Generalized fragmentation functions for fractal jet observables
Received: May
Generalized fragmentation functions for fractal jet observables
Benjamin T. Elder 0 2 4 5 6
Massimiliano Procura 0 2 3 5 6
Jesse Thaler 0 2 4 5 6
Wouter J. Waalewijn 0 1 2 5 6
Open Access 0 2 5 6
c The Authors. 0 2 5 6
0 University of Amsterdam
1 Nikhef, Theory Group
2 77 Massachusetts Ave. , Cambridge, MA 02139 , U.S.A
3 Theoretical Physics Department , CERN
4 Center for Theoretical Physics, Massachusetts Institute of Technology
5 Science Park 105 , 1098 XG, Amsterdam , The Netherlands
6 Science Park 904 , 1098 XH 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
observables; Jets; QCD Phenomenology

from gluon jets.
Contents
1 Introduction
2 Formalism
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
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
with Ei the energy of branch i.2 With these de nitions, the leadingorder evolution
equation of the corresponding GFF takes the simpli ed form
d Fi(x; ) =
1 X Z
dz dx1 dx2
Pi!jk(z) Fj (x1; ) Fk(x2; ) [x
x^(z; x1; x2)];
Weighted energy fractions
Node products
E.3 Fulltree observables
Introduction
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.
objects that describe the ow of momentum from a fragmenting quark or gluon into an
identi ed
nalstate hadron [1{7]. Since the momentum of a single hadron is not collinear
safe, cross sections for singlehadron observables have singularities beginning at O( s).
These collinear singularities are absorbed by the FFs order by order in
. From this
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,
nary FF case. Motivated by the structure of the DGLAP equations, we de ne fractal jet
wellsuited to numerical evaluation.1
Speci cally, we focus on observables de ned using hierarchical binary clustering trees
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 + E2
p~; x = x^(z; x1; x2)
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.
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
to the semiclassical parton shower approach.
The class of fractal jet observables described by eq. (1.1) is surprisingly rich, allowing
eq. (1.3) describes the evolution of weighted energy fractions,
x =
where wa is a weight factor that depends on nonkinematic quantum numbers such as
charge or
> 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
by charged particles [15, 16]; and the observable pTD used by the CMS experiment for
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
appropriately clustered in the collinear limit. An example of a nonassociative fractal
observable is given by nodebased energy products,
x =
X (4zLzR) =2 ;
where the observable depends on the momentum fractions carried by the left and right
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
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
fractal observables in terms of subjets of radius Rsub. In this case, the initial conditions
x) at the initial
scale sub = EjetRsub
QCD, such that no nonperturbative input is needed. By evolving
the GFFs to
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
equations are nonlinear. The key di erence here is that fractal jet observables are not
by nonkinematic quantum numbers (e.g. charge). As discussed in ref. [14] for the case
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
and a description of the numerical RG implementation to the appendices.
To motivate the de nition of fractal jet observables, it is instructive to rst review the
which serves as a preamble to the explicit constructions in section 3.
Review of standard fragmentation
Ordinary FFs, denoted by Dih(x; ), are nonperturbative objects that describe the number
from the fragmentation of a parton of type i. They are the
nalstate counterpart to
conservation sum rule
dx x Dih(x; ) = 1 :
At leading order, the FFs are independent of the factorization scheme (see e.g. [37]).
The eldtheoretic de nition of the bare unpolarized quark FF is given by [6, 7]
Dih(x; ) =
2 h0j i(y+; 0; y?)jhXihhXj i(0)j0i ;
where we are working in a frame with quark transverse momentum p~
gauge choice A
ph with p
the quark
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
= w
= n w;
w+ = n w;
where n is a lightlike vector along the direction of the energetic parton, and n is de ned
= 2Ejet. Gauge invariance
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,
! hX) =
Ci(z; s; ) Dih(x=z; );
where x = 2Eh= s
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
referred to as the total FF, in which case Dih(x; ) is called the parton FF.
! hX) = F h(x; ) is sometimes
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
bative convergence in eq. (2.4), the renormalization scale
should be chosen close to ps.
