Generalized fragmentation functions for fractal jet observables
Received: May
Generalized fragmentation functions for fractal jet observables
Benjamin T. Elder 0 1 3 6 7
Massimiliano Procura 0 1 3 4 7
Jesse Thaler 0 1 3 6 7
Wouter J. Waalewijn 0 1 2 3 5 7
Kevin Zhou 0 1 3 6 7
Geneva 0 1 3 7
Switzerland 0 1 3 7
0 Science Park 904 , 1098 XH Amsterdam , The Netherlands
1 University of Amsterdam
2 Nikhef, Theory Group
3 77 Massachusetts Ave. , Cambridge, MA 02139 , U.S.A
4 Theoretical Physics Department , CERN
5 Institute for Theoretical Physics Amsterdam and Delta Institute for Theoretical Physics
6 Center for Theoretical Physics, Massachusetts Institute of Technology , USA
7 Science Park 105 , 1098 XG, Amsterdam , The Netherlands
We introduce a broad class of fractal jet observables that recursively probe the collective properties of hadrons produced in jet fragmentation. To describe these collinearunsafe observables, we generalize the formalism of fragmentation functions, which are important objects in QCD for calculating cross sections involving identi ed hadrons. Fragmentation functions are fundamentally nonperturbative, but have a calculable renormalization group evolution. Unlike ordinary fragmentation functions, generalized fragmentation functions exhibit nonlinear evolution, since fractal observables involve correlated subsets of hadrons within a jet. Some special cases of generalized fragmentation functions are reviewed, including jet charge and track functions. We then consider fractal jet observables that are based on hierarchical clustering trees, where the nonlinear evolution equations also exhibit treelike structure at leading order. We develop a numeric code for performing this evolution and study its phenomenological implications. As an application, we present examples of fractal jet observables that are useful in discriminating quark jets from gluon jets.
Jets; QCD Phenomenology

nalstate
3.1
3.2
4.1
4.2
4.3
4.4
4.5
1 Introduction
2 Formalism
2.1
Review of standard fragmentation
2.2 Introducing generalized fragmentation
2.3 Introducing fractal observables
3 Fractal observables via clustering trees
Construction
Requirements
3.3 Evolution equations
4
Weighted energy fractions
Associativity
Extraction of GFFs
Evolution of GFFs
Limits
Moment space analysis
5 Treedependent observables
5.1
Node products
5.2 Fulltree observables
6 Application in quark/gluon discrimination
7 Fractal observables from subjets
8 Conclusions
A Generalized fragmentation in inclusive jet production
B A nonfractal example: sums of weighted energy fractions
C Software implementation
C.1 Running coupling
C.2 Discretization
C.3 RungeKutta algorithm
D Numerical stability
{ 1 {
E.1
E.2
1This should not be confused with \extended fractal observables" recently introduced in ref. [12], which
are based on determining the fractal dimension of a jet.
2While it would be more accurate to call eq. (1.2) the \energy fraction", we use momentum fraction
since that is more common in the fragmentation function literature.
{ 2 {
Fragmentation functions (FFs) have a long history in QCD for calculating cross sections
safe, cross sections for singlehadron observables have singularities beginning at O( s).
These collinear singularities are absorbed by the FFs order by order in
s
. From this
singularity structure, one can derive the renormalization group (RG) evolution for FFs,
leading to the wellknown DGLAP equations [8{11]. This evolution is linear, since FFs
depend only on the momentum of a single hadron in the nal state.
In this paper, we present a formalism for generalized fragmentation functions (GFFs),
which describe the ow of momentum from a fragmenting quark or gluon into subsets of
nalstate hadrons. Because GFFs depend on correlations between
nalstate hadrons,
their evolution equations are nonlinear and therefore more complicated than in the
ordinary FF case. Motivated by the structure of the DGLAP equations, we de ne fractal jet
observables where the evolution, albeit nonlinear, takes a special recursive form that is
wellsuited to numerical evaluation.1
Speci cally, we focus on observables de ned using hierarchical binary clustering trees
that mimic the leadingorder treelike structure of the evolution equations. A fractal jet
observable x can then be de ned recursively according to gure 1 as
where x1 and x2 are the values of the observable on the branches of a 1 ! 2 clustering
tree, and z is the momentum sharing between branches, de ned by
x = x^(z; x1; x2);
E1
E1 + E2
z
s( )
p~; x = x^(z; x1; x2)
p~1
, x1
p~2, x
2
recursive step, the value x for the mother is expressed in terms of the momentum fraction z and
the value x1 and x2 of the observable for the daughters.
where Fi(x; ) is the GFF for parton i = fu; u; d; : : : ; gg, Pi!jk(z) is the 1 ! 2 QCD
splitting function, and
is the MS renormalization scale. This evolution equation has the
same structure as a 1 ! 2 parton shower, which is su ciently straightforward to implement
numerically. Although we mostly restrict ourselves to lowest order in perturbation theory,
our framework allows for the systematic inclusion of higherorder corrections, in contrast
to the semiclassical parton shower approach.
The class of fractal jet observables described by eq. (1.1) is surprisingly rich, allowing
for many collinearunsafe observables to be calculated with the help of GFFs. For example,
eq. (1.3) describes the evolution of weighted energy fractions,
HJEP06(217)85
X
a2jet
x =
where wa is a weight factor that depends on nonkinematic quantum numbers such as
charge or
avor,
> 0 is an energy weighting exponent, and the sum extends over all
jet constituents. These observables are de ned by associative recursion relations, such
that their value is independent of the choice of clustering tree. Examples of weighted
energy fractions include weighted jet charge [13], whose nonlinear evolution was rst studied
in ref. [14]; track functions which characterize the fraction of a jet's momentum carried
by charged particles [15, 16]; and the observable pTD used by the CMS experiment for
quark/gluon discrimination [17, 18], whose nonlinear evolution was rst studied in ref. [19].
adapts to hadronic collisions with jets of transverse momentum pjTet.
While we focus on the case of e+e
collisions with jets of energy Ejet, our formalism easily
In addition to performing a more general analysis of weighted energy fractions, we
also present examples of fractal observables with nonassociative recursion relations. These
quantities depend on the details of the clustering tree used to implement eq. (1.1), providing
a complementary probe of jet fragmentation. In particular, while eq. (1.1) does not involve
any explicit angular separation scales, the clustering tree does introduce an implicit angular
dependence. Remarkably, the details of the clustering do not a ect the leadingorder RG
evolution in eq. (1.3) considered in this paper, beyond the requirement that particles are
appropriately clustered in the collinear limit. An example of a nonassociative fractal
observable is given by nodebased energy products,
(1.4)
(1.5)
x =
X (4zLzR) =2 ;
nodes
{ 3 {
where the observable depends on the momentum fractions carried by the left and right
branches at each node in the clustering tree. We also study observables de ned entirely in
terms of eq. (1.1), with no obvious simpli cation. This sensitivity to the tree structure
allows nonassociative observables to probe parton fragmentation from a di erent perspective
than previouslystudied jet observables. As one application, we consider the
discrimination between quark and gluoninitiated jets (see e.g. [19{27] for recent studies). We nd
that fractal observables are e ective for this purpose, in some cases yielding improved
quark/gluon separation power compared to weighted energy fractions.
For clustering trees obtained from the Cambridge/Aachen (C/A) algorithm [28, 29],
the depth in the tree is directly related to the angular separation scale between subjets.
This opens up the possibility of modifying the recursion relation x^ in eq. (1.3) to be a
function of angular scale. For example, starting from a jet of radius R, one can introduce
a subjet radius parameter Rsub
R such that evolution equation takes a di erent form
below and above Rsub. A particularly simple case is if the weighted energy fraction with
= 1 is measured on the branches below Rsub, since this e ectively amounts to de ning
fractal observables in terms of subjets of radius Rsub. In this case, the initial conditions
for the GFF leadingorder evolution is simply given by Fi(x; sub) = (1
x) at the initial
scale sub = EjetRsub
QCD, such that no nonperturbative input is needed. By evolving
the GFFs to
= EjetR, we achieve the resummation of leading logarithms of Rsub=R.
Related evolution techniques have been used to resum logarithms of the jet radius R in
inclusive jet cross sections [30{32].
The formalism of GFFs is reminiscent of other multihadron FFs in the literature. This
includes dihadron fragmentation functions which describe the momentum fraction carried
by pairs of nalstate hadrons [33, 34], and fracture functions which correlate the properties
of one initialstate and one nalstate hadron [35, 36]. In all of these cases, the RG evolution
equations are nonlinear. The key di erence here is that fractal jet observables are not
based on a xed number of hadrons, but rather allow for arbitrary hadron multiplicities.
Depending on the observable, this may require that all hadrons can be consistently labeled
by nonkinematic quantum numbers (e.g. charge). As discussed in ref. [14] for the case
of weighted jet charge, the nth moment of GFFs can sometimes be related to moments
of nhadron FFs. At the level of the full distribution, though, GFFs are distinct from
multihadron FFs, and thereby probe complementary aspects of jet fragmentation.
The rest of this paper is organized as follows. In section 2, we review the theoretical
underpinnings of ordinary parton fragmentation and explain how to extend the formalism
to generalized fragmentation and fractal observables. We then construct generic fractal jet
observables using clustering trees in section 3. In section 4, we treat the case of weighted
energy fractions, exploring their RG evolution for a range of parameters. We introduce two
new sets of nonassociative fractal observables in section 5  node products and fulltree
observables  and motivate their application in quark/gluon discrimination in section 6.
We brie y explain how our formalism also applies to fractal observables based on subjets
rather than hadrons in section 7. We conclude in section 8, leaving calculational details
and a description of the numerical RG implementation to the appendices.
{ 4 {
Formalism
To motivate the de nition of fractal jet observables, it is instructive to rst review the
formalism of standard fragmentation and then generalize it to arbitrary collinearunsafe
observables. We give a general de nition of fractal jet observables at the end of this section,
which serves as a preamble to the explicit constructions in section 3.
2.1
Review of standard fragmentation
Ordinary FFs, denoted by Dih(x; ), are nonperturbative objects that describe the number
density of hadrons of type h carrying momentum fraction x among the particles resulting
from the fragmentation of a parton of type i. They are the
nalstate counterpart to
parton distribution functions (PDFs). For any parton avor i, they satisfy the momentum
where we are working in a frame with quark transverse momentum p~
gauge choice A
ph with p
h
= 0. The jetlike state jhXi contains an identi ed hadron h of momentum
xp , and X refers to all other hadrons in that state. The factor 1=(2NC ),
where NC = 3 is the number of colors, accounts for averaging over the color and spin of
the quark
eld
of avor i. Here and in the rest of the paper, we adopt the following
convention for decomposing a fourvector w in lightcone coordinates:
? = 0 and using the
(2.1)
(2.2)
w
= w
n
2
+ w+ n
2
+ w ;
?
w
= n w;
w+ = n w;
(2.3)
where n is a lightlike vector along the direction of the energetic parton, and n is de ned
such that n2 = n2 = 0 and n n = 2. Thus at leading order, p
= 2Ejet. Gauge invariance
requires adding eikonal Wilson lines in eq. (2.2) (see e.g. [38]), which we suppress here for
notational convenience. An analogous de nition applies for the gluon FF.
