Disentangling Heavy Flavor at Colliders

MIT-CTP/4880

Philip Ilten,1, ∗ Nicholas L. Rodd,2, † Jesse Thaler,2, ‡ and Mike Williams1, § 1Laboratory for Nuclear Science, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. 2Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A. We propose two new analysis strategies for studying charm and beauty quarks at colliders. The first strategy is aimed at testing the kinematics of heavy-flavor quarks within an identified jet. Here, we use the SoftDrop jet-declustering algorithm to identify two subjets within a large-radius jet, using subjet flavor tagging to test the heavy-quark splitting functions of QCD. For subjets containing a J/ψ or Υ, this declustering technique can also help probe the mechanism for quarkonium production. The second strategy is aimed at isolating heavy-flavor production from gluon splitting. Here, we introduce a new FlavorCone algorithm, which smoothly interpolates from well-separated heavy- quark jets to the gluon-splitting regime where jets overlap. Because of its excellent ability to identify charm and beauty hadrons, the LHCb detector is ideally suited to pursue these strategies, though similar measurements should also be possible at ATLAS and CMS. Together, these SoftDrop and FlavorCone studies should clarify a number of aspects of heavy-flavor physics at colliders, and provide crucial information needed to improve heavy-flavor modeling in parton-shower generators.

I. INTRODUCTION

The production of charm and beauty quarks at the Large Hadron Collider (LHC) is studied both as a fun- damental probe of Standard Model (SM) phenomenology, and as an important component of searches for physics beyond the SM. For example, heavy-flavor tagging is used (a) Gluon Splitting (b) Flavor Excitation to test the properties of the SM Higgs boson, whose largest branching fraction is to a pair of beauty quarks [1]. Similarly, identifying large-radius jets with double-flavor- tagged substructure enables searching for new physics scenarios involving high-pT Higgs bosons [2–7]. To ad- dress SM backgrounds in both cases, it is essential to understand the mechanisms for heavy-flavor production at the LHC within quantum chromodynamics (QCD). Of (c) s-channel Flavor (d) t-channel Flavor particular importance is the process of gluon splitting to Creation Creation heavy-quark pairs g QQ¯, where Q denotes a b or c → FIG. 1. Leading diagrams in QCD that contribute to heavy-flavor quark, which is challenging to study both theoretically production at the LHC: (a) gluon splitting, where a QQ¯ pair arises and experimentally. from a time-like off-shell gluon; (b) flavor excitation, where Q is In this article, we present two analysis strategies aimed excited from an incoming proton; and (c,d) flavor creation, where at testing key features of heavy-flavor production at the the QQ¯ pair originates directly from the hard scattering. The pre- LHC. First, we use a jet-declustering method to study cise αs order at which these diagrams appear depends on whether one is working in a 3-, 4-, or 5-flavor scheme for parton distribution heavy-flavor kinematics within identified jets, with the functions (PDFs). Note that at higher orders, there is no gauge- goal of testing the well-known but as-of-yet-unmeasured invariant distinction between these categories. massive 1 2 splitting functions of QCD. Second, we arXiv:1702.02947v2 [hep-ph] 4 Apr 2017 introduce a→ new jet algorithm designed to enable disentangling the various QCD-production processes for heavy flavor (see Fig.1), with an emphasis on studying the con- prospects. In the appendices, we also present results for ATLAS and CMS, which cover η [ 2.5, 2.5]. Qualita- tribution from gluon splitting. Both of these analyses can ∈ − in principle be performed by any of the LHC experiments. tively, the results of both proposed analyses are the same Here, we focus on the LHCb detector, which covers the for LHCb and for ATLAS/CMS. pseudorapidity range η [2, 5], since its excellent heavy- Our jet-declustering method is based on the SoftDrop flavor-identification capabilities∈ offer the best short-term algorithm [8] and its precursor, the (modified) MassDrop tagger [9–11]. Starting from a single large-radius jet, the declustering method strips away soft peripheral radiation and forces the groomed jet to have 2-prong substructure. ∗ philten@cern.ch As shown in Ref. [12], the kinematics of the two resulting † [email protected] subjets match the famous Altarelli-Parisi 1 2 split- ‡ [email protected] ting functions for massless QCD [13]. SoftDrop→ has been § [email protected] used by CMS [14] and STAR [15] in the context of heavy- 2 ion collisions, and a related strategy was proposed to test are always nonperturbative corrections from hadroniza- the dead cone effect for boosted top quarks [16]. Here, we tion and the underlying event, but jet-level measurements extend the analysis to QCD with heavy-flavor quarks, ex- are generally expected to be closer to parton-level pertur- ploiting the ability to flavor-tag subjets to test the split- bative calculations. In any case, jet-based and hadron- ting kinematics of Q Qg and g QQ¯. Because the based analyses provide complementary information and SoftDrop algorithm works→ equally well→ on tagged and un- both should be pursued when studying heavy flavor. tagged jets, we can compare our massive results directly We validate the performance of these methods at to the massless case. In addition, this method can be the 13 TeV LHC using parton-shower generators. Our applied to quarkonium states like the J/ψ and Υ, poten- primary focus is on Pythia 8.212 [26–28], which in- tially providing new insights into the puzzle of quarko- cludes heavy-quark mass effects using matrix-element nium polarization and fragmentation [17–22]. corrections [29], and allows a leading-order classification Our gluon-splitting study is based on a new jet al- of events into gluon-splitting and non-gluon-splitting gorithm, referred to as FlavorCone, that identifies con- topologies. For the SoftDrop study, we compare Pythia ical jets by centering the jet axes along the flight di- to Herwig++ 2.7.1 [30, 31] in order to test the robust- rections of well-identified flavor-tagged hadrons. Unlike ness of the 1 2 subjet kinematics to different showering → 1 standard jet algorithms like anti-kt [23], the FlavorCone and hadronization models. For the FlavorCone study, method allows two jet axes to become arbitrarily close. we also consider alternative perturbative-shower results This feature, partially inspired by the XCone jet algo- from Vincia 2.0.01 [32, 33] and Dire 0.900 [34], as well rithm [24, 25], is ideal for studying gluon splitting to as matched next-to-leading-order (NLO) results from heavy flavor, where the two outgoing heavy quarks are PowhegBox v2 [35–38], all using Pythia for hadroniza- often more collimated than the jet radius R. In standard tion.2 Where needed, we use FastJet 3.1.2 [39] for jet jet analyses, overlapping heavy-flavor jets are typically finding and the RecursiveTools fjcontrib 1.024 [39] merged, with a precipitous drop in efficiency at angular for SoftDrop. scales smaller than R. In the FlavorCone method, by The remainder of the article is organized as follows. contrast, heavy-flavor jet axes can be arbitrarily close, In Sec.II, we show how SoftDrop declustering can be with the separate jet constituents determined by nearest- used to study heavy-flavor kinematics within