Generalized fragmentation functions for fractal jet observables

Journal of High Energy Physics, Jun 2017

We introduce a broad class of fractal jet observables that recursively probe the collective properties of hadrons produced in jet fragmentation. To describe these collinear-unsafe observables, we generalize the formalism of fragmentation functions, which are important objects in QCD for calculating cross sections involving identified final-state 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 tree-like 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.

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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 collinear-unsafe 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 tree-like 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 - nal-state 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 Tree-dependent observables 5.1 Node products 5.2 Full-tree observables 6 Application in quark/gluon discrimination 7 Fractal observables from subjets 8 Conclusions A Generalized fragmentation in inclusive jet production B A non-fractal example: sums of weighted energy fractions C Software implementation C.1 Running coupling C.2 Discretization C.3 Runge-Kutta 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 single-hadron 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 well-known 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 nal-state hadrons. Because GFFs depend on correlations between nal-state 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 well-suited to numerical evaluation.1 Speci cally, we focus on observables de ned using hierarchical binary clustering trees that mimic the leading-order tree-like 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 higher-order corrections, in contrast to the semi-classical parton shower approach. The class of fractal jet observables described by eq. (1.1) is surprisingly rich, allowing for many collinear-unsafe 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 non-kinematic 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 non-associative 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 leading-order 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 non-associative fractal observable is given by node-based 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 non-associative observables to probe parton fragmentation from a di erent perspective than previously-studied jet observables. As one application, we consider the discrimination between quark- and gluon-initiated 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 leading-order 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 multi-hadron FFs in the literature. This includes dihadron fragmentation functions which describe the momentum fraction carried by pairs of nal-state hadrons [33, 34], and fracture functions which correlate the properties of one initial-state and one nal-state 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 non-kinematic quantum numbers (e.g. charge). As discussed in ref. [14] for the case of weighted jet charge, the n-th moment of GFFs can sometimes be related to moments of n-hadron FFs. At the level of the full distribution, though, GFFs are distinct from multi-hadron 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 non-associative fractal observables in section 5 | node products and full-tree 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 collinear-unsafe 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 nal-state 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 jet-like 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 four-vector w in light-cone 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 light-like 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 semi-inclusive 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 tree-level 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 { process-dependent perturbative functions that encode the physics of the hard subprocess. The FFs Dih(x; ) are universal, process-independent 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 next-order 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 lowest-order 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 leading-order 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 nal-state 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 nal-state 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, plus-function regulators are required at both endpoints when integrating over the entire range 0 z { 6 { fraction and S is de ned by non-kinematic 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 phase-space 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 gluon-initiated 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 space-time 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 multi-hadron FFs [33, 34], with the key di erence that multi-hadron 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 nal-state 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 center-of-mass 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 next-to-next-to leading order and (0) denotes the tree-level 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 xed-order calculations and to extract GFFs beyond leading order. At next-to-leading 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 x-dependence 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 next-tonext-to-leading 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 track-based 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 higher-order 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 next-to-leading order in appendix A. As in refs. [14, 15], we expect that the absorption of collinear divergences by GFFs can be carried out order-by-order in s due to the universality of the collinear limits in QCD. { 8 { The above generalized fragmentation formalism works for any collinear-unsafe (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 (leading-order) 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 higher-order evolution to future work, and focus primarily on the leading-order 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 next-to-leading-order 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 non-leaf node. The observable values at the non-leaf 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 nal-state 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 heavy-quark 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 leading-order RG evolution equations. Our construction is based on the following three ingredients, as shown in gure 2: 1. Weights wa for each nal-state hadron; 2. An IRC-safe 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. anti-kt [45] in the studies below. As the initial boundary HJEP06(217)85 condition for the observable, each nal-state hadron within the jet is assigned a weight wa (possibly zero) based on some non-kinematic 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 nal-state hadrons are then used as inputs to an IRC-safe 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 generalized-kt family of jet clustering algorithms [45], which are designed to follow the leading-order 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 tree-dependence of the observable, with p = f 1; 0; 1g corresponding to the fanti-kt [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, generalized-kt 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 single-particle 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 non-zero, 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 leading-order evolution equations do not depend on the clustering algorithm. When explicit -dependence is included in the x^ function, this sometimes results in a fully IRC-safe 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 generalized-kt 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 one-to-one 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 one-loop renormalization factor and one-loop Fj (and tree-level Fk). Away from the strongly-ordered 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 higher-order 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 one-loop F 's and one tree-level F . Finally, the 1= UV from the one-loop virtual contribution to the 1 ! 