In the context of e+e
annihilation, FFs are crucial ingredients in the factorization
formula for the semiinclusive cross section at leading power in
QCD=ps,
1 d
(0) dx
(e+e
! hX) =
X Z 1 dz
i
x
z
Ci(z; s; ) Dih(x=z; );
(2.4)
where x = 2Eh= s
p
1 is the hadron energy fraction, (0) is the treelevel cross section and
X represents all other nal state particles in the process.3 The coe cients Ci(z; s; ) are
3In the literature (see e.g. [39]), the cross section 1= (0) d =dx(e+e
referred to as the total FF, in which case Dih(x; ) is called the parton FF.
! hX) = F h(x; ) is sometimes
{ 5 {
processdependent perturbative functions that encode the physics of the hard subprocess.
The FFs Dih(x; ) are universal, processindependent functions, which appear (with
appropriate PDF convolutions) in related channels such as ep ! hX or pp ! hX. Since the
coe cients Ci contain logarithms of s= 2, in order to avoid terms that could spoil
perturbative convergence in eq. (2.4), the renormalization scale
should be chosen close to ps.
While computing the FFs themselves requires nonperturbative information about the
hadronic matrix elements in eq. (2.2), their scale dependence is perturbatively calculable.
This allows us to, for example, take FFs extracted from
ts to experimental data at one
scale and evolve them to another perturbative scale. The RG evolution of FFs is described
by the DGLAP equations [8{11],
Here, the splitting kernels Pji(z) can be calculated in perturbation theory,
Pji(z) = Pj(i0)(z) +
s Pj(i1)(z) + : : : ;
2
and are at lowest order the same as the splitting kernels for PDF evolution. The nextorder
splitting function Pj(i1) arises from 1 ! 3 splittings as well as loop corrections to 1 ! 2
splittings.
In order to motivate the transition to generalized fragmentation, it is convenient to
rewrite the lowestorder splitting function explicitly as a 1 ! 2 process:
Pj(i0)(z)
Pi!jk(z);
where the parton j carries momentum fraction z, e.g. Pg!gg(z) or Pq!qg(z) = Pq!gq(1 z).
With this notation, we can rewrite the leadingorder DGLAP equation in a suggestive form4
(2.5)
(2.6)
(2.7)
d
d
Though we have written eq. (2.8) as an integral over both x1 and x2, corresponding to the
two nal state branches from the i ! jk splitting, the FFs only require information about
one single nalstate hadron in each term, so the evolution simpli es to the linear form in
eq. (2.5). This will no longer be the case with generalized fragmentation, which depends
on correlations between the nalstate hadrons.
2.2
Introducing generalized fragmentation
We now extend the FF formalism to handle the distribution of quantities x carried by a
subset S of collinear particles, where x can be more general than the simple momentum
4Because the splitting functions are divergent as z ! 1 and as z ! 0, plusfunction regulators are
required at both endpoints when integrating over the entire range 0
z
{ 6 {
fraction and S is de ned by nonkinematic quantum numbers. For example, we will consider
observables de ned on all particles within a jet, but also on charged particles only. For a
given observable x, there is a GFF for each parton species i, which we denote by Fi(x; ).
At lowest order in
s, the GFF is the probability density for the particles in S to yield a
value of the observable x from jets initiated by a parton of type i. The GFF automatically
includes information about hadronization
uctuations. Being a probability density, the
GFFs are normalized to unity for each parton type,
Z
2 h0j i(y+; 0; y?)jSXihSXj i(0)j0i ;
(2.10)
to be compared with eq. (2.2). Here, jSXi is the asymptotic
nal state divided into
the measured subset S and unmeasured subset X, and x~(p ; S) is the functional form
of the quantity being observed, which can depend on the overall jet momentum and any
information from S. We stress that, in contrast to the standard FFs, a GFF involves a
sum over polarizations and a phasespace integration over all detected particles in S; if the
measured set S consists of a single hadron, then eq. (2.10) reduces to eq. (2.2) for a quark
FF. The de nition for gluoninitiated jets is
Fg(x; ) =
(d
1
2)(NC2
1)p
Z
?
dy+d2y eip y+=2 X
[x
x~(p ; S)]
SX
h0jG ;a(y+; 0; y?)jSXihSXjG ;a; (0)j0i;
(2.11)
where G ;a = n Ga is the gluon eld strength tensor for generator T a, the factor of
1=(d
2) comes from averaging over the gluon polarizations in d spacetime dimensions,
and the factor of 1=(NC2
1) comes from averaging over the color of the gluon.
The de nitions in eqs. (2.10) and (2.11) extend the ones introduced in ref. [15] for track
functions. In the track function case, x is the momentum fraction carried by the charged
particles in the nal states, irrespective of their individual properties or multiplicities. As
mentioned in the introduction, GFFs are reminiscent of multihadron FFs [33, 34], with
the key di erence that multihadron FFs describe a xed number of identi ed
hadrons (i.e. two in the case of dihadron FFs), whereas GFFs allow for a variable number
of nalstate hadrons in the subset S.
With these GFFs in hand, we can calculate the cross section di erential in the fractal
observable x for an inclusive jet sample with radius parameter R
1. Letting zJ be the
fraction of the centerofmass energy carried by the measured jet (zJ
2Ejet=Ecm), we
{ 7 {
have
1
d
dz dx1 dx2 Ji(!1)jk(z; EjetR; ) Fj (x1; ) Fk(x2; ) [x x^(z; x1; x2)]
dz dx1 dx2 Ji(!2)jk(y0; z; EjetR; ) Fj (x1; ) Fk(x2; ) [x
x^(z; x1; x2)] + : : : ;
where the ellipsis includes further terms at nexttonextto leading order and (0) denotes
the treelevel cross section. There is a similar version of eq. (2.12) for pp and ep collisions
with the inclusion of PDFs, where the jet rapidity would appear in the Ci coe cients.
As in eq. (2.4), the e ects of the hard interaction producing a parton i are encoded in
At leading order, the jet only consists of parton i, thus C(0)(zJ ) =
i
the coe cients Ci, which can be expanded perturbatively and depend on zJ and Ecm.
(1
zJ ) and the
dependence on the fractal observable x arising from parton production and hadronization
is described simply by Fi. For most of the paper, we restrict ourselves to leading order,
though we stress that eq. (2.12) provides the tools to interface our GFF formalism with
xedorder calculations and to extract GFFs beyond leading order.
At nexttoleading order in eq. (2.12), the parton i can undergo a perturbative splitting
(2)
into partons j and k. If only j is inside the jet then zJ < 1, as described by the perturbative
coe cient Ji(!1)j that can be derived from ref. [31], and the xdependence is described by
Fj . If both partons belong to the jet then again zJ = 1, but the observable x now follows
from combining the values x1 and x2 of the GFFs for partons j and k with the momentum
fraction z of the perturbative splitting described by the Ji!jk from ref. [14]. At
nexttonexttoleading order, there are even more contributions, including one with three partons
(1)
in the jet involving Ji!jk`. In eq. (2.12), we displayed only the term with two partons
belonging to the jet, since it is the rst term that directly correlates zJ and z. The natural
scale of the coe cients Ji!j ; Ji!jk; : : : ; is the typical jet invariant mass EjetR, so we
conclude that the GFFs should be evaluated at
' EjetR to minimize the e ect of
higherorder corrections. If R & 1, then Ci and J can be combined, and the natural scale to
evaluate the GFF would be
' Ejet.
It is important to note that eq. (2.12) really combines two di erent formalisms. The
rst is the formalism for GFFs discussed initially in refs. [14, 15] for trackbased observables
and further developed here. The second is the formalism for fragmentation in inclusive jet
production of refs. [32, 40], which builds upon work on fragmentation in exclusive jet
samples refs. [41{44]. Both of these formalisms are needed to perform higherorder jet
calculations, though at leading order, the GFF formalism alone su ces. For the interested reader,
we provide all details of the matching for e+e
! jet + X at nexttoleading order in
appendix A. As in refs. [14, 15], we expect that the absorption of collinear divergences by GFFs
can be carried out orderbyorder in s due to the universality of the collinear limits in QCD.
{ 8 {
The above generalized fragmentation formalism works for any collinearunsafe (but
softsafe) observable. The RG evolution for a generic Fi(x; ), however, can be very
complicated. In order to deal with numerically tractable evolution equations, we focus on
observables whose RG evolution simpli es to a nonlinear version of eq. (2.8). Speci cally,
we want to
nd the most general form of the function x~(p ; S) in eqs. (2.10) and (2.11)
such that the RG evolution of Fi(x; ) depends only on itself and other GFFs for the same
observable, and does not mix with other functions. An example of an observable that
involves GFF mixing is given in appendix B, where the evolution equation is considerably
more complicated than considered below.
We de ne fractal observables as those whose GFFs obey the (leadingorder) RG
equation in eq. (1.3), repeated here for convenience:
d
where x^(z; x1; x2) is a function related to x~(p ; S), which now depends on the momentum
p only through the momentum sharing z. As advertised, the evolution of Fi(x; ) depends
only on GFFs for the same observable x, and no other nonperturbative functions. We leave
a detailed discussion of higherorder evolution to future work, and focus primarily on the
leadingorder evolution here. As a consistency check, the
function in eq. (2.13) ensures
that the RG evolution automatically preserves the GFF normalization,
d Z
d
dx Fi(x; ) =
1 X Z
2 j;k
dz s( )
Z
Z
where we used the fact that Pj;k R dz Pi!jk(z) = 0.
As a simple example of a fractal observable, consider the momentum fraction x carried
by a subset S of hadrons of a common type. This case has already been studied in the
context of track functions [15, 16], where S corresponded to charged particles. Treating
the states jSXi in eqs. (2.10) and (2.11) partonically, the nexttoleadingorder bare GFF
in dimensional regularization with d = 4
2 satis es
Fi(1)(x) =
1 X Z
2 j;k
Z
dz s( )
2
1
UV
1
IR
Here, the function x^(z; x1; x2) is the form of x~(p ; S) written in terms of two subjets,
where x1 and x2 are the momentum fractions carried by particles belonging to subjets 1
and 2 within S, and z is the momentum fraction carried by subjet 1, as de ned in eq. (1.2).
Renormalizing the UV divergences in eq. (2.15) in the MS scheme leads directly to the RG
{ 9 {
w1
p
1
x12
p1
+
p2
p
2
w2
p
3 +
p
4
w3
p
3
x34
p
4
w4
edge has a momentum value pi, which is used to calculate the momentum fraction z of the splitting
at each nonleaf node. The observable values at the nonleaf nodes are given by the x^(z; x1; x2)
recursion relation. The
nal value of the observable measured on the tree as a whole is the value
obtained at the root node.
equation in eq. (2.13). Thus, the momentum fraction x carried by the nalstate subset S
is indeed a fractal observable.
In the above analysis, we implicitly assumed massless partons, since otherwise the
parton mass m would regulate the 1= IR divergence. As long as m
EjetR, it is consistent
to take the m ! 0 limit, which resums the large logarithms of EjetR=m in the cross section
for the fractal observable. At the scale
= m, one has to match the GFF evolution onto
the appropriate heavyquark description.
3
Fractal observables via clustering trees
We now present a straightforward way to build a broad class of fractal observables that
have the desired RG evolution in eq. (2.13). The idea is to use recursive clustering trees
that mimic the structure of the leadingorder RG evolution equations. Our construction is
based on the following three ingredients, as shown in gure 2:
1. Weights wa for each nalstate hadron;
2. An IRCsafe binary clustering tree;
3. The recursion relation x^(z; x1; x2).
By implementing the function x^ directly on recursive clustering trees, the resulting
observable is guaranteed to have fractal structure.