large-radius neighbor partitioning. In this way, the FlavorCone al- jets, including the kinematics of quarkonium production. gorithm enables a full exploration of the heavy-flavor In Sec.III, we define the FlavorCone jet algorithm and production phase space, interpolating between the tra- demonstrate how it can be used to disentangle heavy- ditional regime of well-separated jets to the overlapping flavor production processes in QCD. We do not include regime dominated by gluon splitting. detector-response effects on the distributions presented here, though we do discuss the prospects for apply- Like standard approaches to studying high-pT heavy- flavor production at the LHC, the SoftDrop and Fla- ing these methods in the realistic LHCb environment in vorCone strategies involve tagging (sub)jets that contain Sec.IV. We conclude in Sec.V, leaving additional plots heavy-flavor hadrons. As we will see below, however, to the appendices. both the SoftDrop and FlavorCone analyses require a definition of flavor tags that is more closely tied to heavy- flavor hadrons than typically required for tagging appli- II. SOFTDROP JET DECLUSTERING TO cations. Specifically, it will be essential to reconstruct PROBE HEAVY-FLAVOR KINEMATICS the flight directions of heavy-flavor hadrons. For Soft- Drop, these flight directions are used to define flavor- The goal of our jet-declustering analysis is to study the tagged subjet categories. For FlavorCone, these flight collinear-splitting kernels of QCD appropriate for mas- directions directly determine the central jet axes. In this sive quarks.3 These kernels form the basis for parton way, the experimental requirements for—and challenges showers like Pythia, so we expect jet-declustering mea- of—performing both analyses are largely shared. surements will help improve theoretical predictions in the As an alternative to (sub)jet tagging, one could per- collinear regime. We also present results for quarkonium form exclusive reconstruction of heavy-flavor hadrons. production within an identified jet. The current Pythia From the experimental perspective, tagging is typically more efficient than reconstruction, since there are relatively few heavy-flavor decay modes that can be fully 1 reconstructed. From the theoretical perspective, analy- Because we are focusing on relatively low-pT jets at LHCb, we ses based on flavor-tagged jets are less sensitive to non- generate minimum bias events, which precludes the use of NLO perturbative physics than those directly based on heavy- generators. 2 flavor hadrons. To the extent that the typical jet scale These programs are not compatible with a common underlying event model; therefore, we turn off multiple parton interactions pTR is larger than the heavy-flavor-hadron masses, the (MPI) in Pythia for the FlavorCone study to focus on perturba- properties of heavy-flavor jets can be reliably calculated tive physics. In the case of Pythia, we tested that the addition in (resummed) perturbative QCD, without the use of of MPI does not impact our conclusions. heavy-flavor fragmentation functions. Of course, there 3 For related work, see Ref. [40]. 3 models for J/ψ and Υ production are known to be in- hadron requirements, and then use the following analysis complete, so measurements of the quarkonium-splitting workflow. kinematics should provide valuable information. In this section, we use the SoftDrop algorithm along with heavy- We identify all flavor-tagged hadrons with pT > • 2 GeV and treat their flight directions as ghost par- flavor tagging to reveal the massive-quark splitting ker- 4 nels. ticles [57] for the purposes of jet clustering. For the case of charm tagging, we require that the c- hadron does not come from a b decay. A. Review of SoftDrop We cluster the hadrons and the ghosts into anti-kt • fat jets with R = 1.0. SoftDrop is a jet-grooming technique that removes The hardest jet is required to have η [3, 4], so wide-angle soft radiation from a jet. This algorithm is • ∈ a generalization of the (modified) MassDrop tagger from that the full nominal jet cone is within LHCb ac- Refs. [9–11], with an additional angular exponent β that ceptance, and pT > 20 GeV, which is a typical jet controls the degree of grooming. In general, SoftDrop scale in LHCb. In App.C, we show results relevant reduces the dependence of the jet kinematics on other for ATLAS and CMS using a larger pT threshold. aspects of the full event, such as the underlying event, We apply the SoftDrop jet-declustering algorithm color correlations to the initial state, and pileup contam- • to the hardest jet, taking the SoftDrop parameters ination. Here, we will be primarily interested in using to be β = 0 and zcut = 0.1. Note that with this SoftDrop to define 1 2 splitting kinematics. → choice of parameters SoftDrop acts identically to SoftDrop starts from a jet of radius R that has been the modified MassDrop tagger with µ = 1 [11]. clustered with some jet algorithm, typically anti-kt. Re- gardless of the clustering algorithm used to form the ini- For each flavor-tagged hadron that is kept after tial jet, one builds a Cambridge-Aachen (C/A) [41, 42] • SoftDrop, we calculate clustering tree from the jet constituents. Working back- tag wards from the top of the tree, SoftDrop recursively pT ztag = , (2) checks whether the two branches of the tree satisfy the pT1 + pT2 following condition, set by the the grooming parameters zcut and β: where pT1 and pT2 are the transverse momenta of the two SoftDrop subjets. To count as a flavor tag β min(pT1, pT2) R12 in the classification scheme below, we require ztag > > zcut , (1) pT1 + pT2 R 0.05. where pTi are the transverse momenta of the two The resulting SoftDrop subjets, and their flavor labels, branches and R12 is their rapidity-azimuth separation. form the basic objects of interest for subsequent analy- If the condition in Eq. (1) is not satisfied, then the softer sis. We choose ztag to be half the value of zcut in order of the two branches is dropped and the procedure is re- to reduce kinematic dependence on the tagging condi- peated on the next node down the C/A tree. The proce- tion, though one could further optimize the relationship dure terminates once the SoftDrop condition is satisfied, between ztag and zcut to balance perturbative control and the two final branches define the two SoftDrop sub- against dependence on heavy-flavor fragmentation. Ex- jets. perimentally, the method used to tag the hadron flavor The SoftDrop algorithm has proven to be a valuable must provide a measurement of the hadron flight direc- tool for the study of jets; see, for example, Refs. [16, 43– tion, using, for example, the vector formed by connecting 56]. As already mentioned, SoftDrop has been shown the pp-collision point to the hadron-decay vertex. both theoretically [12] and experimentally [14, 15] to ex- These selection requirements are loose, and require pose the basic splitting functions of massless QCD. Using some care to implement properly in event generators. a parton-shower analysis, we argue below that SoftDrop Within the LHCb acceptance, we often find that the fat can also be used to directly study the massive QCD split- jet comes not from the hard-scattering process but from ting functions. underlying event activity. For this reason, we only test the Pythia and Herwig++ event generators, since they have full implementations of the underlying event includ- B. Event Selection and Flavor Classification ing MPI.