While computing the FFs themselves requires nonperturbative information about the
This allows us to, for example, take FFs extracted from
ts to experimental data at one
by the DGLAP equations [8{11],
Dih(x; ) =
Pji(z)Djh(x=z; ):
Here, the splitting kernels Pji(z) can be calculated in perturbation theory,
Pji(z) = Pj(i0)(z) +
splitting function Pj(i1) arises from 1 ! 3 splittings as well as loop corrections to 1 ! 2
In order to motivate the transition to generalized fragmentation, it is convenient to
rewrite the lowestorder splitting function explicitly as a 1 ! 2 process:
With this notation, we can rewrite the leadingorder DGLAP equation in a suggestive form4
Dih(x; ) =
1 X Z 1
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
on correlations between the nalstate hadrons.
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
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
includes information about hadronization
uctuations. Being a probability density, the
GFFs are normalized to unity for each parton type,
dx Fi(x; ) = 1:
For any collinearunsafe (but softsafe) observable x, we can give an operator de nition
= 0 is de ned as
Fi(x; ) =
dy+d2y eip y+=2 1
2 h0j i(y+; 0; y?)jSXihSXj i(0)j0i ;
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
information from S. We stress that, in contrast to the standard FFs, a GFF involves a
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; ) =
dy+d2y eip y+=2 X
SX
h0jG ;a(y+; 0; y?)jSXihSXjG ;a; (0)j0i;
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
the key di erence that multihadron FFs describe a xed number of identi ed
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
(0) dzJ dx
! jet + X) =
Ci(zJ =y0; Ecm; )
Ji(!1)j (y0; EjetR; ) Fj (x; )
dz dx1 dx2 Ji(!1)jk(z; EjetR; ) Fj (x1; ) Fk(x2; ) [x x^(z; x1; x2)]
x^(z; x1; x2)] + : : : ;
where the ellipsis includes further terms at nexttonextto leading order and (0) denotes
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.
zJ ) and the
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
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
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
nexttoin 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
It is important to note that eq. (2.12) really combines two di erent formalisms. The
we provide all details of the matching for e+e
! jet + X at nexttoleading order in
apcan be carried out orderbyorder in s due to the universality of the collinear limits in QCD.
safe) 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
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
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 Fi(x; ) =
1 X Z
dz dx1 dx2
Pi!jk(z) Fj(x1; ) Fk(x2; ) [x
x^(z; x1; x2)];
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
leadingorder evolution here. As a consistency check, the
function in eq. (2.13) ensures
that the RG evolution automatically preserves the GFF normalization,
dx Fi(x; ) =
1 X Z
dx2 Fk(x2; ) = 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
the states jSXi in eqs. (2.10) and (2.11) partonically, the nexttoleadingorder bare GFF
2 satis es
Fi(1)(x) =
1 X Z
dx1 dx2 Fj(0)(x1; ) Fk(0)(x2; ) [x
x^(z; x1; x2)]:
Here, the function x^(z; x1; x2) is the form of x~(p ; S) written in terms of two subjets,
x^(z; x1; x2) = z x1 + (1
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).
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.
is indeed a fractal observable.
In the above analysis, we implicitly assumed massless partons, since otherwise the
EjetR, it is consistent
for the fractal observable. At the scale
the appropriate heavyquark description.
Fractal observables via clustering trees
have the desired RG evolution in eq. (2.13). The idea is to use recursive clustering trees
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).
able is guaranteed to have fractal structure.
a suitable jet algorithm, e.g. antikt [45] in the studies below. As the initial boundary
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
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,
jet. For our studies, we use the generalizedkt family of jet clustering algorithms [45], which
e+e collisions, these algorithms have the pairwise clustering metric
dij = min[Ei2p; Ej2p] i2j ;
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
paira 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
genfact responsible for the evolution in eq. (1.3).
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
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
lim x^(z; x1; x2) = x1;
lim x^(z; x1; x2) = x2;
such that an arbitrarily soft branch in the clustering tree has no impact on the values of
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
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
informa
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
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
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; )
depends only on GFFs of the same observable. Schematically, this can be written as
d Fi =
represents a convolution. This equation includes 1 ! n splittings at order sn 1.
