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 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 collinear-unsafe 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 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 with Ei the energy of branch i.2 With these de nitions, the leading-order 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 Full-tree 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 nal-state hadron [1{7]. Since the momentum of a single hadron is not collinear 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 . 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 nal-state hadrons. Because GFFs depend on correlations between nal-state hadrons, nary FF case. Motivated by the structure of the DGLAP equations, we de ne fractal jet well-suited 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 semi-classical 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 non-kinematic 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 non-associative fractal observable is given by node-based 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 multi-hadron FFs in the literature. This equations are nonlinear. The key di erence here is that fractal jet observables are not by non-kinematic quantum numbers (e.g. charge). As discussed in ref. [14] for the case 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 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 nal-state 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 eld-theoretic 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 four-vector w in light-cone coordinates: ? = 0 and using the = w = n w; w+ = n w; where n is a light-like 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 semi-inclusive 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 tree-level 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, process-independent 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 lowest-order splitting function explicitly as a 1 ! 2 process: With this notation, we can rewrite the leading-order 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 nal-state hadron in each term, so the evolution simpli es to the linear form in on correlations between the nal-state 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, plus-function 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 collinear-unsafe (but soft-safe) 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 gluon-initiated 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 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 the key di erence that multi-hadron FFs describe a xed number of identi ed 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 (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 next-to-next-to 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 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 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 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-toin 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 next-to-leading order in apcan be carried out order-by-order 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 (leading-order) 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 leading-order 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 next-to-leading-order 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 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. 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 heavy-quark 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 nal-state hadron; 2. An IRC-safe binary clustering tree; 3. The recursion relation x^(z; x1; x2). able is guaranteed to have fractal structure. a suitable jet algorithm, e.g. anti-kt [45] in the studies below. As the initial boundary 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 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, jet. For our studies, we use the generalized-kt 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 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 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 single-particle 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 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 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 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 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 tree-level Fk). Away from the strongly-ordered 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 higher-order 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 one-loop F 's and one tree-level F . Finally, the 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. 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 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 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 generalized-kt clustering, particularly in the collinear limit. 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. Extraction of GFFs using perturbatively calculated values for the coe cients Ci, Ji!j , Ji!jk, . . . . For the 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 ! gg processes in Pythia 8.215 [57], switching o initial-state radiation. We 1 (i.e. the version of anti-kt [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 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; ) = 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 quark-singlet 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 non-singlet case to section 4.5. gure 4, we show the extracted gluon and quark-singlet 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 quark-singlet 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 higher-order 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 quark-singlet 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 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. We can check whether this jet-energy 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 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 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 leading-logarithmic 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 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 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 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 of its behavior. Conclusions To date, the bulk of analytic jet physics studies are based on either single-hadron fragfunctions that absorb collinear singularities order by order in s, which not only restores 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, 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 non-associative 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 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 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 next-to-leading 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 left-hand 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 (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 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 : (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 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) 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] 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 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 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 y2-independent 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 anti-quark 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 non-fractal 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 leading-order 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 equal-width 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 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 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: 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 plus-function the following primitives then their integrals over the n-th 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 x-bin is less than 0.06%. Runge-Kutta algorithm 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. error. The error estimated this way applies to the fourth-order value yk+1, but we take the 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 = 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 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, 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 −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) quark-singlet 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 Euler-Mascheroni 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 node-product observables with (top row) and (bottom row) = 4 for the generalized-k = 1 extracted from the parton shower average at E + 0(N + 1). These lution equation in eq. (4.13) for non-integer . 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 quark-singlet 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 non-singlet GFFs, so these still evolve according to for higher moments. 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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, 1-54, DOI: 10.1007/JHEP06(2017)085