Because we want to compare the splitting kinematics for jets that contain different numbers of heavy-flavor- 4 A ghost is a particle with infinitesimal energy, but well-defined tagged hadrons, we define an event selection that is inde- direction, that is clustered for the purpose of (sub)jet heavy- pendent of the heavy-flavor content. We start from large- flavor tagging. See Sec.IV for a discussion of the experimental radius merged jets without applying any flavor-tagged aspects of flavor-tagged hadrons. 4

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FIG. 2. Event displays for SoftDrop subjets with (a) no flavor-tagged hadrons, (b,c,d) c-tagged hadrons, and (e,f,g) b-tagged hadrons. The fat anti-kt jet axis defines ∆y = ∆φ = 0. The filled black boxes represent particles clustered by anti-kt, where the area of the box is proportional to the pT of the particle. The flavor-tagged ghost particles are shown as open circles, with blue for charm and red for beauty. The (darker) leading and (lighter) subleading SoftDrop subjets are displayed as shaded gray regions. Because these are relatively low-pT jets that are heavily contaminated by underlying event activity, the SoftDrop procedure often terminates at the first stage of declustering such that no particles are removed from the anti-kt jet.

To study the splitting kinematics, we use the flavor- Two single tags (1,1): Here, both subjets are asso- tagged ghosts to label the SoftDrop subjets. Because • ciated with heavy flavor, which usually arises from the ghosts participate in the original anti-kt clustering gluon splitting, g QQ¯. and subsequent SoftDrop declustering, they can become → constituents of the subjets. By counting the number of One double tag (0,2): Double-tagged subjets come • from cascaded splittings such as g gg followed ghosts within each subjet that satisfy ztag > 0.05, we as- by g QQ¯, making their interpretation→ in terms sociate flavor labels and interpretations to the fat jets. In → cases where the ghost particles are removed by the Soft- of splitting functions more complicated. Drop procedure, we simply ignore them when assigning We also rarely find jets with n + n > 2, which are 5 1 2 flavor labels. ignored in the analysis presented below. For simplicity, Specifically, we label our jets as (n1, n2), where n1 is we only treat one flavor label at a time, such that the the number of heavy-flavor hadrons tagged in the first c-tagged categories do not include an explicit veto on b- subjet and n2 the number tagged in the second subjet, tagged objects, and vice versa. In Fig.2, we show some defined such that n1 n2. The cases we focus on are: ≤ example SoftDrop event displays. No tagged subjets (0,0): Labeling light quarks • generically as q, this category comes from g gg, C. Splitting Function Interpretation q qg, and g qq¯. → → → One single tag (0,1): This most likely arises from Because the above flavor-tagged categories are based • a heavy quark emitting a gluon, Q Qg. on identified hadrons, they can be directly implemented → in an experimental analysis. Of course, at the level of idealized partons, there can be category migration if one of the flavor tags is removed by the SoftDrop procedure 5 The dropped ghost tags are interesting in their own right, since or fails the ztag condition, and this has to be accounted they can be used to diagnose counterintuitive kinematical sce- for when interpreting the observed distributions. For ex- narios. The main reason not to consider them for this study is ample, a (0,2) jet with one dropped tag becomes a (0,1) to avoid a proliferation of curves on the following plots. jet. In addition, soft g QQ¯ splittings can contaminate → 5 the flavor labels, though this effect is highly suppressed Here, the mass ratio is by the z condition. Being mindful of migration, it is tag m2 instructive to discuss the expected kinematical distribu- µ2 = Q , (8) tions for each of the flavor-tagged categories. Qg m2 m2 Qg − Q We are specifically interested in the momentum sharing and m is the invariant mass of the heavy quark plus z between the SoftDrop subjets, and adopt a modified Qg g gluon system. Taking the m 0 limit and symmetriz- definition of z compared to the literature Q g ing z 1 z, one recovers→ Eq. (6) as expected. By → − pT1 comparing the (0,1) and (0,0) distributions, it is possible zg , (3) 7 ≡ pT1 + pT2 to test the splitting-function form in Eq. (7). For the (1,1) category with one flavor tag in each sub- where the 1 and 2 subjet labels are derived from the jet, the dominant process is g QQ¯. The quasi-collinear (n1, n2) flavor-tagged label, instead of being ordered by splitting function for this case→ is [58] pT, such that zg [zcut, 1 zcut]. In cases where n1 = n2, 2 2 2 we randomize the∈ ordering− of the subjets resulting in a P ¯ (z) = z + (1 z) + µ ¯ , (9) g→QQ − QQ z distribution that is symmetric about z = 1/2. g g the mass ratio is For the (0,0) case, which has no flavor tags, this is 2 essentially massless QCD with Nf = 3. As shown in 2 2mQ Ref. [12] and experimentally measured by CMS [14] and µQQ¯ = 2 , (10) mQQ¯ STAR [15], the zg distribution is closely related to the massless-QCD splitting kernels. Specifically, for β = 0 8 and mQQ¯ is the invariant mass of the heavy-quark pair. and to lowest order in αs, the probability distribution Note the absence of any singular behavior in the z 0 for zg is given by or z 1 limits, as expected since this process does→ not have→ a soft singularity. P i(zg) Finally, the (0,2) category, where one subjet has two pi(zg) = Θ(z > zcut), (4) R 1/2 0 0 flavor tags, does not have a simple interpretation in terms z dz P i(z ) cut of 1 2 splitting functions. In a parton shower, this where i labels the initiating parton for the jet. Here, configuration→ can be obtained from g gg followed by ¯ → P i(z) are symmetrized versions of the QCD splitting g QQ. More intuitively, one can think of the double- functions for parton i summed over all final state par- tagged→ subjet as being a color-octet configuration that tons, radiates soft gluons via (QQ¯)8 (QQ¯)8g. In this color- octet interpretation, the (0,2) distribution→ is expected to X P i(z) = Pi→jk(z) + Pi→jk(1 z) . (5) look like the (0,1) case with the replacement mQ 2mQ, − → jk since the different Casimir factors do not appear in P (z) at lowest order. It is of particular interest to compare Because we are not distinguishing between quark and the (0,2) category to the quarkonium case studied in gluon (sub)jets in this analysis, the measured p(zg) dis- Sec.IIE. tribution probes a combination of all massless splittings: In addition to zg, the other natural kinematic observ- g gg, q qg, and g qq¯. For N = 3, the sym- f able for SoftDrop jets is Rg, the opening angle between metrized→ splitting→ functions→ for quarks and gluons are the two subjets. For massless partons, the Rg distribu- identical to this order, tion was calculated to next-to-leading-logarithmic accu- 1 z z 1 racy in Ref. [8]. The Rg distribution for massive par- P q(z) P g(z) − + + . (6) tons has not been calculated in the literature, though ' ' z 1 z 2 − Ref. [16] used a variant of Rg to test the dead cone effect for boosted top quarks. We do not show the perturbative Note that P does not include the Casimir factor (Cq = predictions here, since for the jet p range of interest for 4/3 and Cg = 3), which drops out from the p(zg) distri- T LHCb, the R distribution is dominated by nonpertur- butions at lowest order in αs. g For the (0,1) case of one flavor tag, the dominant con- bative physics and is relatively insensitive to the flavor tribution comes from Q Qg. In this case, the zg distribution depends on the quasi-collinear→ splitting function [58], which is not symmetrized over the two subjets:6 7 Pythia implements the heavy-flavor splitting functions using a matrix-element correction [29] instead of the −2µ2 term in 1 z z 2 Qg PQ→Qg(z) = − + 2µQg. (7) Eq. (7). z 2 − 8 In the implementation, the µ2 term is multiplied by Pythia QQ¯ an additional factor of 4z(1 − z)[28]. This explains why the (1, 1) category in Fig.3 exhibits a downturn towards z → 0 and z → 1. As one goes to higher jet pT, this additional factor is less 6 Note that we are using the reversed convention of z versus 1 − z important, and one recovers the expected upturn from Eq. (9); in the splitting function in order to match the definition of zg. see Figs. 13(a) and 13(b). 6

