{2}=-x_{a}\,\sqrt{s}\,p_{t}\,e^{-y}\,,$
(64b) $\displaystyle\hat{u}$
$\displaystyle=\left(p_{b}-p_{j}\right)^{2}=-x_{b}\,\sqrt{s}\,p_{t}\,e^{y}\,.$
(64c)

For the process $gg\to Hg$ (i.e. channel $(\delta_{3})$) we have
Ravindran:2002dc

$\displaystyle\frac{\mathrm{d}\sigma_{0,\delta_{3}}}{\mathrm{d}\mathcal{B}_{\delta_{3}}}$
$\displaystyle=\frac{\mathrm{G_{F}}}{\sqrt{2}}\,\tau^{2}\,F^{2}\,\frac{\pi}{\mathrm{C_{F}}}\left(\frac{\alpha_{s}}{4\pi}\right)^{3}\frac{1}{\hat{s}}\,\times$
$\displaystyle\times\frac{\hat{s}^{4}+\hat{t}^{4}+\hat{u}^{4}+m^{8}}{\hat{s}\,\hat{t}\,\hat{u}}\,\delta\left(\hat{s}+\hat{t}+\hat{u}-M_{H}^{2}\right),$
(65)

where $\tau=7.65$ and $F=0.09$.

The total Born cross-section $\sigma_{0}$ is simply the integral of
$\mathrm{d}\sigma_{0,\delta}/\mathrm{d}\mathcal{B}_{\delta}$ (including the
kinematical-cuts function $\Xi_{\mathcal{B}}$) over $\mathcal{B}_{\delta}$,
summed over possible $\delta$, where we have

$\mathrm{d}\mathcal{B}_{\delta}=\mathrm{d}x_{a}\,\mathrm{d}x_{b}\,f_{a}(x_{a},\mu_{\mathrm{F}}^{2})\,f_{b}(x_{b},\mu_{\mathrm{F}}^{2})\,\mathrm{d}p_{t}^{2}\,\mathrm{d}y\,,$
(66)

where $f_{i}$ denotes the parton density function for the corresponding
incoming parton $(i)$ evaluated at a factorisation scale $\mu_{\mathrm{F}}$.

Substituting the Mandelstam variables into the delta function in the integrand
we obtain

$\delta\left[x_{a}\,x_{b}\,s-x_{a}\,\sqrt{s}\,p_{t}\,e^{-y}-x_{b}\,\sqrt{s}\,p_{t}\,e^{y}-M^{2}\right].$
(67)

This delta function can be used to perform the integration over one of the
$x$’s, say $x_{b}$, and thus we set

$x_{b}=\frac{x_{a}\,e^{-y}\,p_{t}/\sqrt{s}+M^{2}/s}{x_{a}-e^{y}\,p_{t}/\sqrt{s}}\,,$
(68)

and multiply the integrand by

$\Theta(1-x_{b})\,\Theta(x_{b})/\left(x_{a}\,s-\sqrt{s}\,p_{t}\,e^{y}\right).$
(69)

Since $x_{b}>0$ then $x_{a}>e^{y}\,p_{t}/\sqrt{s}$ and
$y<\ln(\sqrt{s}/p_{t})$. Additionally, since $x_{b}<1$ then
$y>-\ln(\sqrt{s}/p_{t})$ and

$x_{a}>\frac{e^{y}\,p_{t}/\sqrt{s}+M^{2}/s}{1-e^{-y}\,p_{t}/\sqrt{s}}\,.$ (70)

The latter inequality overrules $x_{a}>e^{y}\,p_{t}/\sqrt{s}$, and
furthermore, since $x_{a}<1$ we deduce that

$\cosh y<\left(1-\frac{M^{2}}{s}\right)\frac{1}{2}\,\frac{\sqrt{s}}{p_{t}}\,,$
(71)

which also overrules the condition $|y|<\ln\sqrt{s}/p_{t}$.

We perform the integration over $p_{t}$, $y$ and $x_{a}$, either in the Born
cross-section or in the jet mass distribution, numerically via Monte Carlo
method.

##  Appendix B Resummed Global form factor

The Sudakov global form factor that resums global logarithms is given in eq.
(28). The radiator $\mathcal{R}$ is composed of contributions from double-
logarithmic soft-collinear and single-logarithmic hard-collinear emissions
from the outgoing leg $(j)$, and from single-logarithmic soft wide-angle
emissions from all legs. The leading-order soft wide-angle contribution that
we calculated in section 3.1 simply exponentiates to all orders. Additionally,
the non-global and clustering corrections appear as a factorised part that
multiplies the resummed global form factor. The remaining soft-collinear and
hard-collinear contributions may be obtained using the general formalism of
resummation introduced in ref. Banfi:2004yd  as we show below.