3.1
Construction
For this discussion, we start with a collection of hadrons from an identi ed jet, found using
a suitable jet algorithm, e.g. antikt [45] in the studies below. As the initial boundary
HJEP06(217)85
condition for the observable, each
nalstate hadron within the jet is assigned a weight
wa (possibly zero) based on some nonkinematic quantum number associated with that
hadron. This weight controls how much each type of hadron contributes to the value of the
jet observable. For example, to construct an observable that only depends on the charged
particles in the jet, all charged particles would be given weight 1 and all neutral particles
weight 0. It is crucial that wa is independent of the energy and direction of the hadron,
otherwise the NLO GFF would not take the form in eq. (2.15).
These nalstate hadrons are then used as inputs to an IRCsafe binary clustering tree,
which is in general di erent from any clustering algorithm used to determine the identi ed
jet. For our studies, we use the generalizedkt family of jet clustering algorithms [45], which
are designed to follow the leadingorder structure of the parton shower. In the context of
e+e collisions, these algorithms have the pairwise clustering metric
dij = min[Ei2p; Ej2p] i2j ;
(3.1)
where the exponent p parametrizes the treedependence of the observable, with p =
f 1; 0; 1g corresponding to the fantikt [45]; C/A [28, 29]; kt [46, 47]g clustering
algorithms, and
i2j is a measure of the angular separation between two constituent's momenta
scaled by the jet radius parameter R.5 For any value of p, generalizedkt provides a
pairwise clustering structure that directly mimics eq. (2.13). For pp collisions, one insteads use
a form of eq. (3.1) based on transverse momenta pT and distance
Rij in azimuthal angle
and rapidity.
From this clustering tree, one can determine the observable x by applying the recursion
relation x^(z; x1; x2) at each stage of the clustering. Speci cally, the value of x at each node
depends on the momentum fraction z given by the 2 ! 1 merging kinematics as well as on
the x1 and x2 values determined from the corresponding daughter nodes (which might be
the initial weights wa). When all nodes are contained in a single connected tree, the root
node represents the entire jet, and the root value of x determines the nal observable.
Even though the clustering tree is IRC safe, the resulting fractal observable x is
generally collinear unsafe. These collinear divergences are absorbed into the GFFs, and are in
fact responsible for the evolution in eq. (1.3).
3.2
Requirements
There are a few fundamental limitations on the choice of x^(z; x1; x2) dictated by the fact
that this same function will appear in eq. (2.13). First, the recursion relation must be
symmetric under the exchange z $ 1
z, x1 $ x2, since the assignment of these labels
is unphysical.6 Second, the recursion relation has to be IR safe, since the GFF formalism
5Since we start with the constituents of an identi ed jet, all of the particles are (re)clustered into a
single tree. For this reason, the singleparticle distance measure and the jet radius parameter R in the
(re)clustering algorithm are irrelevant.
6In the case of jets with heavy
relations (see e.g. [48]). We do not give a separate treatment of heavy avor GFFs in this work, and instead
assume to always work in the mb;c
EjetR limit.
avor, one could use heavy avor tags to de ne asymmetric recursion
only regulates collinear (and not soft) divergences. In order that an emission with z ! 0
does not change the observable, IR safety translates into the conditions
z!1
lim x^(z; x1; x2) = x1;
z!0
lim x^(z; x1; x2) = x2;
(3.2)
such that an arbitrarily soft branch in the clustering tree has no impact on the values of
x. Third, the recursion relation has to have unambiguous limits. As a counterexample,
x^(z; x1; x2) = xzx1 z satis es eq. (3.2) when x1 and x2 are nonzero, but not when they
1 2
vanish. Apart from these limitations, any choice of x^(z; x1; x2) (along with starting weights
and a clustering tree) de nes a fractal observable.
The tree traversal prescription, along with the requirement in eq. (3.2), helps ensure IR
safety to all s orders. As a counterexample, consider the sum over all tree nodes of some
function f (z) which vanishes as z ! 0 or z ! 1. In that case, the resulting observable would
receive no contribution from a single in nitely soft splitting, but subsequent nite z
splittings that followed the soft one would not be suppressed, violating IR safety. By contrast,
eq. (3.2) requires the contribution from an entire soft branch to be suppressed, as desired.
In this paper, we mainly focus on recursion relations that do not depend explicitly
on the opening angle
between branches in the clustering tree. In section 7, we do
discuss how the recursion relation gets modi ed if a threshold value for
is introduced
(i.e. thr = Rsub
R). Of course, fractal observables depend indirectly on angular
information through the structure of the clustering tree, but as discussed below, the leadingorder
evolution equations do not depend on the clustering algorithm. When explicit dependence
is included in the x^ function, this sometimes results in a fully IRCsafe observable,
requiring a di erent type of evolution equation that is beyond the scope of the present work (see
e.g. [49]).
3.3
Evolution equations
The generalizedkt clustering tree has an obvious mapping to a parton branching tree,
such that at order s, the RG evolution is given precisely by eq. (2.13), with the avor of
the GFF matching the avor of the jet's initiating parton. More formally, as discussed in
section 2.3, the NLO calculation of the bare GFF shows that the same recursion relation
x^(z; x1; x2) appears in eq. (2.15), as desired.
In fact, to order s, the evolution in eq. (2.13) is insensitive to the clustering tree, as
long as it is IRC safe, even if the fractal observable itself depends on the clustering order.
We explicitly test this surprising feature in section 5. Note that if the clustering tree is not
collinear safe, in the sense that particles with collinear momenta are not clustered with each
other rst, then the collinear divergences in the GFF will not cancel against the collinear
divergences in the hard matching coe cients of eq. (2.12). If the clustering tree is not IR
safe, then the observable x is not IR safe, and the GFF formalism does not apply.
We stress that the evolution in eq. (2.13) is only valid to lowest order in s. At higher
orders in
s, the evolution of fractal observables is more complicated, but, as discussed
more in the paragraph below, still satis es the property that the evolution of Fi(x; )
HJEP06(217)85
depends only on GFFs of the same observable. Schematically, this can be written as
d
d Fi =
s
represents a convolution. This equation includes 1 ! n splittings at order sn 1.
There is no longer a onetoone correspondence between pairwise clustering trees and GFF
evolution trees, and one has to explicitly carry out the calculation in eq. (2.15) to higher
orders to determine the evolution. In particular, there will be di erent clusterings of the
1 ! n splitting into a binary tree when integrating over phase space, which depend on the
choice of clustering algorithm. Because our speci c realization of fractal observables in this
section is based on recursive clustering trees, this guarantees that eq. (3.3) depends only
on GFFs of the same type as Fi at all perturbative orders.
To justify the structure of eq. (3.3) in a bit more detail, it is instructive to take a
closer look at the 1= UV poles of Fi. As usual, the anomalous dimension of the GFFs
is determined by the single 1= UV poles. At order
s, we get (1= UV)Pi!jk, as shown
in eq. (2.15). At order
s2, the 1 ! 3 splitting factorizes into a sequence of two 1 ! 2
splittings when the angles of the splittings are strongly ordered. This leads to a term
like (1= 2UV)Pi!jk
Pj!`m which does not contribute to the GFF's anomalous dimension.
However, it does justify attaching Fj and Fk to the external splittings in eq. (2.13), as
it corresponds to the cross term between a oneloop renormalization factor and oneloop
Fj (and treelevel Fk). Away from the stronglyordered limit, the 1 ! 3 splitting does
have a genuine 1= UV divergence, contributing to the second term in eq. (3.3). The precise
structure of this term depends on how the clustering algorithm maps the three partons to
a binary tree. The justi cation for attaching GFFs to each of the three external partons
follows again by considering higherorder corrections with some strong ordering.
For
example, consider a 1 ! 5 splitting that is strongly ordered such that it factorizes in a
1 ! 3 splitting, in which two partons undergo 1 ! 2 splittings. Such a term would have
a 1= 3UV divergence, corresponding to the cross term of the renormalization factor for the
1 ! 3 splitting term at order
s2 with two oneloop F 's and one treelevel F . Finally, the
1= UV from the oneloop virtual contribution to the 1 ! 2 splitting gives a higherorder
correction to the rst term in eq. (2.13). For the remainder of this paper, we focus on the
leadingorder evolution, leaving an analysis at higher orders to future work.
4
Weighted energy fractions
The procedure outlined in section 3 is very general, but for special choices of x^(z; x1; x2),
the de nition of a fractal observable can simplify greatly. In this section, we consider the
recursion relation
x^(z; x1; x2) = x1 z + x2 (1
z) ;
where
> 0 is an energy exponent. As we will see, for any choice of pairwise clustering
tree, the resulting observable simpli es to a sum over the hadrons in a jet,
(4.1)
(4.2)
X
a2jet
x =
(4.4)
(A)
(B)
(C)
associative observables studied in section 4, the order of the clustering does not a ect the
nal
observable. The ordering of the clustering will matter for the nonassociative observables studied
is the same as in eq. (4.1), and wa is the hadron weight factor. We call these
observables weighted energy fractions.
Several examples of weighted energy fractions have already been studied in the
literature. The weighted jet charge is de ned for any
> 0 and weights given by the electric
charges of nalstate hadrons [13, 14, 50]. This quantity has, for example, been used in
forwardbackward asymmetry measurements at e+e
experiments [51, 52], as well as to
infer the charge of quarks [53{55]. Recently, the scale dependence of the average jet charge
was observed in pp ! dijets [56]. Track fractions correspond to the case of
charged particles are given weight 1 and neutral particles given weight 0 [15, 16]. Jet pTD is
= 1, where
a weighted energy fraction with
= 2 and all particles given weight 1 [17, 18]. Weighted
energy fractions with arbitrary
> 0 and wa = 1 for all particles were studied in ref. [19]
for applications to quark/gluon discrimination.
4.1
Associativity
Weighted energy fractions have an associative recursion relation, meaning that the order of
the clustering tree does not a ect the nal observable. To see this, consider the case of just
three particles with weights fw1; w2; w3g and respective momentum fractions fz1; z2; z3g.
splittings, labeled as A, B, and C.7 The corresponding observables are
As shown in
gure 3, there are three clustering trees that can be built using only 1 ! 2
xA = x^ z1; w1; x^
xB = x^ z2; w2; x^
xC = x^ z3; w3; x^
z2 + z3
z2
z3
z3 + z1
z1
z1 + z2
; w2; w3
; w3; w1
; w1; w2
;
;
:
xA = xB = xC = w1 z1 + w2 z2 + w3 z3 ;
Using eq. (4.1) and the fact that z1 + z2 + z3 = 1, it is straightforward to prove that
7Of course, for a speci c choice of kinematics, not all of these trees will be possible from generalizedkt
clustering, particularly in the collinear limit.
owing to the fact that the recursion relation has homogenous scaling with z. This argument
generalizes to an arbitrary numbers of particles, so the weighted energy fractions are indeed
independent of the clustering tree.8
Of course, there are other observables that have nonassociative recursion relations,
where the observable does not simplify to a sum over nalstate hadrons and the full tree
traversal is necessary. We explore some nonassociative observables in section 5.
In general, to extract GFFs, one has to numerically match the cross section in eq. (2.12)
using perturbatively calculated values for the coe cients Ci, Ji!j , Ji!jk, . . . . For the
parton shower studies in this paper, we limit ourselves to leading order where C(0)(zJ ) =
i
zJ ), and we use partonshower truth information to assign the parton label i. To
generate pure samples of quark and gluoninitiated jets, we use the e+e
!
=Z
! qq and
! H
! gg processes in Pythia 8.215 [57], switching o initialstate radiation. We
nd jets using FastJet 3.2.0 [58], with the eegeneralized kt algorithm with p =
1 (i.e. the
version of antikt [45]) and then determine the various weighted energy fractions on
the hardest jet in the event. At leading order, the normalized probability distributions for
the weighted energy fractions directly give the corresponding GFF Fi(x; ).