σ(Pythia)[µb] σ(Herwig++)[µb] the important caveat that zg is defined here to be sym- 2 2 metric about zg = 1/2 (instead of stopping at zg = 1/2). (0, 0)c 9.96 × 10 5.28 × 10 1 1 For the (0,1) category, the zg distribution agrees qual- (0, 1)c 7.56 × 10 2.64 × 10 0 0 itatively with the Q Qg splitting function, peaking (1, 1)c 6.87 × 10 2.87 × 10 towards z 0 as expected→ from Eq. (7). The (1,1) cat- 1 0 (0, 2)c 1.00 × 10 5.64 × 10 egory, which→ is largely due to g QQ¯, has no singular −1 −1 otherc 8.86 × 10 2.47 × 10 structures near z 0 or z 1. Figure→ 4 shows that the 3 2 → → (0, 0)b 1.07 × 10 5.52 × 10 analogous distributions obtained using Herwig++ are 1 0 (0, 1)b 1.34 × 10 9.58 × 10 similar to those obtained from Pythia, with some small −1 −1 (1, 1)b 8.40 × 10 5.03 × 10 differences observed near the endpoints. −1 −1 All three categories exhibit distinct behavior that qual- (0, 2)b 9.50 × 10 5.94 × 10 −2 −3 itatively agrees with predictions from perturbative QCD. otherb 1.13 × 10 7.75 × 10 −1 While the 1 2 splitting functions of massive QCD are (0, 1)J/ψ 3.03 × 10 – → −2 well known theoretically, they have never been probed (0, 1)Υ 1.54 × 10 – in such a direct manner experimentally. By exploiting the ability to flavor-tag SoftDrop subjets, our approach provides the opportunity to directly probe the splitting TABLE I. The cross sections for each of the fat-jet flavor- kinematics of Q Qg and g QQ¯. Having confirmed tagged categories determined from Pythia and Herwig++, → → where the total cross section is normalized to the nominal that our parton-shower results agree qualitatively with inelastic cross section of 100 mb. Because we only consider the expected theoretical predictions in Sec.IIC, we look one flavor label at a time, the sum of the c-categories equals forward to tests of these zg distributions in data. the sum of the b-categories. We also show cross sections for quarkonium production in Pythia. E. Results: Quarkonium Production content of the jets. For completeness, we show the R g The SoftDrop jet-declustering strategy is also appli- distributions in App.A. cable to fat jets containing identified quarkonium states. In the analysis below, we treat the jet fragmentation In terms of flavor content, jets with a quarkonium-tagged process as if it were rotationally symmetric about the subjet are similar to the (0,2) category defined above, so jet axis. As recently discussed in Ref. [40], though, it is it is interesting to compare their zg distributions to see interesting to study the angle between the jet produc- whether the underlying physics is similar or not. Tradi- tion plane and the subjet decay plane. For the case of tionally, quarkonium production within a jet is studied g QQ¯, this angle is sensitive to gluon polarization, mo- → using fragmentation functions, which describe the mo- tivating future multi-differential studies of the full Soft- mentum fraction carried by a J/ψ or Υ hadron within a Drop subjet decay phase space. resolved jet. Here, we pursue a complementary approach using zg, which gives the momentum fraction carried by a J/ψ -tagged or Υ-tagged subjet within a SoftDrop fat D. Results: Heavy-Quark Splittings jet. As a preamble, it is worth emphasizing that there are Using this SoftDrop jet-declustering strategy, we first considerable theoretical uncertainties in both the pro- consider the inclusive cross section for each of the flavor- duction and splittings associated with the J/ψ and Υ. tagged categories in TableI. Quantitatively, the predic- A standard approach to study these quarkonium states tions obtained from Pythia and Herwig++ do not is to use nonrelativistic QCD (NRQCD) [59–61], though agree: both the absolute and relative cross sections show there is a long-standing quarkonium-polarization puzzle sizable discrepancies. There is qualitative agreement, associated with this approach; see, for example, Ref. [21] however, as both generators predict that the (0, 0) cate- and references therein. More recently, Refs. [62, 63] used gory with no flavor tags dominates the total rate, followed the method of fragmenting jet functions (FJF) [64] as an by the (0, 1) category which is largely due to Q Qg. alternate