First we write the definition of the observable $\varrho$ in terms of the
transverse momentum $k_{t}^{(\ell)}$, rapidity $\eta^{(\ell)}$, and azimuth
$\phi^{(\ell)}$ of a single soft-collinear emission with respect to the
direction of the hard leg $(\ell)$. Emissions that are collinear to the
incoming legs $(a)$ and $(b)$ do not end up inside the jet, so they do not
contribute to its mass, hence $\varrho^{(a)}=\varrho^{(b)}=0$. For emissions
that are collinear to the outgoing leg $(j)$, we introduce a coordinate
rotation that takes the momentum of leg $(j)$ to the $z$ axis, where the
momenta of the jet and the soft emission become

$\displaystyle p_{j}^{(j)}$ $\displaystyle=p_{t}\cosh y\left(1,0,0,1\right),$
(72a) $\displaystyle k^{(j)}$
$\displaystyle=k_{t}^{(j)}\left(\cosh\eta^{(j)},\cos\phi^{(j)},\sin\phi^{(j)},\sinh\eta^{(j)}\right).$
(72b)

The jet mass observable (being invariant under rotations) is then given by

$\varrho^{(j)}=2\,\frac{k^{(j)}\cdot
p_{j}^{(j)}}{p_{t}^{2}}=2\,\frac{k_{t}^{(j)}}{p_{t}}\,e^{-\eta^{(j)}}\,\cosh
y\,.$ (73)

Comparing the definition of the normalised invariant jet mass (73) to the
general parametrisation of observables from ref. Banfi:2004yd

$V=d_{\ell}\left(\frac{k_{t}^{(\ell)}}{Q}\right)^{a_{\ell}}e^{-b_{\ell}\,\eta^{(\ell)}}\,g_{\ell}(\phi^{(\ell)})\,,$
(74)

where $Q=p_{t}$ is the hard scale, we see that $a_{j}=b_{j}=g_{j}=1$ and
$d_{j}=2\cosh y$.

Employing the master formula for resummation (eq. (3.6) from ref. Banfi:2004yd
) we obtain the expression of the radiator $\mathcal{R}_{\delta}(\rho)$, for a
given Born channel $\delta$, in the $\overline{\mathrm{MS}}$ renormalisation
scheme

$\displaystyle\mathcal{R}_{\delta}(\rho)$
$\displaystyle=C_{j}\left[L\,g_{1}(\alpha_{s}L)+g_{2}(\alpha_{s}L)+g_{2,\mathrm{coll}}(\alpha_{s}L)\right]+$
$\displaystyle+g_{2,\mathrm{wide}}(\alpha_{s}L)\left[\mathcal{C}_{ab}\,\frac{R^{2}}{2}+\left(\mathcal{C}_{aj}+\mathcal{C}_{bj}\right)h(R)\right],$
(75)

with $h(R)$ being given in eq. (26). Here $C_{j}$ is the colour factor
associated with leg $j$, $C_{j}=\mathrm{C_{F}}$ for outgoing (anti-) quark jet
and $C_{j}=\mathrm{C_{A}}$ for outgoing gluon jet, and $\mathcal{C}_{i\ell}$
is the colour factor for dipole $(i\ell)$ introduced in the main text. We have

$\displaystyle
g_{1}=\frac{1}{2\pi\beta_{0}\lambda}\left[(1-2\lambda)\ln(1-2\lambda)-2(1-\lambda)\ln(1-\lambda)\right],$
(76a) $\displaystyle
g_{2}=\frac{\mathrm{K}}{4\pi^{2}\beta_{0}^{2}}\left[2\ln(1-\lambda)-\ln(1-2\lambda)\right]+\frac{\beta_{1}}{2\pi\beta_{0}^{3}}\times$
$\displaystyle\left[\frac{1}{2}\ln^{2}(1-2\lambda)-\ln^{2}(1-\lambda)+\ln(1-2\lambda)-2\ln(1-\lambda)\right],$
(76b) $\displaystyle
g_{2,\mathrm{coll}}=-B_{j}\,\frac{1}{\pi\beta_{0}}\,\ln(1-\lambda)\,,$ (76c)
$\displaystyle g_{2,\mathrm{wide}}=-\frac{1}{2\pi\beta_{0}}\ln(1-2\lambda)\,,$
(76d)

with $\lambda=\alpha_{s}(R\,p_{t})\,\beta_{0}\,\ln(R^{2}/\rho)$. The factor
$B_{j}$ accounts for corrections due to hard-collinear emissions to the
outgoing jet $j$ and is given by