As discussed in section 2.2, for jets of a given energy Ejet and radius R, the
characteristic scale for GFFs is expected to be
= EjetR;
(4.5)
which is roughly the scale of the hardest possible splitting in the jet. By varying Ejet and R
but keeping
xed, we can estimate part of the uncertainty in the extraction of the GFFs.
In addition, we assess the uncertainty from using di erent parton shower models. Here,
since our primary interest is in the perturbative uncertainty in di erent shower evolution
equations, we test the native Pythia parton shower along with the Vincia 2.0.01 [59] and
Dire 0.900 [60] parton shower plugins. A further source of uncertainty would be given by
the hadronization model, which enters the boundary conditions used for GFF evolution.
This is not included in our present study, since we decided to interface all of the showers
above with the Lund string model. In the context of an experimental analysis, one would
also have statistical and systematic uncertainties from the extraction of GFFs from data.
For each observable x, there are 11 GFFs, corresponding to 5 quark avors fu; d; s; c; bg,
5 antiquark
avors, and the gluon. To avoid a proliferation of curves, it is convenient to
de ne singlet (denoted by hQuarki in the gures below) and nonsinglet combinations for
the quark GFFs, respectively,
S(x; ) =
1
2nf
X
i2fu;u;d;:::bg
Fi(x; );
Nij (x; ) = Fi(x; )
Fj (x; ):
(4.6)
8Remember that this tree is one obtained from reclustering the particles in the jet. The value of a
jet observable of course depends on the choice of initial jet algorithm, which may itself be a clustering
algorithm.
HJEP06(217)85
PS = Vincia
R = { 0.3, 0.6, 0.9}
= 2, with all particles given starting weight 1. These distributions were extracted at the
scale
= 100 GeV. The left column shows results from the Vincia parton shower, with uncertainty
bands from varying R = f0:3; 0:6; 0:9g while keeping
xed. The right column shows the
xed
jet radius R = 0:6, with uncertainty bands from testing three di erent parton showers: Pythia,
Vincia, and Dire. In this and subsequent gures, hQuarki always refers to the quarksinglet
combination S(x; ) de ned in eq. (4.6).
For the observables we study, the antiquark GFFs are either identical to the quark GFFs
!
or simply involve the replacement x
x, due to charge conjugation symmetry.
We
start by showing numerical results for the gluon GFF and the quarksinglet combination,
postponing a discussion of the nonsinglet case to section 4.5.
gure 4, we show the extracted gluon and quarksinglet GFFs at
= EjetR =
100 GeV for the weighted energy fractions with wa = 1, comparing
Since gluon jets have roughly a factor of CA=CF larger hadron multiplicity than quark jets,
the mean of the gluon GFF is roughly a factor of (CA=CF )1
higher than the mean of
the quarksinglet GFF. In the left column, we show the impact of changing the jet radius
R = f0:3; 0:6; 0:9g, leaving
xed. The envelope from changing R is very small, indicating
that
= EjetR is an appropriate de nition for the RG scale. In the right column, we show
the impact of switching between the Pythia, Vincia, and Dire parton shower models.
The envelope is larger, but still reasonably narrow, giving us con dence in the extraction of
the GFFs, at least as far as changing the perturbative shower model is concerned. Though
not shown here, we checked that the GFFs for the
as well (see section 4.4 below).
We now use these extracted GFFs as boundary conditions for the RG evolution in eq. (2.13).
In appendix C, we describe in detail the numeric implementation of the evolution.
Formally, the evolution equations work equally well running up or down in , but in practice
downward evolution is numerically unstable, as discussed further in appendix D. As a proof
of principle for our RG evolution code, we show upward evolution from
= 4 TeV, comparing our RG evolution in eq. (2.13) to that obtained from parton showers.
In gures 5 and 6, we present the evolution results for gluon and quarksinglet GFFs
respectively, for the weighted energy fractions with
= f0:5; 1:0; 2:0g.
We test three
di erent choices for the particle weights: wa = 1 for all particles, wa = 1 (wa = 0) for
charged (neutral) particles, and wa = Qa with Qa being the particle's electric charge. The
initial conditions extracted from the parton showers at
= 100 GeV are the same as those
shown in gure 4, with the same color scheme of red for gluon GFFs and blue for
quarksinglet GFFs. As described in section 4.2, the uncertainty bands are given by the envelope
of values obtained both from varying the jet radius/energy (keeping
xed) and from using
di erent parton showers. The evolved distributions to
= 4 TeV are shown in orange for
the gluon GFFs and light blue for the quarksinglet GFFs, where the uncertainty bands
show the spread in nal values due to the spread in initial conditions.
For comparison, we show in dashed lines the GFFs extracted at
= 4 TeV, averaged
over the three parton showers and three R values.9 Overall, our numerical GFF evolution
agrees well with parton shower evolution, with both methods giving the same shift in the
peak locations. As previously seen in ref. [14], the two evolution methods agree best for
1, with larger di erences seen in the widths of the distributions when
< 1. This
is likely because
< 1 is more sensitive to collinear fragmentation, with larger expected
corrections from higherorder perturbative e ects. Note the expected function when
=
1 and wa = 1 for all particles, since the sum of the energy fractions for all particles in the
jet equals 1. The
! 1 limit of weighted energy fractions is discussed in section 4.4 below.
9The uncertainties from varying the jet radius/energy and changing parton showers at
= 4 TeV are
similar to the ones shown at
= 100 GeV.
HJEP06(217)85
0.6
Fg 0.4
Fg
(c)
0
x
x
(f )
0.0
x
(i)
Fg
= 2. Shown are distributions involving (left column) all particles, (middle
column) just charged particles, and (right column) charged particles weighted by their charge. The
GFFs extracted from parton showers at
= 100 GeV are shown in solid red. The result of evolving
these initial conditions to
= 4 TeV are plotted in solid orange, to be compared to the average
distribution obtained from parton showers at that value, plotted in dashed orange. The uncertainties
come from both varying R and the choice of parton shower (i.e. both variations shown in gure 4).
Fg 1.0
−0.2
−0.5
S
S
0.6
0.4
0.2
0.0
0
1.5
1.0
0.5
0.0
0.0
20
15
10
5
0
0.0
WEF RG: κ = 2
WEF RG: κ = 2
WEF RG: κ = 2
(c)
0
x
x
(e)
x
(h)
0.4
0.1
0.2
0.3
0.2
0
−0.2
gure 5 but for quarksinglet GFFs, where the distributions extracted from
parton showers at
= 100 GeV are shown in solid blue, the evolved distribution are shown in solid
light blue, and the distributions extracted at
= 4 TeV are shown in dashed light blue.
S 1.0
S
S
2.0
1.5
x
(f )
0.0
x
(i)
1.0
−1.0
−0.5
0.5
1.0
Vincia, μ = 100 GeV
κ = 2, p = 0
ξ = 0, (pTD)
ξ = 2
ξ = 4
ξ = 6
t
t
a
t
s
s
i
i
o
u
u
l
l
Vincia, μ = 4 TeV
κ = 2, p = 0
ξ = 0, (pTD)
ξ = 2
ξ = 4
ξ = 6
1.0
0.2
0.4
0.6
Quark Efficiency
quark jets against the mistag rate for gluon jets. These plots are obtained from Vincia,
comparing the discrimination performance at
we show variants of the nodeproduct observables de ned on C/A trees for
recalling that
= 2 is the same as 2(1
pTD). The node product with
better discrimination power than
= 2, especially at
= 4 TeV. The discrimination
power does continue increasing (slowly) with lower , but approaching the
the observable becomes IR unsafe and the GFF formalism no longer applies.
! 0 limit,
We can check whether this jetenergy dependence is reasonable using the RG evolution
equations, as shown in gure 16. For
= 1, the discrimination power does indeed increase
with increasing , but not as much as predicted by the parton showers. This could have
already been anticipated from the results in gure 11b, where the RGevolved gluon GFF does
not shift as dramatically as predicted in the parton showers. This could either be a sign that
the parton showers are too aggressive in their evolution, or that higherorder terms in the
evolution equation are important for getting the proper shape of the
= 1 distribution. For
= 2, the evolution of the ROC curves according to eq. (2.13) does match the evolution in
the parton shower, but this evolution is very slight, less than the spread in the ROC curves
at either scale from varying R and the parton shower. For
= 4, the discrimination power is
poor at all scales, but the evolution matches well between eq. (2.13) and the parton showers.
pTD at
We next turn to the fulltree observables in gure 17, using a C/A tree with
= 2 on all
particles. We compare
yields comparable performance to pTD at
= f0; 2; 4; 6g, where
= 0 is identical to pTD. The
= 4 observable
= 100 GeV, but performs somewhat better than
= 4 TeV. Note that the quark/gluon discrimination power is not monotonic as
a function of . We can again check whether this evolution is reasonable using the RG
equations, as shown in
gure 18. For all three
values, the evolution of the ROC curves
in eq. (2.13) matches the parton shower, but the evolution is extremely slow.
= f1; 2; 4g,
= 1 exhibits much
1.0
e0.8
t
s
i
M0.4
n
κ = 2
p = 0
a
M0.4
n
u
l
G0.2
g0.6 ξ = 4
R
a
κ = 2
p = 0
Full Tree ROC Curve RG
Full Tree ROC Curve RG
Full Tree ROC Curve RG
= 6. The = 0 case is identical to pTD, shown in gure 16b.
As emphasized in ref. [19], predicting the quark/gluon discrimination power from
rst
principles is a much more challenging task than predicting the distributions themselves.
Because the ROC curve shapes depend sensitively on the overlap between the quark and
gluon distributions, small changes in the distribution shapes can lead to large changes
in the predicted discrimination power. This is especially evident in
gure 18, where the
uncertainties in the ROC curves at the same scale are generally larger than the
evolution between scales. This highlights the importance of precision calculations for correctly
predicting quark/gluon discrimination behavior.
7
Fractal observables from subjets
As our nal investigation into the structure of fractal jet observables, we now consider the
possibility that the recursion relation in eq. (1.1) is modi ed to depend on the angular scale
of the clustering. For simplicity, we only consider observables de ned on angularordered
C/A clustering trees, since in that case the depth in the C/A tree is directly associated with
an angular scale . This opens up the possibility to de ne a modi ed recursion relation
with
dependence, for example,
x^(z; x1; x2) =
x^1(z; x1; x2) if < Rsub,
x^2(z; x1; x2) if > Rsub.
As shown in gure 19, the nodes as de ned by x^1 become the starting weights for the
subsequent nodes de ned by x^2.
It is straightforward to implement the leadinglogarithmic resummation of an
observable de ned by eq. (7.1). Starting from a lowenergy boundary condition, this involves an
initial evolution to the scale
using eq. (2.13) with the recursion relation x^1, followed by an evolution to
= EjetR using
x^2 instead. The discontinuity in anomalous dimensions of the evolution equations across
sub = EjetRsub
> Rsub
x^1
x^1
x^1
scale Rsub. When using a C/A tree, it is possible to switch the recursion relation from x^1 to x^2
for angular scales
> Rsub. This is equivalent to determining the observable x^1 on all subjets of
radius Rsub and then using these as initial weights for the tree with x^2.