method to calculate J/ψ production. Specifi- The (1, 1) and (0, 2) categories, which arise from g → QQ¯ cally, Ref. [63] considered the kinematics of J/ψ produc- and cascaded splittings, respectively, are predicted→ to tion within a resolved jet, finding that their theoretical have similar rates, while events with n1 + n2 > 2 are predictions for the showering, and hence splitting func- rare as expected. tions, associated with the J/ψ disagreed with those im- We next show SoftDrop distributions for zg, as defined plemented in Pythia. In Pythia, a J/ψ produced in in Eq. (3), for both c-tagged and b-tagged fat jets. We fo- the color-octet channel is treated as a loosely bound cc¯ cus on the (0,0), (0,1), and (1,1) categories here, leaving state, and its total showering is the sum of the showers the (0,2) category to Sec.IIE. In Fig.3, we show results originating from either quark. By contrast, in the FJF from Pythia. The (0,0) distribution, which has no flavor approach, a produced J/ψ is showered through DGLAP tags, agrees with those already found in Ref. [12], with evolution of the splitting kernels from 2mc up to the 7

LHCb: pT > 20 GeV, ztag > 0.05, η [3, 4] LHCb: pT > 20 GeV, ztag > 0.05, η [3, 4] 5 ∈ 5 ∈

c-hadron tag (0, 0)c b-hadron tag (0, 0)b

4 Pythia (0, 1)c 4 Pythia (0, 1)b ) ) g R = 1.0 (1, 1) g R = 1.0 (1, 1)

z c z b

d 3 d 3 σ/ σ/

)(d 2 )(d 2 /σ /σ (1 (1 1 1

0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

zg zg (a) (b)

FIG. 3. The SoftDrop zg distributions for the (a) c-tagged and (b) b-tagged categories. Shown here are the results for the (0,0), (0,1), and (1,1) categories obtained from Pythia.

LHCb: pT > 20 GeV, ztag > 0.05, η [3, 4] LHCb: pT > 20 GeV, ztag > 0.05, η [3, 4] 5 ∈ 5 ∈

c-hadron tag (0, 0)c b-hadron tag (0, 0)b

4 Herwig++ (0, 1)c 4 Herwig++ (0, 1)b ) ) g R = 1.0 (1, 1) g R = 1.0 (1, 1)

z c z b

d 3 d 3 σ/ σ/

)(d 2 )(d 2 /σ /σ (1 (1 1 1

0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

zg zg (a) (b)

FIG. 4. Same as Fig.3 but for Herwig++. jet energy scale. There have been other treatments of from heavy-ﬂavor quarks primarily diﬀers from soft-gluon quarkonia showers discussed in the literature; see, for ex- radiation from quarkonium only in the overall color fac- ample, Ref. [65]. Given these theoretical uncertainties, tor. The quarkonium distributions are not included in we expect measurements of zg to help clarify the mech- the equivalent Herwig++ plot shown in Fig.6, since anism of quarkonium production within jets. Further- quarkonium production is not available in the version of more, LHCb recently published a study of prompt J/ψ Herwig++ used in these studies. Since the zg observ- production in jets [66]. Their results are consistent with able is insensitive to Casimir factors at lowest order, the the predictions of the FJF-based approach, and in stark distributions in Fig.5 are all similar. Given the calcula- disagreement with Pythia, providing additional motiva- tion in Ref. [63], and the LHCb results in Ref. [66], it will tion to measure zg for quarkonium production. be fascinating if this prediction is borne out in data.

In TableI we give the predicted rates for J/ψ -tagged and Υ-tagged jet production. These rates are more than an order of magnitude smaller than those of the III. FLAVORCONE JETS TO DISENTANGLE double-flavor-tagged (0, 2) category with the same quark HEAVY-FLAVOR PRODUCTION flavor. In Fig.5, we compare the zg distributions for quarkonium-tagged jets to the (0, 1) and (0, 2) cate- We now transition from studying heavy-flavor within gories. In the context of Pythia, soft-gluon radiation a single jet to heavy-flavor production in the event as a 8

LHCb: pT > 20 GeV, ztag > 0.05, η [3, 4] LHCb: pT > 20 GeV, ztag > 0.05, η [3, 4] 7 ∈ 7 ∈ c-hadron tag (0, 1) b-hadron tag (0, 1) 6 c 6 b Pythia (0, 2)c Pythia (0, 2)b

) 5 ) 5 g R = 1.0 g R = 1.0 (0, 1) z (0, 1)J/ψ z Υ d 4 d 4 σ/ σ/

)(d 3 )(d 3 /σ /σ

(1 2 (1 2

1 1

0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

zg zg (a) (b)

FIG. 5. Same as Fig.3 for Pythia, but comparing quarkonium-tagged jets to ﬂavor-tagged jets in the (0,1) and (0,2) categories. Note that the solid line for the (0,1) category matches Fig.3.