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
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
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
Fj (and treelevel Fk). Away from the stronglyordered limit, the 1 ! 3 splitting does
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.
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
1 ! 3 splitting term at order
s2 with two oneloop F 's and one treelevel F . Finally, the
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.
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
> 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,
x =
associative observables studied in section 4, the order of the clustering does not a ect the
in section 5.
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
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
energy fractions with arbitrary
for applications to quark/gluon discrimination.
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^
xA = xB = xC = w1 z1 + w2 z2 + w3 z3 ;
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.
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.
Extraction of GFFs
using perturbatively calculated values for the coe cients Ci, Ji!j , Ji!jk, . . . . For the
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
! gg processes in Pythia 8.215 [57], switching o initialstate radiation. We
1 (i.e. the
version of antikt [45]) and then determine the various weighted energy fractions on
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;
which is roughly the scale of the hardest possible splitting in the jet. By varying Ejet and R
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,
This is not included in our present study, since we decided to interface all of the showers
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; ) =
i2fu;u;d;:::bg
Nij (x; ) = Fi(x; )
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
PS = Vincia
R = { 0.3, 0.6, 0.9}
hQuarki GFF
PS = Vincia
R = { 0.3, 0.6, 0.9}
hQuarki GFF
PS = { Pythia, Vincia, Dire}
R = 0.6
hQuarki GFF
PS = { Pythia, Vincia, Dire}
R = 0.6
hQuarki GFF
= 0:5 and
xed. The right column shows the
Vincia, and Dire. In this and subsequent gures, hQuarki always refers to the quarksinglet
combination S(x; ) de ned in eq. (4.6).
or simply involve the replacement x
x, due to charge conjugation symmetry.
postponing a discussion of the nonsinglet case to section 4.5.
gure 4, we show the extracted gluon and quarksinglet GFFs at
= EjetR =
= 0:5 and
= 2.
higher than the mean of
R = f0:3; 0:6; 0:9g, leaving
xed. The envelope from changing R is very small, indicating
not shown here, we checked that the GFFs for the
as well (see section 4.4 below).
Evolution of GFFs
! 1 limits behave sensibly
of principle for our RG evolution code, we show upward evolution from
= 100 GeV to
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
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
quarkof 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
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
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
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.
−1
= 0:5, (middle row)
= 1,
and (bottom row)
GFFs extracted from parton showers at
these initial conditions to
−0.2
−0.5
−1
−0.2
gure 5 but for quarksinglet GFFs, where the distributions extracted from
parton showers at
light blue, and the distributions extracted at
−1.0
−0.5
Full Tree ROC Curves
Full Tree ROC Curves
Quark Efficiency
Quark Efficiency
= 2 and
= 0 case is identical to pTD.
= f0; 2; 4; 6g. Note
comparing the discrimination performance at
= 100 GeV to
= 4 TeV. In
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.
We can check whether this jetenergy dependence is reasonable using the RG evolution
equations, as shown in gure 16. For
evolution equation are important for getting the proper shape of the
= 1 distribution. For
at either scale from varying R and the parton shower. For
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
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
p = 0
Full Tree ROC Curve RG
Full Tree ROC Curve RG
Full Tree ROC Curve RG
As emphasized in ref. [19], predicting the quark/gluon discrimination power from
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
evolupredicting quark/gluon discrimination behavior.
Fractal observables from subjets
an angular scale . This opens up the possibility to de ne a modi ed recursion relation
dependence, for example,
p = 0
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
observinitial 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
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
100 GeV → 4 TeV
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
is the weighted energy fraction of all particles with
each of the branches, so the GFFs at the scale sub are
Fi(x; sub) = (1
which are then the input for the fractal observable x^2 for
> Rsub. This e ectively removes
lytically, as long as the scale sub is perturbative. An example of this kind of observable is
gure 20, where the observable is clustered using the recursion relation eq. (4.1)
= 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
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
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
of its behavior.
Conclusions
To date, the bulk of analytic jet physics studies are based on either singlehadron
fragfunctions that absorb collinear singularities order by order in
s, which not only restores
universal, but still selfsimilar, and has the schematic form in eq. (3.3).