Subjets: R = 0.6, Rsub = 0.015
Vincia: μ = 100 GeV
100 GeV → 4 TeV
Vincia: μ = 4 TeV
μ sub = 100 GeV
Fg 2 κ1 = 1, κ2 = 2
and x^2 are given by weighted energy fractions measured on all particles with 1 = 1 and 2 = 2,
respectively.
the threshold
sub will be compensated by a xedorder correction at that scale, but this
only enters at nexttoleadinglogarithmic order.
One interesting case is when the observable de ned at small angular scales
< Rsub
is the weighted energy fraction of all particles with
= 1. This observable is simply 1 for
each of the branches, so the GFFs at the scale sub are
Fi(x; sub) = (1
x) ;
(7.3)
which are then the input for the fractal observable x^2 for
> Rsub. This e ectively removes
the sensitivity to nonperturbative physics, allowing us to calculate fractal observables
analytically, as long as the scale sub is perturbative. An example of this kind of observable is
shown in
gure 20, where the observable is clustered using the recursion relation eq. (4.1)
with
= 1 for angles
< Rsub and
= 2 for
> Rsub. The spike at x = 1 persists in
the numerical evolution, even with very ne bins and a large amount of computing time.17
This feature is not seen in the Vincia evolution, which at every stage in the parton shower
uses a scale closer to
' z Ejet , where z and
are the momentum fraction and opening
angle of the splitting. Compared to our choice of
= EjetR for the shower as a whole, we
would expect the Vincia scale, which corresponds to a larger coupling, to accelerate the
depletion of the
function in the evolution. It will be interesting to see if this behavior
persists with higherorder evolution equations.
An alternative way of viewing the above prescription is that we can build fractal jet
observables not just out of hadrons but also out of subjets of radius Rsub, thus enlarging the
range of applicability of the GFF framework. By taking Rsub not too small, the observable
becomes perturbative. On the other hand, we still want Rsub
R, such that the leading
logarithms of R=Rsub dominate the observable and eq. (2.13) gives a reliable description
of its behavior.
8
Conclusions
To date, the bulk of analytic jet physics studies are based on either singlehadron
fragmentation functions or IRCsafe jet shapes. In this paper, we emphasized the intermediate
possibility of IRsafe but collinearunsafe jet observables de ned on a subset of hadrons. We
started by introducing the framework of Generalized Fragmentations Functions (GFFs),
which are applicable to general collinearunsafe jet observables. The GFFs are universal
functions that absorb collinear singularities order by order in
s, which not only restores
calculational control, but also implies that the GFFs evolve under a nonlinear version of
the DGLAP equations. We then discussed fractal jet observables, de ned recursively on
an IRCsafe clustering tree with certain initial hadron weights, which satisfy a selfsimilar
RG evolution at leading order given by eq. (2.13). The higher order evolution is no longer
universal, but still selfsimilar, and has the schematic form in eq. (3.3).
The simplest fractal jet observables are those with associative recursion relations,
whose value does not depend on the choice of clustering tree. This is indeed the case for the
T
weighted energy fractions, studied in section 4, which include several observables already
in use at colliders, including pD, weighted jet charge, and track fractions. More exotic
fractal jet observables depend on the clustering sequence, including the nodeproduct and
fulltree observables studied in section 5. Remarkably, the structure of the RG evolution
for these observables is independent of the clustering tree at leading order.
As one potential application of fractal observables, we studied whether nonassociative
observables could be useful for quark/gluon discrimination. Indeed, we found examples in
section 6 which do perform better than the weighted energy fraction pTD currently used by
17The generating functional approach (see e.g. ref. [72]) provides an alternative implementation of the
evolution in eq. (1.3) that can be used to resum (sub)jet radius logarithms [30]. This approach may be
more amenable to an initial condition with a delta function.
CMS. Though the GFF formalism does not allow us to predict the absolute discrimination
power of collinearunsafe observables, it does allow us to predict the RG evolution of
the discrimination power, a feature that is further exploited in ref. [49]. To gain more
perturbative control, one can work with fractal observables de ned on subjets (instead of
hadrons), as brie y discussed in section 7.
Looking to the future, the next step for fractal jet observables is pushing beyond the
leadingorder evolution equations. This will require computing the bare GFFs to higher
s, as well as extracting GFFs using the matching scheme sketched in eq. (2.12),
and presented in detail at nexttoleading order for e+e collisions in appendix A. More
ambitiously, one would like to study correlations between two or more fractal jet observables,
which would require multivariate GFFs. Such correlations are known to be important for
improved quark/gluon discrimination [19, 21, 26], though even for IRCsafe jet shapes, there
are relatively few multivariate studies [73{75]. Together with the work in this paper,
higherorder and correlation studies would facilitate a deeper understanding of jet fragmentation,
with important consequences for analyses at the LHC and future collider experiments.
Acknowledgments
We thank Christopher Frye and Andrew Larkoski for helpful comments on quark/gluon
discrimination. The work of B.E., J.T., and K.Z. is supported by the DOE under grant
contract numbers DESC00012567 and DESC00015476. M.P. is supported by a Marie
Curie IntraEuropean Fellowship of the European Community's 7th Framework Programme
under contract number PIEFGA2013622527. W.W. is supported by the European
Research Council under grant ERCSTG2015677323, and the DITP consortium, a program
of the Netherlands Organization for Scienti c Research (NWO) that is funded by the Dutch
Ministry of Education, Culture and Science (OCW).
A
Generalized fragmentation in inclusive jet production
In this appendix, we explicitly verify eq. (2.12) at O( s). We rst calculate the lefthand
side of this equation for the measurement of the fractal variable x together with the fraction
of the centerofmass energy carried by the jet, zJ
2Ejet=Ecm. Assuming that R is not
so large that all nalstate partons get clustered into one jet, we get
d
Z
1
d
= (0)
(R
ij) (zJ yk)Fk(0)(x; )
+ (zJ yi yj)
dx1 dx2 Fi
(0)(x1; ) Fj(0)(x2; ) x x^
(zJ yi) Fi(0)(x; ) :
yi + yj
yi ; x1; x2
(A.1)
Here, i; j = 1; 2; 3 and yi is the parton momentum fraction normalized such that y1 + y2 +
y3 = 2. In the following calculations, we identify parton 1 with the quark, 2 with the
antiquark, and 3 with the gluon. The angle ij between partons i and j is given by
ij = arccos 1
2(1
yk)
yi yj
;
and k denotes the parton di erent from i and j. Although the angle ij becomes ambiguous
when yi or yj is zero, IR safety ensures that the measurement is not. The term in eq. (A.1)
with ij < R describes the situation where partons i and j are clustered in a jet but parton
k is in a separate jet. The
nal term, where all ij > R, corresponds to the situation
where all partons are in separate jets. Each of the three partons has a leadingorder GFF
attached to it. The squared matrix element that enters in eq. (A.1) is given up to O( s) by
(A.2)
(A.3)
(A.4)
+
(A.5)
ln(1 z)
1 z
+
(A.6)
1
(0) dy1dy2
where
= (1 y1) (1 y2) +
+ (1 y2)
+ (y1 $ y2) ;
Pq!qg(y) = CF
1 + y2
1
y
+
:
Let us now focus on the righthand side of eq. (2.12). In our case, the coe cients Ci are
the standard ones for inclusive fragmentation in e+e
collisions [4, 76, 77] since the only
kinematic variable appearing on the lefthand side of eq. (A.1) is the jet energy fraction zJ :
2
(1 y3)(y12 + y22)
2(1 y1)+(1 y2)+
+
2
2
Pq!qg(y1)
CF
1
IR
+ ln y1Ec2m
2
+ (1 + y12)
4
(1 y1) (1 y2)
ln(1 y1)
1 y1
+
+ 1 y1
Cq(z; Ecm; ) = (1
z) +
Cg(z; Ecm; ) =
+
2
2 ln z
z
1
s
Pq!qg(1
2
s
3
1
2 (1
z) +
z) ln
Pq!qg(z) ln
2
Ec2m + CF (1 + z2)
+ (1
z)
2
Ec2m + ln(1
ln(1
1
z +
5
2
z)
z
;
(1) (1)
The coe cients Jq!qg and Jq!gq for an e+e
using the MS scheme in ref. [14],
kT like jet algorithm were calculated
Jq(!1)qg(z; EjetR; ) =
2
s
2CF L
2 (1 z) + 2Pq!qg(z) 3CF (1 z) L + CF 4z
Jq(!1)gq(z; EjetR; ) = Jq(!1)qg(1 z; EjetR; ) ;
1 + z2
1 z
ln z + 1 z
(1 z)
;
2
12
Jq(!1)q(z; EjetR; ) =
Jq(!1)g(z; EjetR; ) = Jq!q 1
(1)
2
s CF (1
z)
2LPq!qg(z)
2L2 + 3L +
2CF (1 + z2)
1
L
ln
EjetR;
;
EjetR
:
2
12
ln(1
1
z)
CF (1
z) ;
The coe cients for antiquarks are identical. Note that the relation between Jq(!1)q and
z, because the jet energy Ejet rather than the energy of
xed. Since Jq(!1)q and Jq!qg describe the same splitting
(1)
in complementary regions of phase space (injet versus outofjet), their sum vanishes in
Jq(!1)qg(z; EjetR; ) + Jq(!1)q(z; z EjetR; ) = 0 :
The nal ingredient we need is the renormalized oneloop expression for the GFF (see
eq. (2.15)),
while Jq(!1)q and Jq(!1)g are given by the nite terms of eq. (2.34) and eq. (2.35) in ref. [31]
(R
ij )
R2
4
yk ;
Fi(x) = Fi(0)(x)
1
X Z
2 IR j;k
dz
s( )
2
where
Jq(!1)g is not simply z $ 1
the initiating parton is held
dimensional regularization,
Z
Z
+
Z
(A.7)
(A.8)
(A.9)
(A.11)
(A.12)
(A.13)
Let us rst verify the cancellation of IR divergences between left and righthand sides
in eq. (2.12). On the latter, these solely come from Cq(0)(zJ ; Ecm; )[Fq(1)(x; ) + Fq(1)(x; )].
On the lefthand side, we nd
1
(0) dzJ dx IR div
dy1 dy2
2
1
IR
(1
y1)Pq!qg(y2)
(zJ
1) Fq(0)(x; )
dx1 dx2 Fq(0)(x1; ) Fg(0)(x2; )
x
x^(y2; x1; x2)
+ (q $ q)
= (zJ
1)[Fq(1)(x; ) + Fq(1)(x; )];
which demonstrate the cancellation of the IR divergences. Note that the term on the rst
line of eq. (A.11) proportional to Fi
and
straint in the small R limit as
(0) does not contribute here because it is y2independent
Z
dy2 Pq!qg(y2) = 0 :
To verify that also the nite terms match in eq. (2.12), we expand the angular
conwhich implies yk
which gives
1
(0) dzJ dx 13
sCF Z
2
yi. We rst consider the (R
13) term in eq. (A.1),
+ (1
R2
4
1
y1 (1
Z
CF
1 + y12
(1
1
y1) + (1
y2) +
Pq!qg(y1) ln y1Ec2m + (1 + y12)
2
+ ( 2
8) (1
y1) (1
ln(1
1
y1)
y1
+
y1
(zJ
1)Fq(0)(x; )
+ (zJ
1)
dx1 dx2 Fq(0)(x1; ) Fg(0)(x2; ) x
x^(y1; x1; x2)
=
s Z
2
+ (zJ
1)
Z
Z
dz Pq!qg(z) ln
z)
z2Ej2etR2
2
+ CF 2(1 + z2)
(zJ
1)Fq(0)(x; )
ln(1
1
z)
+
dx1 dx2 Fq(0)(x1; ) Fg(0)(x2; ) x
dz dx1 dx2 Jq(!1)qg(z; EjetR; ) Fq(x1; ) Fg(x2; )
[x
x^(z; x1; x2)] + (: : : )13:
(A.14)
As the integral over y2 yields a ln(1 y1), the resulting ln(1 y1)=(1 y)+ is not properly
regularized, leaving the coe cient of (1
z) undetermined. As we will see, however, this
ambiguity cancels exactly against the one arising from Jq(!1)q, due to eq. (A.9). The (R
23)
term gives the corresponding contribution with quark and antiquark interchanged, whereas
the (R
12) term is O(R2) suppressed due to the e+e
! qqg squared matrix element.