2 splitting gives a higher-order correction to the rst term in eq. (2.13). For the remainder of this paper, we focus on the leading-order 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 non-associative 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 nal-state hadrons [13, 14, 50]. This quantity has, for example, been used in forward-backward 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 generalized-kt 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 non-associative recursion relations, where the observable does not simplify to a sum over nal-state hadrons and the full tree traversal is necessary. We explore some non-associative 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 parton-shower truth information to assign the parton label i. To generate pure samples of quark- and gluon-initiated jets, we use the e+e ! =Z ! qq and ! H ! gg processes in Pythia 8.215 [57], switching o initial-state radiation. We nd jets using FastJet 3.2.0 [58], with the ee-generalized kt algorithm with p = 1 (i.e. the version of anti-kt [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 anti-quark 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 non-singlet 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 quark-singlet combination S(x; ) de ned in eq. (4.6). For the observables we study, the anti-quark 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 quark-singlet combination, postponing a discussion of the non-singlet case to section 4.5. gure 4, we show the extracted gluon and quark-singlet 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 quark-singlet 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 quark-singlet 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 quark-singlet 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 higher-order 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 quark-singlet 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 node-product 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 jet-energy 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 RG-evolved 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 higher-order 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 full-tree 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 angular-ordered 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 leading-logarithmic resummation of an observable de ned by eq. (7.1). Starting from a low-energy 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 xed-order correction at that scale, but this only enters at next-to-leading-logarithmic 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 higher-order 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 single-hadron fragmentation functions or IRC-safe jet shapes. In this paper, we emphasized the intermediate possibility of IR-safe but collinear-unsafe 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 collinear-unsafe 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 IRC-safe clustering tree with certain initial hadron weights, which satisfy a self-similar RG evolution at leading order given by eq. (2.13). The higher order evolution is no longer universal, but still self-similar, 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 node-product and full-tree 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 non-associative 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 collinear-unsafe 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 leading-order 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 next-to-leading 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 IRC-safe 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 DE-SC-00012567 and DE-SC-00015476. M.P. is supported by a Marie Curie Intra-European Fellowship of the European Community's 7th Framework Programme under contract number PIEF-GA-2013-622527. W.W. is supported by the European Research Council under grant ERC-STG-2015-677323, and the D-ITP 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 left-hand side of this equation for the measurement of the fractal variable x together with the fraction of the center-of-mass energy carried by the jet, zJ 2Ejet=Ecm. Assuming that R is not so large that all nal-state 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 leading-order 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 right-hand 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 left-hand 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 anti-quarks 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 (in-jet versus out-of-jet), their sum vanishes in Jq(!1)qg(z; EjetR; ) + Jq(!1)q(z; z EjetR; ) = 0 : The nal ingredient we need is the renormalized one-loop 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 right-hand sides in eq. (2.12). On the latter, these solely come from Cq(0)(zJ ; Ecm; )[Fq(1)(x; ) + Fq(1)(x; )]. On the left-hand 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 y2-independent 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 anti-quark 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 right-hand 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 anti-quark 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 non-fractal example: sums of weighted energy fractions While eq. (1.1) is rather general, there are of course many collinear-unsafe 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 leading-order evolution, the running of s is strictly speaking only required at leading-logarithmic 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 equal-width 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 n1-th and n2-th 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 x-bin 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 plus-function regulator must be handled correctly for the endpoint bins. If the regulated functions have the following primitives then their integrals over the n-th 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) quark-singlet 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 x-bin is less than 0.06%. C.3 Runge-Kutta algorithm After the discretization in eq. (C.4), the RG evolution is performed with an embedded fthorder Runge-Kutta 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 fth-order 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 fourth-order Runge-Kutta 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 fourth-order value yk+1, but we take the (more accurate) fth-order value. This ensures that our solution is actually slightly more accurate than our error indicates. Estimating the error on this fth-order solution would require calculating a still-higher 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 fourth-order 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 quark-singlet 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 ill-posed. 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 Runge-Kutta 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) quark-singlet 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 non-associative 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 Euler-Mascheroni 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 node-product observables with (top row) and (bottom row) = 4 for the generalized-k 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 quark-singlet (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 non-integer . 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 quark-singlet 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 non-singlet 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 quark-singlet 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 Full-tree observables For full-tree 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 plus-prescription 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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Benjamin T. Elder, Massimiliano Procura, Jesse Thaler, Wouter J. Waalewijn, Kevin Zhou. Generalized fragmentation functions for fractal jet observables, Journal of High Energy Physics, 2017, 85, DOI: 10.1007/JHEP06(2017)085