LHCb: pT > 20 GeV, ztag > 0.05, η [3, 4] LHCb: pT > 20 GeV, ztag > 0.05, η [3, 4] 7 ∈ 7 ∈ c-hadron tag (0, 1) b-hadron tag (0, 1) 6 c 6 b Herwig++ (0, 2)c Herwig++ (0, 2)b

) 5 ) 5 g R = 1.0 g R = 1.0 z z d 4 d 4 σ/ σ/

)(d 3 )(d 3 /σ /σ

(1 2 (1 2

1 1

0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

zg zg (a) (b)

FIG. 6. Same as Fig.5 but for Herwig++. Quarkonium production is not available in Herwig++. whole. There are multiple production channels for heavy A. Traditional Approach to Heavy-Flavor Rates flavor in QCD, which leads to various complications in predicting heavy-flavor rates. Typically, one considers the three main production mechanisms shown in Fig.1, At the LHC, it is well-known that the gluon-splitting with the caveats that the αs order at which these di- and flavor-excitation processes can dominate the total agrams appear depends on the PDF scheme employed, rate over flavor creation; see, for example, Refs. [67–69]. and that at higher orders there is no gauge-invariant def- These two dominant processes are challenging to calcu- inition of these categories. Still, making a gluon-splitting late since, as emphasized by their 2 3 representations → versus non-gluon-splitting distinction provides helpful in- in Figs. 1(a) and 1(b), they involve multiple emission tuition about the relevant phase-space regions populated scales. The cross sections for these channels have been by these diagrams. The main challenge of using tradi- calculated at NLO in perturbative QCD; see, for exam- tional jet algorithms in the gluon-splitting regime is jet ple, Refs. [70–78] and references therein. Fixed-order re- merging. In this section, we first review the jet-merging sults have been interfaced with parton-shower programs issue, and then introduce our FlavorCone jet algorithm to provide predicted rates for the LHC [79–81]. Pertur- designed to resolve overlapping heavy-flavor jets. bative QCD predictions for heavy-flavor production have been tested experimentally for b quarks—at the Tevatron [82–85], and at the LHC by ATLAS [86–91], CMS [92– 97], and LHCb [98]—and for a combination of flavors at 9

ATLAS [68]. Many of these studies have noted a tension By construction, the FlavorCone algorithm does not between the theoretical predictions and the experimental include a merging step, so there are always exactly n data, especially in the gluon-splitting regime. jets in the event, regardless of whether the n flavor- The phase-space region where the disagreement is tagged hadrons are widely separated or almost collinear. largest is also where analysis strategies based on tra- In the well-separated regime, the resulting jets are per- ditional jet algorithms break down. In gluon-splitting fect cones centered on the flight directions of the flavor- events, the heavy quarks tend to have small angular sep- tagged hadrons, yielding results that are similar to anti- aration due to the gluon propagator. Standard jet algo- kt. Crucially, there is no limitation on the jet axes getting rithms, such as anti-kt, have difficulty in this situation, closer than R, and abutting jet regions are determined since events where two heavy quarks fall into a single jet by nearest-neighbor partitioning. Of course, this feature are typically cut from the analysis.9 This limitation is relies fundamentally on the ability of experiments like unavoidable for anti-kt, since the jet axes cannot get any LHCb to accurately reconstruct the flight direction of closer than the jet radius R, usually taken to be R = 0.4 heavy-flavor hadrons, even when the hadrons are almost or 0.5.10 Despite this, almost all of the LHC analyses ref- collinear, as discussed further in Sec.IV. erenced above use anti-kt, and suffer the associated loss The partitioning scheme used for FlavorCone is mo- of performance. tivated by the XCone jet algorithm [24, 25], which is It is worth mentioning an alternative strategy for also designed to yield a fixed, predetermined number of studying heavy-flavor production based on flavored jet al- jets. Beyond just the simplicity of the FlavorCone algorithms [80, 100]. These algorithms, which work equally gorithm, its main advantage over XCone is that flavor- well on heavy-flavor quarks or hadrons, attempt to neu- tagged hadrons always end up in separate jets, whereas tralize gluon-splitting topologies through a recursive clus- XCone can still allow merging. The XCone algorithm is tering strategy. In this way, events with g QQ¯ are not infrared and collinear (IRC) safe, since it starts from a even categorized as heavy-flavor production,→ giving com- set of IRC-safe seed-jet axes and then applies an iterative plementary information to FlavorCone jets. procedure to find the jet regions that (locally) minimize an N-jettiness measure [101]. The FlavorCone algorithm is also IRC safe, since additional soft particles or collinear B. A New Approach: FlavorCone splittings cannot impact the flight direction of a flavor- 12 tagged hadron, to the extent that mb,c ΛQCD. We now introduce a simple jet algorithm aimed at re- In general, the FlavorCone jet axis and the FlavorCone constructing the gluon-splitting phase space. In an event jet momentum are not aligned. In this respect, the Fla- with n heavy-flavor-tagged hadrons, we define n Flavor- vorCone axes are similar in spirit to the winner-take-all Cone jets of radius R as follows. axes [102–104], especially since flavor-tagged hadrons often carry a large fraction of the jet momentum in the 1. The flight direction of each flavor-tagged hadron non-gluon-splitting regime. If desired, one could apply defines a separate jet axis. an iterative procedure to find n mutually stable cones 2. Any particle that is further away than R from a jet using the n flavor-tagged-hadron directions as seed axes. axis is left unclustered. As mentioned in Sec.IIIE, we find better performance by simply using flavor-tagged-hadron axes directly, since 3. Each remaining particle is clustered to the nearest this avoids the issue of abutting jets repelling (or merg- jet axis. ing into) each other after iteration, which tends to wash 4. The momentum of each jet is then determined by out the gluon-splitting phase space. the summed momenta of its constituents.