For the last contribution in eq. (A.1), we rewrite
( 12
R) ( 13
R) ( 23
R) = 1
(R
12)
(R
13)
(R
23) : (A.15)
where the rst term in the sum corresponds to the calculation of the matching coe cients
for inclusive fragmentation, thus yielding the Ci(zJ ; Ecm; )Fi(x; ) contribution on the
righthand side of eq. (2.12). For the remaining terms, we can follow the same strategy as
in eq. (A.14). For example, the
(R
13) term gives
1
(0) dzJ dx 13
1 + y12
1
(1 y1) + (1 y2) +
+ (1
Pq!qg(y1) ln y1Ec2m + (1 + y12)
2
sCF Z
2
R2
4
ln(1
y1)
y1
+
+ 1
y1
(zJ
y1)Fq(0)(x; )
=
+ (zJ
2
s Z
1)Fq(0)(x; ) + (zJ
1 + y1)Fg(0)(x; )
dz Pq!qg(z) ln
z2Ec2mR2
+ CF 2(1 + z2)
ln(1
1
z)
z
+
= Jq(!1)q(zJ ; EjetR; ) Fq(x; ) + Jq(!1)g(zJ ; EjetR; ) Fg(x; ) (: : : )13:
in eq. (A.7) together with eq. (A.9) make this straightforward to verify. The (: : : )13 term
cancels in the sum with eq. (A.14). The
(R
quark and antiquark interchanged and the
(R
by O(R2). This completes the check of eq. (2.12) at O( s).
23) term corresponds to the term with
12) contribution is again suppressed
A nonfractal example: sums of weighted energy fractions
While eq. (1.1) is rather general, there are of course many collinearunsafe observables
that are not fractal jet observables. In this appendix, we give an explicit example of an
observable that does not satisfy the requirements in section 2.3.
Consider two weighted energy fractions
x = X wi zi ;
i2jet
y = X vi zi ;
i2jet
and . Individually, x and y are
described by the evolution equation in eq. (2.13). On the other hand, their sum
is not a fractal jet observable, though it still can be described by a GFF.
To see this, consider the GFF for t, Fi(t), which can be written in terms of a joint
GFF for x and y as
Z
Fi(t) =
dx dy Fi(x; y) [t
x
y]:
The evolution equation for the joint GFF follows from the analysis in eq. (2.15), leading to
d Fi(x; y; ) =
s( ) X Z
2
j;k
dz dx1 dx2 dy1 dy2 Pi!jk(z) Fj(x1; y1; ) Fk(x2; y2; )
x
z x1
(1
z) x2
y
z y1
(1
z) y2 :
Plugging eq. (B.4) into eq. (B.3), we can insert a factor of
Z
1
dt1 dt2 [t1
x1
x2
y2]
to perform the integrals over y1 and y2. The resulting equation is
(A.16)
(B.1)
(B.2)
(B.3)
(B.4)
(B.5)
+ (zJ
1 + z)Fg(0)(x; )
d
d Fi(t; ) =
s( ) X Z
2
j;k
dz dt1 dt2 dx1 dx2 Pi!jk(z) Fj(x1; t1
x1) Fk(x2; t2
x2)
t z t1
(1
z) t2
z )x1
z)
z) x2 : (B.6)
As written, this is a valid GFF evolution equation, but the GFF for t explicitly involves
the joint GFF for x and y, so we do not get an evolution equation of the form of eq. (2.13).
If and only if
=
, can we cancel the x1 and x2 terms inside of the
function in
eq. (B.6). In that case, we can rewrite the joint probabilities as probability densities for
the sums t1 = x1 + y1 and t2 = x2 + y2, so that the evolution equation is of the desired
fractal form. Of course,
=
just corresponds to a regular weighted energy fraction with
weights wi + vi, so this is not a new fractal observable.
C
Software implementation
The software to perform the RG evolution in this paper is available from the authors upon
request. In this paper, we discuss some of the speci cs of its implementation. A public
version of the code is planned for a release some time in the future.
HJEP06(217)85
C.1
Running coupling
Because we only perform leadingorder evolution, the running of
s is strictly speaking
only required at leadinglogarithmic accuracy. In our implementation, though, the running
of the strong coupling is included using the
CATF nf
4CF TF nf :
d s( )
2 s
0
0 =
CA
11
3
4
s
4
3 TF nf ;
s( ) =
function at O( s3),
s 2
4
1 =
;
3
3
0
1
L
02 L12 ln L ;
The running coupling at the scale
is given by solving eq. (C.1) iteratively to order O( s3),
where L = ln 2QCD
C.2
Discretization
2 . Using the PDG value s(MZ ) = 0:1181 gives the boundary condition
QCD = 0:2275 GeV. The group theory factors for QCD are CF = 43 , TF = 12 , and CA = 3.
For applications to the LHC running at 13 TeV, the number of quark avors is nf = 5.
The evolution equation in eq. (2.13) can be solved by binning the values of the GFFs in
the x variable. If the GFF domain is partitioned into N bins, eq. (2.13) becomes a set of
(2nf + 1)N coupled ordinary di erential equations. The evolution equation for the binned
GFF for bin n, Fei(n; ), is given by18
d ln
Fei(n; )
d ln
N
=
2 j;k
Z n=N
(n 1)
N
(n 1)
N
dx
dx Fi(x; )
X Z n1=N
n1;n2
(n1 1)
N
Z n2=N
dx1 (n2 1)
N
0
dx2
dz Pi!jk(z)
(C.1)
(C.2)
(C.3)
(C.4)
18This equation is written for N equalwidth bins for simplicity of notation. The generalization to unequal
bins is straightforward, and the software implementation is set up to handle variable bin widths if desired.
2 j;k
(n 1)
N
X Z 1
n1;n2 0
dz Pi!jk(z)Fej (n1; ) Fek(n2; ) x
x^(z; xn1 ; xn2 ) ;
where xn1 and xn2 are the positions of the midpoints of the n1th and n2th bins. Note
that eq. (C.4) is written in terms of ln
instead of , since this is how the evolution was
implemented numerically to make the step size and numerical errors more consistent. In
principle, the
function could be used to carry out the z integral exactly. In practice, it is
easier to discretize the z integral and use the
function to choose the xbin corresponding
to each triplet (z; x1; x2). This is because inverting x^ to solve for z analytically for general
x1 and x2 is not possible. Doing so in advance separately for each value of x, x1 and x2
can be prohibitively memory intensive for large numbers of bins.
The splitting functions are approximated by the analytic value of their integral over
the width of the bin. For our analysis, we need the following splitting functions:
HJEP06(217)85
Pq!gq(z) = Pq!qg(1
z) = CF
Pg!qq(z) = TF z2 + (1
Pg!gg(z) = 2CA
1
z
1 + (1
[z] ;
0
2
+ z(1
z)
[1
z] + [z] ;
(C.5)
where Pq!gq(z) is the splitting function for a quark radiating a gluon with momentum
fraction z, the integration constant for integrals of the plus distributions are xed by
(C.6)
(C.7)
(C.8)
and 0 is given in eq. (C.2).19 When performing the integration, terms with a plusfunction
regulator must be handled correctly for the endpoint bins. If the regulated functions have
the following primitives
then their integrals over the nth bin are implemented by
Z z+0:5 z
z 0:5 z
dz0 f (z0)
z0
z 0:5 z
dz0
=
F (z + 0:5 z)
(G(z + 0:5 z)
G(z
0:5 z)
F (z + 0:5 z)
F (z
0:5 z) n 6= 0;
G(z
0:5 z) n 6= n nal;
n = 0;
n = n nal:
In our implementation, the integration range z 2 [0; 1] is divided into nrough bins,
and the rst and last bin are then further subdivided by a factor of n ne. The user can
19The 1=z+ and (z) terms in Pq!gq(z) and Pg!gg(z) are necessary because the evolution in eq. (1.3)
requires distributions that are also regulated at z = 0.
Z 1 dz
0 z+
= 0 ;
0 (1
z) +
= 0 ;
dF (z)
dz
f (z)
z
=
1
z
Residuals Against nfine = 1000
ΔFg 0.02
Δn1
Δn10
Δn25
= 100 GeV to 4 TeV on the choice of ne bin width.
Shown are the (left) gluon GFF and (right) quarksinglet GFF for the weighted energy fraction
with
= 0:5. The curves labeled
nX are the di erence between the result using n ne = X and
the result using n ne = 1000. For the default value of n ne = 100 used in this paper, the results are
indistinguishable by eye.
specify these two parameters. For the results presented in this paper, the values used were
nrough = 1000 and n ne = 100. The
ner division of the endpoint bins is necessary to
accurately capture the singular behavior of the splitting functions near z = 0 and z = 1.
For many GFFs, this is not necessary, but consider the weighted energy fractions, whose
recursion relation satis es
z) x2 =)
1x1
(1
z)
1x2):
(C.9)
For
< 1, there are poles in the derivative of x^ at z = 0 and z = 1, resulting in a noticeable
dependence on n ne. This is shown in
gure 21 for the case of
= 0:5, with all particle
weights one. Once we increase n ne = 100 ! 1000, the maximum change in the value of
the evolved GFFs in a single xbin is less than 0.06%.
C.3
RungeKutta algorithm
After the discretization in eq. (C.4), the RG evolution is performed with an embedded
fthorder RungeKutta method adapted from ref. [78]. This method requires six evaluations
of the right side of eq. (2.13), which on the kth step can be combined to give a fthorder
estimate yk+1 of the desired function after a step of size hk. These computations can be
recombined with di erent coe cients to give a fourthorder RungeKutta estimate yk+1.
The di erence between these two methods then gives an estimate of the local truncation
error. The error estimated this way applies to the fourthorder value yk+1, but we take the
(more accurate) fthorder value. This ensures that our solution is actually slightly more
accurate than our error indicates. Estimating the error on this fthorder solution would
require calculating a stillhigher order step.
Once a step hk is taken, with an error Ek, we would like to choose an appropriate trial
value for our next step. This fourthorder error estimate scales as O(h5), so we choose the
next step, hk+1, to be
hk+1 =
(
S hkj EkE+k1 j0:20
S hkj EkE+k1 j0:25
Ek+1 > Ek;
Ek+1 < En:
Here, Ek+1 is the projected error in the (k + 1)th step, and S is a safety factor taken to
be 0:9. This formula allows the step size to grow if the error is much smaller than our
tolerance. If the error is larger than the tolerance, the step fails, and is retried with a
It is important that the algorithm be able to dynamically change step size in order to
evolve a solution e ciently while keeping errors within desired limits. At low scales, the
strong coupling grows large, and the solution changes rapidly. Numerical precision therefore
requires small step sizes in this region. At high scales, asymptotic freedom ensures that
the solutions change slowly, so much larger step sizes result in the same level of accuracy.