The simplest fractal jet observables are those with associative recursion relations,
in use at colliders, including pD, weighted jet charge, and track fractions. More exotic
for these observables is independent of the clustering tree at leading order.
As one potential application of fractal observables, we studied whether nonassociative
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.
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
hadrons), as brie y discussed in section 7.
Looking to the future, the next step for fractal jet observables is pushing beyond the
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
amwith important consequences for analyses at the LHC and future collider experiments.
Acknowledgments
discrimination. The work of B.E., J.T., and K.Z. is supported by the DOE under grant
Ministry of Education, Culture and Science (OCW).
Generalized fragmentation in inclusive jet production
In this appendix, we explicitly verify eq. (2.12) at O( s). We rst calculate the lefthand
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
(0) dzJ dx
= (0)
dx1 dx2 Fi
(0)(x1; ) Fj(0)(x2; ) x x^
+ ( 12 R) ( 13 R) ( 23 R) X
antiquark, and 3 with the gluon. The angle ij between partons i and j is given by
ij = arccos 1
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
attached to it. The squared matrix element that enters in eq. (A.1) is given up to O( s) by
ln(1 z)
(0) dy1dy2
= (1 y1) (1 y2) +
Pq!qg(y) = CF
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 :
(1 y3)(y12 + y22)
2(1 y1)+(1 y2)+
+ (1 + y12)
Cq(z; Ecm; ) = (1
Cg(z; Ecm; ) =
Pq!qg(z) ln
z) + 2 ln 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 (1 z) + 2Pq!qg(z) 3CF (1 z) L + CF 4z
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)
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]
Fi(x) = Fi(0)(x)
X Z
dx1 dx2 Fj(0)(x1; ) Fk(0)(x2; ) [x
x^(z; x1; x2)] :
Jq(!1)g is not simply z $ 1
the initiating parton is held
dimensional regularization,
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
y1)Pq!qg(y2)
1) Fq(0)(x; )
dx1 dx2 Fq(0)(x1; ) Fg(0)(x2; )
x^(y2; x1; x2)
= (zJ
1)[Fq(1)(x; ) + Fq(1)(x; )];
line of eq. (A.11) proportional to Fi
straint in the small R limit as
(0) does not contribute here because it is y2independent
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
yi. We rst consider the (R
Pq!qg(y1) ln y1Ec2m + (1 + y12)
2
dx1 dx2 Fq(0)(x1; ) Fg(0)(x2; ) x
x^(y1; x1; x2)
z + (: : : ) (1
dx1 dx2 Fq(0)(x1; ) Fg(0)(x2; ) x
x^(z; x1; x2)
= (1
dz dx1 dx2 Jq(!1)qg(z; EjetR; ) Fq(x1; ) Fg(x2; )
x^(z; x1; x2)] + (: : : )13:
ularized, 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
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
R) = 1
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
in eq. (A.14). For example, the
13) term gives
Pq!qg(y1) ln y1Ec2m + (1 + y12)
2
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
quark and antiquark interchanged and the
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
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 ;
y = X vi zi ;
t = x + y
for particle weights wi and vi, and energy exponents
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
Fi(t) =
dx dy Fi(x; y) [t
d Fi(x; y; ) =
dz dx1 dx2 dy1 dy2 Pi!jk(z) Fj(x1; y1; ) Fk(x2; y2; )
Plugging eq. (B.4) into eq. (B.3), we can insert a factor of
dt1 dt2 [t1
to perform the integrals over y1 and y2. The resulting equation is
d Fi(t; ) =
dz dt1 dt2 dx1 dx2 Pi!jk(z) Fj(x1; t1
x1) Fk(x2; t2
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
fractal form. Of course,
just corresponds to a regular weighted energy fraction with
weights wi + vi, so this is not a new fractal observable.
Software implementation
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.