$\displaystyle B_{q}$ $\displaystyle=-\frac{3}{4}$ $\displaystyle\text{for
quark jets}\,,$ (77) $\displaystyle B_{g}$
$\displaystyle=-\frac{11\,\mathrm{C_{A}}-4\,\mathrm{T_{R}}\,\mathrm{n_{f}}}{12\,\mathrm{C_{A}}}=-\frac{\pi\beta_{0}}{\mathrm{C_{A}}}\quad$
$\displaystyle\text{for gluons jets}\,,$

with $\mathrm{T_{R}}$ the Dynkin index (normalisation constant) for the
$\mathrm{SU}(\mathrm{N_{c}})$ generators, $\mathrm{T_{R}}=1/2$, and
$\mathrm{n_{f}}=5$ the number of active quark flavours. Additionally we have

$\begin{split}\mathrm{K}&=\mathrm{C_{A}}\left(\frac{67}{18}-\frac{\pi^{2}}{6}\right)-\frac{5}{9}\,\mathrm{n_{f}}\,,\\\
\beta_{0}&=\frac{11\,\mathrm{C_{A}}-2\,\mathrm{n_{f}}}{12\,\pi}\,,\\\
\beta_{1}&=\frac{17\,\mathrm{C_{A}^{2}}-5\,\mathrm{C_{A}}\,\mathrm{n_{f}}-3\,\mathrm{C_{F}}\,\mathrm{n_{f}}}{24\pi^{2}}\,.\end{split}$
(78)

In the master formula we excluded the single-logarithmic soft wide-angle term
referred to in ref. Banfi:2004yd  as $\ln S(T)$, and we calculated it manually
in section 3.1. It appears as the last term in the radiator (B).

The derivative of the radiator $\mathcal{R}$ with respect to $L$, relevant in
the expression (28), is given by

$\mathcal{R}^{\prime}=\frac{\partial\mathcal{R}}{\partial
L}=\frac{C_{j}}{\pi\beta_{0}}\left[\ln(1-\lambda)-\ln(1-2\lambda)\right].$
(79)

##  Appendix C Fixed-order expansion

For the sake of matching we need the fixed-order expansion of the resummed
form factor (49). We can cast the latter in the form

$\displaystyle\frac{\mathrm{d}\Sigma_{\delta}(\rho)}{\mathrm{d}\mathcal{B}_{\delta}}$
$\displaystyle=\frac{\mathrm{d}\sigma_{0,\delta}}{\mathrm{d}\mathcal{B}_{\delta}}\,C_{\delta}(\rho)\,\exp\left[\sum_{n=1}^{\infty}\sum_{m=1}^{n+1}G_{nm}\,\bar{\alpha}_{s}^{n}\,L^{m}\right]$
$\displaystyle=\frac{\mathrm{d}\sigma_{0,\delta}}{\mathrm{d}\mathcal{B}_{\delta}}\,\sum_{n=0}^{\infty}\sum_{m=0}^{2n}H_{nm}\,\bar{\alpha}_{s}^{n}\,L^{m},$
(80)

where the expansion coefficients in the exponent, $G_{nm}$, up to
$\mathcal{O}(\bar{\alpha}_{s}^{2})$, are

$\displaystyle G_{12}$ $\displaystyle=-\frac{C_{j}}{2}\,,$ $\displaystyle
G_{11}$
$\displaystyle=-B_{j}\,C_{j}-\mathcal{C}_{ab}\,\frac{R^{2}}{2}-\left(\mathcal{C}_{aj}+\mathcal{C}_{bj}\right)h(R)\,,$
$\displaystyle G_{23}$ $\displaystyle=-\frac{\pi\beta_{0}}{2}\,C_{j}\,,$
$\displaystyle G_{22}$
$\displaystyle=-\frac{C_{j}}{4}\left(K+2\pi\beta_{0}\,B_{j}\right)-\pi\beta_{0}\left(\mathcal{C}_{aj}+\mathcal{C}_{bj}\right)h(R)+$
$\displaystyle-\pi\beta_{0}\,\mathcal{C}_{ab}\,\frac{R^{2}}{2}+\frac{\mathcal{F}_{2}^{\delta}}{2}-\frac{\mathcal{S}_{2}^{\delta}}{2}-\frac{\zeta_{2}}{2}\,.$
(81)

and the expansion coefficients in the series, $H_{nm}$, are

$\displaystyle H_{12}=G_{12}\,,\qquad H_{11}=G_{11}\,,\qquad
H_{10}=C^{(1)}_{\delta}\,,$ $\displaystyle H_{24}=\frac{G_{12}^{2}}{2},\qquad
H_{23}=G_{23}+G_{12}\,G_{11},$ $\displaystyle
H_{22}=G_{22}+\frac{G_{11}^{2}}{2}+\left(C^{(1)}_{\delta}\right)G_{12}.$ (82)

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