This procedure requires a prescription for the maximal acceptable error. For a system of
M
considered a failure unless every equation is within its error tolerance. The error Ekm for
(2nf + 1)n coupled ODEs, there is a separate Ekm for each m 2 M . The step is
the mth equation on the kth step is required to satisfy
m
Ek
jykmj + jhk dykm=d ln
j + 10 6
< :
(C.10)
(C.11)
The value
is an overall upper limit which was set to 10 9 for the GFF evolution. The
last numerical term in the denominator is required to avoid arti cially large errors when
the domain of the GFFs input into the program exceeds the actual support of the GFF.
As an additional constraint, our algorithm sets a maximum step size of d ln
0:4. Note
that the same step size is used for every equation in the system.
D
Numerical stability
All of the RG results in this paper are based on the numerical solution of eq. (2.13) for
upwards evolution in the scale . The reason is because downward evolution is numerically
unstable, in the sense that small irregularities in the initial conditions amplify into large
uctuations, especially for the gluon GFFs. This behavior is illustrated in gure 22, where
gluon and quarksinglet GFFs are evolved downward from 4 TeV to 100 GeV.
Heuristically, if evolution upwards in scale is analogous to convolution of the GFFs,
evolution downwards is akin to deconvolution, a problem known to be illposed. To verify
that the instability is inherent to the di erential equation, and not merely a numerical
artifact, we checked that the envelope shown in
gure 22 is not a ected by choosing a
smaller step size or more stringent error bound in the RungeKutta algorithm. To get a
15
−0.2
0
Downward Evolution: κ = 2.0
Downward Evolution: κ = 2.0
4 TeV → 100 GeV
HJEP06(217)85
x
(d)
PS: μ = 4 TeV
4 TeV → 100 GeV
PS: μ = 100 GeV
hQuarki GFF
WEF
Fg
Fg
4
2
0
−2
PS: μ = TeV
4 TeV → 100 GeV
4 TeV → 100 GeV
PS: μ = 100 GeV
= 100 GeV of the (left column) gluon GFF
and (right column) quarksinglet GFF with (top row)
= 0:5 and (bottom row)
= 2:0. The
envelopes of the evolved distributions are constructed as in section 4.2 by varying the jet radius R
and the choice of parton shower, which highlight the numerical instability of downward evolution.
sensible result, one could use a numerical regularization method such as Tikhonov
regularization [79], though we do not do so here. Note that in general, if the evolution in one
direction is stable, such that small uctuations get washed out, the evolution is expected
to be unstable in the reverse direction.
In this appendix, we give details of the moment space analysis from section 4.5, as well as
perform similar analyses for the nonassociative observables from section 5. The moments
of the GFFs are de ned by
where the zeroth moment is just the normalization,
Z
Z
F i(N; ) =
dx xN Fi(x; ) ;
F i(0; ) =
evolution equation,
d
d F i(N; ) =
2
This convention follows the standard nomenclature of probability theory.
Applying
R +1 dx xN to both sides of the evolution equation in eq. (2.13) gives the moment space
dz dx1 dx2 x^(z; x1; x2) N s( )
Pi!jk(z) Fj (x1; ) Fk(x2; ):
In order to proceed further, we need the speci c form of the recursion relation, x^. We now
discuss the details for each of the sets of observables studied in this paper.
E.1
Weighted energy fractions
Inserting the weighted energy fraction recursion relation eq. (4.1) into eq. (E.3) leads to
P q!qg(N ) = CF
P q!gq(N ) = CF
P g!qq(N ) = TF
P g!gg(N ) = 2CA
N 2 + 3N + 4
N (N + 1)(N + 2)
;
N 2 + 3N + 4
(N + 1)(N + 2)(N + 3)
2 0(N + 3) ;
(E.1)
(E.2)
assuming that N is integer and using the binomial theorem. As in eq. (4.12), the moments
of the splitting functions are de ned as
0
P i!j;k(N; M ) =
dz zN (1
z)M Pi!j;k(z) ;
with the convention that P i!j;k(N )
expressed in terms of the digamma function
P i!j;k(N; 0). For any real N > 0, they can be
0(N ) and the EulerMascheroni constant E,
d
d
F i(N; ) =
0
Z
dz z (N M)(1
z) M Pi!jk(z)
dx1 x1
N M
Fj (x1; )
dx2 x2M Fk(x2; );
2(N 2 + 3N + 3)
N (N + 1)(N + 2)(N + 3)
E
0(N + 2)
2
3 TF nf : (E.6)
102
101
101002
1st
1st
105
2nd
2nd
106
107
μ [GeV]
κ = 1, p = −1
κ = 1, p = 0
κ = 1, p = 1
0.7 Node Product Moment Space RG
0.8 Node Product Moment Space RG
1.0 Node Product Moment Space RG
κ = 4, p = −1
κ = 4, p = 0
κ = 4, p = 1
101
101002
1st
1st
105
2nd
2nd
μ [GeV]
hQuarki:
1st
1st
105
1st
1st
2nd
2nd
HJEP06(217)85
(c)
1st
1st
105
2nd
2nd
106
107
2nd
2nd
hQuarki:
1st
1st
2nd
2nd
Figure 23.
Moment space evolution of the nodeproduct observables with (top row)
and (bottom row)
= 4 for the generalizedk
t clustering trees with (left column) p =
= 1
1,
(middle column) p = 0, and (right column) p = 1. Shown are the rst (solid curves) and second
(dashed curves) moments of gluon (red) and quarksinglet (blue) GFFs. The rst (second) moments
extracted from the parton shower average at
= 4 TeV are shown as points (diamonds).
E + 0(N + 1). These
expressions for all positive real numbers are necessary to evaluate the moment space
evolution equation in eq. (4.13) for noninteger . Note that N is shifted up by one from the
expression usually seen in the literature, because our convention for moments in eq. (E.1)
is shifted by one as well compared to Mellin moments.
E.2
Node products
We now insert the recursion relation for the node products from eq. (5.1) into eq. (E.3).
This leads to evolution equations with additional terms compared to those for the weighted
energy fractions. These terms have splitting kernels of the form
0
dz 4z(1
z) azb(1
(E.7)
for a > 0 and b; c
0. These integrals are convergent, so no plus function regulators are
required. They can also be performed analytically for general a, b, and c. Explicitly, the
rst moments of the quarksinglet and gluon GFFs evolve as
S(1; ) !
F g(1; )
s( )
P q!qg( )
P q!gq( )
2nf P g!qq( ) P g!gg( )
S(1; ) !
F g(1; )
+
s( )
The additional constant terms are de ned as
P qN1ode
P gN1ode( )
( )
P qN1ode( )
P gN1ode( )
1 Z 1
2 0
dz Pq!qg(z) + Pq!gq(z) 4z(1
z)
dz 2nf Pg!qq(z) + Pq!gg(z) 4z(1
=2 ;
z)
=2 ;
which can be evaluated in terms of
functions. The additional terms drop out of the
equation for the rst moments of the nonsinglet GFFs, so these still evolve according to
eq. (4.14). The third term in eq. (5.1) leads to several more terms in the evolution equations
for higher moments.
In gure 23, we plot the
evolution of the gluon and quarksinglet GFF moments for
node products with
computed at the scale
= f1; 4g and p = f 1; 0; 1g. The rst and second moments were
= 100 GeV from the GFFs in gure 10, averaged over the di erent
parton showers and R values (as described in section 4.2). These average moments were
evolved to the scale
= 107 GeV using eq. (E.8) and the corresponding second moment
equation. For comparison, the rst and second moments of the GFFs extracted from the
parton shower average at the scale
= 4 TeV are shown as dots and diamonds, respectively.
E.3
Fulltree observables
For fulltree observables with recursion relation given in eq. (5.4), the moment space
evolution equations are of the same general form as for the weighted energy fractions,
FT
P i!j;k(N; M ) F j (N
M; ) F k(M; );
(E.10)
d F i(N; ) =
s( ) X
2
N
X
j;k M=0
N
M
but with di erent splitting kernels,
FT
P i!j;k(N; M )
0
dz eN z(1 z)z (N M)(1
z) M Pi!j;k(z):
(E.11)
To our knowledge, these integrals do not have a closed form solution for general values of
the parameters
and , but it is straightforward to evaluate them numerically. If M = 0
or M = N , these integrals are sensitive to the plusprescription in the splitting functions.
100
10−1
10−2
10−3
10−4
μ [GeV]
10−1502 103 104
10−1502 103 104
105 106 107
10−1502 103 104
105 106 107
hQuarki:
1st
1st
2nd
2nd
ξ = 2, κ = 2, p = −1
1st
1st
2nd
2nd
ξ = 2, κ = 2, p = 0
Gluon:
hQuarki:
1st
1st
2nd
2nd
ξ = 2, κ = 2, p = 1
10−1
10−2
10−3
10−4
0
Z 1
0
0
Z 1
0
10−1
10−2
10−3
10−4
Full Tree Moment Space RG
Full Tree Moment Space RG
1st
1st
2nd
2nd
ξ = −2, κ = 2, p = −1
1st
1st
2nd
2nd
ξ = −2, κ = 2, p = 0
Gluon:
1st
1st
2nd
2nd
ξ = −2, κ = 2, p = 1
+
e z(1 z)z +1
F g(1; ) :
In gure 24, we show the evolution of the rst two moments of the GFFs for
= 2,
= f 2; 2g, and p = f 1; 0; 1g. In this case, the evolution agrees well with the value
extracted from the parton shower average at
= 4 TeV.
Open Access.
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Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
0.1002 103 104
105 106 107
on charged particles, with (top row)
2 and (bottom row) = 2.
Explicitly, for the rst moment in the quarksinglet basis,
d S(1; ) =
d F g(1; ) =
s( )
s( )
2
CF
+ CF
2nf TF
dz
e z(1 z)z (1 + z2) 2
S(1; )
dz e z(1 z)z 1(1 + (1
z)2) F g(1; ) ;
dz e z(1 z)z (z2 + (1
z)2) S(1; )
+ 2CA
dz e z(1 z)(z 1(1
z) + z +1(1
(E.12)
[1] H. Georgi and H.D. Politzer, Quark decay functions and heavy hadron production in QCD,
Nucl. Phys. B 136 (1978) 445 [INSPIRE].
expansion, Phys. Rev. D 18 (1978) 3705 [INSPIRE].
[2] A.H. Mueller, Cut vertices and their renormalization: a generalization of the Wilson
[3] R.K. Ellis, H. Georgi, M. Machacek, H.D. Politzer and G.G. Ross, Perturbation theory and
the parton model in QCD, Nucl. Phys. B 152 (1979) 285 [INSPIRE].
[4] G. Curci, W. Furmanski and R. Petronzio, Evolution of parton densities beyond leading
order: the nonsinglet case, Nucl. Phys. B 175 (1980) 27 [INSPIRE].
[5] G. Altarelli, Partons in quantum chromodynamics, Phys. Rept. 81 (1982) 1 [INSPIRE].
[Erratum ibid. B 213 (1983) 545] [INSPIRE].
(1982) 445 [INSPIRE].
[Yad. Fiz. 20 (1974) 181] [INSPIRE].
[7] J.C. Collins and D.E. Soper, Parton distribution and decay functions, Nucl. Phys. B 194
[8] L.N. Lipatov, The parton model and perturbation theory, Sov. J. Nucl. Phys. 20 (1975) 94
gluons, arXiv:1703.00914 [INSPIRE].
B 136 (1978) 1 [INSPIRE].
[arXiv:1209.3019] [INSPIRE].
[9] V.N. Gribov and L.N. Lipatov, Deep inelastic ep scattering in perturbation theory, Sov. J.