Running coupling
Because we only perform leadingorder evolution, the running of
s is strictly speaking
of the strong coupling is included using the
CATF nf
4CF TF nf :
0 =
s( ) =
function at O( s3),
1 =
The running coupling at the scale
is given by solving eq. (C.1) iteratively to order O( s3),
where L = ln 2QCD
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
N X Z n=N
Z n=N
X Z n1=N
Z n2=N
dz Pi!jk(z)
18This equation is written for N equalwidth bins for simplicity of notation. The generalization to unequal
N X Z n=N
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
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:
Pq!gq(z) = Pq!qg(1
z) = CF
Pg!qq(z) = TF z2 + (1
Pg!gg(z) = 2CA
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
and 0 is given in eq. (C.2).19 When performing the integration, terms with a plusfunction
the following primitives
then their integrals over the nth bin are implemented by
F (z + 0:5 z)
F (z + 0:5 z)
0:5 z) n 6= 0;
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
= 0 ;
= 0 ;
−0.02
100 GeV → 4 TeV
−0.02
hQuarki GFF
100 GeV → 4 TeV
= 0:5. The curves labeled
indistinguishable by eye.
nrough = 1000 and n ne = 100. The
ner division of the endpoint bins is necessary to
recursion relation satis es
x^(z; x1; x2) = z x1 + (1
z) x2 =)
dependence on n ne. This is shown in
gure 21 for the case of
= 0:5, with all particle
the evolved GFFs in a single xbin is less than 0.06%.
RungeKutta algorithm
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.
error. The error estimated this way applies to the fourthorder value yk+1, but we take the
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 =
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
smaller step.
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
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
jykmj + jhk dykm=d ln
is an overall upper limit which was set to 10 9 for the GFF evolution. The
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.
Numerical stability
All of the RG results in this paper are based on the numerical solution of eq. (2.13) for
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,
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
−0.2
4 TeV → 100 GeV
hQuarki GFF
WEF All Particles
4 TeV → 100 GeV
hQuarki GFF
WEF All Particles
−2
−40
−20
4 TeV → 100 GeV
WEF
4 TeV → 100 GeV
= 4 TeV to
and (right column) quarksinglet GFF with (top row)
= 0:5 and (bottom row)
= 2:0. The
larization [79], though we do not do so here. Note that in general, if the evolution in one
to be unstable in the reverse direction.
of the GFFs are de ned by
where the zeroth moment is just the normalization,
F i(N; ) =
F i(0; ) =
dx Fi(x; ) = 1:
evolution equation,
d F i(N; ) =
1 X Z
This convention follows the standard nomenclature of probability theory.
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; ):
discuss the details for each of the sets of observables studied in this paper.
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)
of the splitting functions are de ned as
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,
F i(N; ) =
j;k M=0
2(N 2 + 3N + 3)
N (N + 1)(N + 2)(N + 3)
0(N + 2)
3 TF nf : (E.6)
Node Product Moment Space RG
0.7 Node Product Moment Space RG
0.8 Node Product Moment Space RG 1.0 Node Product Moment Space RG
Moment space evolution of the nodeproduct observables with (top row)
and (bottom row)
= 4 for the generalizedk
= 1
extracted from the parton shower average at
E + 0(N + 1). These
lution equation in eq. (4.13) for noninteger . Note that N is shifted up by one from the
is shifted by one as well compared to Mellin moments.
Node products
We now insert the recursion relation for the node products from eq. (5.1) into eq. (E.3).
energy fractions. These terms have splitting kernels of the form
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
2nf P g!qq( ) P g!gg( )
The additional constant terms are de ned as
dz Pq!qg(z) + Pq!gq(z) 4z(1
dz 2nf Pg!qq(z) + Pq!gg(z) 4z(1
=2 ;
=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
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
evolved to the scale
parton shower average at the scale
lution equations are of the same general form as for the weighted energy fractions,
P i!j;k(N; M ) F j (N
d F i(N; ) =
j;k M=0
but with di erent splitting kernels,
the parameters
Full Tree Moment Space RG
Full Tree Moment Space RG
Full Tree Moment Space RG
Full Tree Moment Space RG
Full Tree Moment Space RG
In gure 24, we show the evolution of the rst two moments of the GFFs for
= 2,
extracted from the parton shower average at
= 4 TeV.
Open Access.
This article is distributed under the terms of the Creative Commons
any medium, provided the original author(s) and source are credited.
= 2 measured
on charged particles, with (top row)
2 and (bottom row)
= 2.
Explicitly, for the rst moment in the quarksinglet basis,
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