As is appropriate for pp collisions, we use the rapidity- C. Event Selection and Classiﬁcation azimuth distance ∆R = p(∆y)2 + (∆φ)2 to determine the unclustered region and the nearest jet axis.11 Exam- Using this FlavorCone algorithm, we now outline ple events clustered with the FlavorCone algorithm are an analysis strategy to disentangle the mechanisms for shown in Fig.7.

9 Alternatively, one could separately perform a subjet analysis on 12 Similar to XCone, FlavorCone can be sensitive to the IRC regime, anti-kt jets with multiple flavor tags [2–5]. This strategy, how- since the algorithm will identify n jets even if the n flavor-tagged ever, does not result in a smooth transition between the gluon- hadrons of interest have very low pT . For this reason, one may splitting and non-gluon-splitting regimes. wish to impose additional requirements on the FlavorCone jets 10 √Other cone-like algorithms can even lead to jet merging below to ensure that one is in the perturbative regime. Here, we set a 2R or 2R; see discussion in Ref. [99]. minimum pT on the flavor tag and then place an additional cut 11 In an experimental analysis, it may be preferable to use pseu- on the overall jet pT, which introduces some mild dependence dorapidity η instead of rapidity y to avoid complications of as- on the b-quark fragmentation function. Alternatively, one could signing masses to reconstructed particles. See Sec.IV for further place a cut on the relative pT of the tag and the jet, in the same discussion. spirit as the ztag condition in Eq. (2). 10

� � � � �� -�� -�� = �� -�� = �� -�� = �� -�� = �� -�� -�� -�� -�� -��

� � � �

� � � � ϕ ϕ ϕ ϕ

-� -� -� -�

-� -� -� -� �� (a) (b) (c) (d)

FIG. 7. Event displays using the FlavorCone algorithm, for (a) a gluon-splitting cc¯ event, (b) a non-splitting cc¯ event, (c) a gluon-splitting b¯b event, and (d) a non-splitting b¯b event. The ﬂavor-tagged ghost particles are shown as small open circles, with blue for charm and red for beauty, and the FlavorCone jet boundaries are shown in the corresponding color. The shaded gray regions show the two hardest ﬂavor-tagged anti-kt jets for comparison. Here we have selected gluon-splitting events where the two anti-kt jets remain resolved, although we emphasize that in many such events the anti-kt jets in fact merge.

heavy-flavor production. Our focus here is on the LHCb This cut avoids highly asymmetric events that are experiment, though we show related distributions for AT- more difficult to predict from perturbation theory. LAS and CMS in App.C. Our analysis workflow is as We perform this analysis separately for b¯b and cc¯ final follows. states. In principle, one could also look at mixed bc We select events that have at least two heavy-flavor events, but we do not pursue that here. • hadrons with η [2.5, 4.5], so that the full Fla- For comparison, we also consider events clustered using ∈ vorCone (with R = 0.5) is within LHCb accep- anti-kt with R = 0.5, where flavor tagging is performed tance. For concreteness in the plots below, we re- by treating the flight directions of the hardest two heavy- quire these hadrons to have pT > 2 GeV, though flavor hadrons as ghost particles in the clustering. The the specific selection criteria would depend on the two resulting anti-kt flavor-tagged jets are then treated implementation details; see Sec.IV. in the same way as the FlavorCone jets, with the same cut on the leading jet pT and subleading zsub. In keeping These heavy-flavor hadrons are ordered in pT, and • with the traditional strategy, anti-kt jets that contain two the two hardest ones are used to form FlavorCone flavor-tagged hadrons are rejected from the analysis. 13 jets with R = 0.5. When using the Pythia parton shower, it is possible to classify generated events as being either splitting or non- The leading FlavorCone jet is required to have pT > • splitting using the event record. If the two flavor-tagged 20 GeV, though larger pT thresholds would likely be needed at ATLAS or CMS. hadrons can be traced back to a common gluon parent from the shower, then the event is labelled as splitting. The subleading FlavorCone jet is required to have Every other event is classified as non-splitting, which in- • zsub > 0.1, where we define cludes prompt production via flavor creation and excitation, as well as gluon-splitting-like events where the two sub pT heavy-flavor hadrons do not come from a common gluon zsub = lead sub . (11) 14 pT + pT parent from the shower. Of course, this splitting/non- splitting distinction is not physical, as it cannot be implemented experimentally and is ambiguous beyond the 13 Although reconstructing the full four-momenta of the hadrons is challenging at the LHC, ordering them by pT is more feasible. That said, events with more than two reconstructed flavor-tagged 14 We also tested alternative classification schemes based on the hadrons are expected to be rare; see Sec.IV. hard-production vertex. We found cases, however, where the 11

cc¯ b¯b σ(cc¯)[µb] σ(b¯b)[µb]