Nucl. Phys. 15 (1972) 438 [Yad. Fiz. 15 (1972) 781] [INSPIRE].
[10] G. Altarelli and G. Parisi, Asymptotic freedom in parton language, Nucl. Phys. B 126 (1977)
298 [INSPIRE].
[11] Y.L. Dokshitzer, Calculation of the structure functions for deep inelastic scattering and e+e
annihilation by perturbation theory in quantum chromodynamics, Sov. Phys. JETP 46 (1977)
641 [Zh. Eksp. Teor. Fiz. 73 (1977) 1216] [INSPIRE].
[12] J. Davighi and P. Harris, Fractal based observables to probe jet substructure of quarks and
[13] R.D. Field and R.P. Feynman, A parametrization of the properties of quark jets, Nucl. Phys.
[14] W.J. Waalewijn, Calculating the charge of a jet, Phys. Rev. D 86 (2012) 094030
pp collisions at p
[15] H.M. Chang, M. Procura, J. Thaler and W.J. Waalewijn, Calculating trackbased
observables for the LHC, Phys. Rev. Lett. 111 (2013) 102002 [arXiv:1303.6637] [INSPIRE].
[16] H.M. Chang, M. Procura, J. Thaler and W.J. Waalewijn, Calculating track thrust with track
functions, Phys. Rev. D 88 (2013) 034030 [arXiv:1306.6630] [INSPIRE].
[17] F. Pandol and D. Del Re, Search for the Standard Model Higgs boson in the
H ! ZZ ! ``qq decay channel at CMS, Ph.D. thesis, ETH, Zurich Switzerland, (2012)
[18] CMS collaboration, Search for a Higgs boson in the decay channel H ! ZZ( ) to qq` `+ in
s = 7 TeV, JHEP 04 (2012) 036 [arXiv:1202.1416] [INSPIRE].
[19] A.J. Larkoski, J. Thaler and W.J. Waalewijn, Gaining (mutual) information about
quark/gluon discrimination, JHEP 11 (2014) 129 [arXiv:1408.3122] [INSPIRE].
[20] J. Gallicchio and M.D. Schwartz, Quark and gluon tagging at the LHC, Phys. Rev. Lett. 107
(2011) 172001 [arXiv:1106.3076] [INSPIRE].
[21] J. Gallicchio and M.D. Schwartz, Quark and gluon jet substructure, JHEP 04 (2013) 090
JHEP 06 (2013) 108 [arXiv:1305.0007] [INSPIRE].
[22] A.J. Larkoski, G.P. Salam and J. Thaler, Energy correlation functions for jet substructure,
[23] B. Bhattacherjee, S. Mukhopadhyay, M.M. Nojiri, Y. Sakaki and B.R. Webber, Associated
jet and subjet rates in lightquark and gluon jet discrimination, JHEP 04 (2015) 131
[arXiv:1501.04794] [INSPIRE].
[24] J.R. Andersen et al., Les Houches 2015: physics at TeV colliders Standard Model working
group report, in 9th Les Houches Workshop on Physics at TeV Colliders (PhysTeV 2015),
[25] D. Ferreira de Lima, P. Petrov, D. Soper and M. Spannowsky, Quarkgluon tagging with
shower deconstruction: unearthing dark matter and Higgs couplings, Phys. Rev. D 95 (2017)
034001 [arXiv:1607.06031] [INSPIRE].
[26] P.T. Komiske, E.M. Metodiev and M.D. Schwartz, Deep learning in color: towards automated
quark/gluon jet discrimination, JHEP 01 (2017) 110 [arXiv:1612.01551] [INSPIRE].
[27] P. Gras et al., Systematics of quark/gluon tagging, arXiv:1704.03878 [INSPIRE].
[28] Y.L. Dokshitzer, G.D. Leder, S. Moretti and B.R. Webber, Better jet clustering algorithms,
JHEP 08 (1997) 001 [hepph/9707323] [INSPIRE].
[29] M. Wobisch and T. Wengler, Hadronization corrections to jet crosssections in deep inelastic
scattering, in Monte Carlo generators for HERA physics. Proceedings, Workshop, Hamburg
Germany, (1998){(1999), pg. 270 [hepph/9907280] [INSPIRE].
[30] M. Dasgupta, F. Dreyer, G.P. Salam and G. Soyez, Smallradius jets to all orders in QCD,
JHEP 04 (2015) 039 [arXiv:1411.5182] [INSPIRE].
[31] Z.B. Kang, F. Ringer and I. Vitev, The semiinclusive jet function in SCET and small
radius resummation for inclusive jet production, JHEP 10 (2016) 125 [arXiv:1606.06732]
[32] L. Dai, C. Kim and A.K. Leibovich, Fragmentation of a jet with small radius, Phys. Rev. D
Phys. Rev. D 22 (1980) 1184 [INSPIRE].
Cim. A 66 (1981) 339 [INSPIRE].
[33] U.P. Sukhatme and K.E. Lassila, Q2 evolution of multihadron fragmentation functions,
[34] I. Vendramin, Twohadron fragmentation functions: a study of their Q2 evolution, Nuovo
[35] L. Trentadue and G. Veneziano, Fracture functions: an improved description of inclusive
hard processes in QCD, Phys. Lett. B 323 (1994) 201 [INSPIRE].
[36] D. Graudenz, One particle inclusive processes in deeply inelastic leptonnucleon scattering,
Nucl. Phys. B 432 (1994) 351 [hepph/9406274] [INSPIRE].
chromodynamics, Z. Phys. C 11 (1982) 293 [INSPIRE].
[37] W. Furmanski and R. Petronzio, Leptonhadron processes beyond leading order in quantum
[38] J.C. Collins, D.E. Soper and G.F. Sterman, Factorization of hard processes in QCD, Adv.
Ser. Direct. High Energy Phys. 5 (1989) 1 [hepph/0409313] [INSPIRE].
[39] Particle Data Group collaboration, C. Patrignani et al., Review of particle physics, Chin.
Phys. C 40 (2016) 100001 [INSPIRE].
[40] Z.B. Kang, F. Ringer and I. Vitev, Jet substructure using semiinclusive jet functions in
SCET, JHEP 11 (2016) 155 [arXiv:1606.07063] [INSPIRE].
[41] M. Procura and I.W. Stewart, Quark fragmentation within an identi ed jet, Phys. Rev. D 81
(2010) 074009 [Erratum ibid. D 83 (2011) 039902] [arXiv:0911.4980] [INSPIRE].
[42] A. Jain, M. Procura and W.J. Waalewijn, Parton fragmentation within an identi ed jet at
NNLL, JHEP 05 (2011) 035 [arXiv:1101.4953] [INSPIRE].
[44] A. Jain, M. Procura and W.J. Waalewijn, Fullyunintegrated parton distribution and
fragmentation functions at perturbative kT , JHEP 04 (2012) 132 [arXiv:1110.0839]
063 [arXiv:0802.1189] [INSPIRE].
[45] M. Cacciari, G.P. Salam and G. Soyez, The antikt jet clustering algorithm, JHEP 04 (2008)
[46] S. Catani, Y.L. Dokshitzer, M.H. Seymour and B.R. Webber, Longitudinally invariant kt
clustering algorithms for hadron hadron collisions, Nucl. Phys. B 406 (1993) 187 [INSPIRE].
[47] S.D. Ellis and D.E. Soper, Successive combination jet algorithm for hadron collisions, Phys.
Rev. D 48 (1993) 3160 [hepph/9305266] [INSPIRE].
arXiv:1702.02947 [INSPIRE].
[48] P. Ilten, N.L. Rodd, J. Thaler and M. Williams, Disentangling heavy avor at colliders,
[49] C. Frye, A.J. Larkoski, J. Thaler and K. Zhou, Casimir meets Poisson: improved
quark/gluon discrimination with counting observables, arXiv:1704.06266 [INSPIRE].
[50] D. Krohn, M.D. Schwartz, T. Lin and W.J. Waalewijn, Jet charge at the LHC, Phys. Rev.
Lett. 110 (2013) 212001 [arXiv:1209.2421] [INSPIRE].
[51] AMY collaboration, D. Stuart et al., Forwardbackward charge asymmetry in e+e
hadron jets, Phys. Rev. Lett. 64 (1990) 983 [INSPIRE].
[52] ALEPH collaboration, D. Decamp et al., Measurement of charge asymmetry in hadronic Z
decays, Phys. Lett. B 259 (1991) 377 [INSPIRE].
[53] FermilabSerpukhovMoscowMichigan collaboration, J.P. Berge et al., Net charge in
deep inelastic antineutrinonucleon scattering, Phys. Lett. B 91 (1980) 311 [INSPIRE].
[54] ALEPH collaboration, D. Buskulic et al., Measurement of BB mixing at the Z using a jet
charge method, Phys. Lett. B 284 (1992) 177 [INSPIRE].
[55] D0 collaboration, V.M. Abazov et al., Experimental discrimination between charge 2e=3 top
quark and charge 4e=3 exotic quark production scenarios, Phys. Rev. Lett. 98 (2007) 041801
[56] ATLAS collaboration, Measurement of jet charge in dijet events from p
s = 8 TeV pp
collisions with the ATLAS detector, Phys. Rev. D 93 (2016) 052003 [arXiv:1509.05190]
[57] T. Sjostrand et al., An introduction to PYTHIA 8:2, Comput. Phys. Commun. 191 (2015)
[hepex/0608044] [INSPIRE].
[arXiv:1111.6097] [INSPIRE].
[58] M. Cacciari, G.P. Salam and G. Soyez, FastJet user manual, Eur. Phys. J. C 72 (2012) 1896
[59] W.T. Giele, D.A. Kosower and P.Z. Skands, A simple shower and matching algorithm, Phys.
Rev. D 78 (2008) 014026 [arXiv:0707.3652] [INSPIRE].
B 267 (1986) 702 [INSPIRE].
B 418 (1998) 214 [hepph/9707393] [INSPIRE].
N3LO+NNLL, Phys. Rev. Lett. 109 (2012) 242002 [arXiv:1209.5914] [INSPIRE].
Phys. J. C 71 (2011) 1661 [arXiv:1012.5412] [INSPIRE].
benchmarks, J. Phys. G 39 (2012) 063001 [arXiv:1201.0008] [INSPIRE].
[arXiv:1311.2708] [INSPIRE].
J. C 75 (2015) 409 [arXiv:1504.00679] [INSPIRE].
CMSPASJME13002, CERN, Geneva Switzerland, (2013).
68 (2003) 014012 [hepph/0303051] [INSPIRE].
highpT jets at the LHC, Phys. Rev. D 79 (2009) 074017 [arXiv:0807.0234] [INSPIRE].
in SCET, JHEP 11 (2010) 101 [arXiv:1001.0014] [INSPIRE].
Phys. Nucl. Phys. Cosmol. 8 (1996) 1 [INSPIRE].
QCD, JHEP 09 (2013) 137 [arXiv:1307.1699] [INSPIRE].
U.S.A., (1977).
[6] J.C. Collins and D.E. Soper , Backtoback jets in QCD, Nucl . Phys. B 193 ( 1981 ) 381 [60] S. Hoche and S. Prestel, The midpoint between dipole and parton showers , Eur. Phys. J . C [61] E.D. Malaza and B.R. Webber , Multiplicity distributions in quark and gluon jets, Nucl . Phys. [62] S. Lupia and W. Ochs , Uni ed QCD description of hadron and jet multiplicities, Phys . Lett. [63] P. Bolzoni , B.A. Kniehl and A.V. Kotikov , Gluon and quark jet multiplicities at