FlavorCone Anti-kt FlavorCone Anti-kt Pythia 2.02 0.97 ∆y 0.09 0.03 0.05 0.02 Vincia 1.40 0.59 ∆φ 0.41 0.30 0.31 0.24 Dire 2.55 0.93 ∆R 0.43 0.31 0.32 0.25 PowhegBox 1.27 0.55

mjj 0.42 0.30 0.29 0.22

TABLE III. The cross section in the LHCb fiducial region, ¯ TABLE II. The discrimination power between splitting and defined with the nominal FlavorCone algorithm, for cc¯ and bb non-splitting events in Pythia for cc¯ and b¯b. For each dijet production from four different predictions. Note that there observable, the values shown correspond to classifier sepa- is a greater disagreement in these rates than there is in the ration δλ from Eq. (12), where larger values indicate better differential shapes in Fig.9. performance. Regardless of the choice of discriminant, the FlavorCone approach outperforms the traditional anti-kt approach. in separating the splitting and non-splitting categories. These observables are, of course, strongly correlated, and we checked that combining them in a multivariate anal- strongly-ordered parton-shower limit. Nonetheless, it al- ysis provides little improvement. TableII clearly shows lows us to isolate the way that Pythia treats the gluon- that the FlavorCone approach outperforms anti-k , yield- splitting regime and test whether the FlavorCone algo- t ing a 30–40% gain in performance as measured by δ . rithm is sufficiently sensitive to the underlying physics. λ This is largely due to heavy-flavor merging by the anti- 16 kt algorithm. D. Results: Clear Separation of Gluon Splitting To highlight the separation power achievable using the FlavorCone algorithm, we show the full distribution for ∆φ in Fig.8, with the other three observables given in The strategy above selects events with two flavor- ¯ tagged jets. To probe the physics of heavy-flavor pro- App.B. We present both bb and cc¯ events, broken down duction, we consider the cross section differential in four into the splitting and non-splitting categories. In or- dijet observables: ∆y, ∆φ, ∆R, each determined using der to isolate the phase-space region dominated by gluon < the jet axes; and the invariant mass of the dijet system splitting, one can select events with ∆φ 1. Note that the ∆φ distribution smoothly approaches∼ zero, with no mjj. For anti-kt, the jet axis and momentum are aligned, whereas for FlavorCone, the jet axis is aligned with the change of behavior as the angle approaches the jet ra- flight direction of the flavor-tagged hadron. dius R = 0.5. It is in this region that the FlavorCone algorithm performs particularly well, while traditional jet In order to determine the potential separation power algorithms result in jet merging (see Sec.IIIE). between splitting and non-splitting events, we first calculate the classifier separation as implemented in In addition to Pythia, we apply this analysis TMVA[105].15 Given two event categories A and B, and to three alternative predictions—Vincia, Dire, and probability distributions pA(λ) and pB(λ) for the observ- PowhegBox—all interfaced to Pythia for hadroniza- able λ, the classifier separation δλ is defined as tion. Already from the total cross sections in TableIII, one can see substantial differences between the genera- Z 2 1 (pA(λ) pB(λ)) tors, but the origin of the disagreement is not clear from δλ = dλ − . (12) 2 pA(λ) + pB(λ) the rates alone. In the normalized distributions in Fig.9, we can see in more detail the different predictions for the The value of δλ always lies within [0, 1], where δλ = 0 splitting-enriched and non-splitting-enriched regions of means that λ has no discriminating power and δλ = 1 phase space. While there is qualitative agreement about indicates ideal separation. the peaks at ∆φ = 0 and π, quantitatively the predictions In TableII, we show the δ values for each of the dijet are sufficiently different that direct comparison to LHC observables as computed from Pythia. As expected, ∆y data should result in improved modeling of heavy-flavor is not a good discriminant, since back-to-back jets from production in parton-shower generators. flavor creation can also have a small rapidity separation. The remaining three observables show good performance

16 In the spirit of footnote9, one could consider double-tagged anti- hard production is labeled as flavor creation or excitation, but the kt jets as a separate event category to be included in the calcu- actual heavy-flavor hadrons within the LHCb acceptance come lation of δλ. Depending on the precise double-tagged selection from a subsequent gluon splitting. criteria one uses, the resulting performance can be comparable 15 An alternative classifier metric is mutual information [106], which to FlavorCone. Alternatively, one could use a smaller jet radius is closely related to classifier separation [107]. to reduce the impact of jet merging. 12

lead lead LHCb: pT > 20 GeV, zsub > 0.1, η [2.5, 4.5] LHCb: pT > 20 GeV, zsub > 0.1, η [2.5, 4.5] 1.4 ∈ 1.4 ∈ c-hadron tag b-hadron tag 1.2 1.2 Pythia Pythia ) 1.0 ) 1.0 φ R = 0.5 φ R = 0.5

d ∆ 0.8 gluon splitting d ∆ 0.8 gluon splitting

σ/ non-splitting σ/ non-splitting 0.6 0.6 )(d total )(d total /σ 0.4 /σ 0.4 (1 (1

0.2 0.2

0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 ∆φ ∆φ (a) (b)

FIG. 8. The azimuthal separation between the two FlavorCone axes seeded by (a) c-tagged and (b) b-tagged hadrons. The normalized distributions are shown for Pythia, which allows a useful (but ambiguous) categorization into splitting and non- splitting events. Low values of ∆φ probe the phase-space region dominated by gluon splitting. The FlavorCone algorithm is ideally suited to study this region, with no anomalous features at the jet radius R = 0.5.

lead lead LHCb: pT > 20 GeV, zsub > 0.1, η [2.5, 4.5] LHCb: pT > 20 GeV, zsub > 0.1, η [2.5, 4.5] 1.4 ∈ 1.4 ∈ c-hadron tag Pythia Pythia b-hadron tag 1.2 1.2 R = 0.5 Dire Dire R = 0.5 ) 1.0 ) 1.0 φ Vincia φ Vincia

d ∆ 0.8 PowhegBox d ∆ 0.8 PowhegBox σ/ σ/ 0.6 0.6 )(d )(d

/σ 0.4 /σ 0.4 (1 (1

0.2 0.2

0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 ∆φ ∆φ (a) (b)

FIG. 9. Same observable and event selection as Fig.8, but now comparing four different predictions. The qualitative behavior is similar between the generators, but quantitatively they differ at a level that should be testable at the LHC. These are normalized distributions; see TableII for the differences in the absolute cross section.

E. Alternative Approaches 3. Anti-kt: The default anti-kt approach using E- scheme recombination, where the jet axes and jet To better understand the behavior of FlavorCone jets, momenta align. it is instructive to make a comparison to alternative re- 4. Q-hadron: Omitting any jet clustering and deter- construction strategies. In Fig. 10, we compare the ∆φ mining ∆φ from the flavor-tagged hadrons alone. distributions obtained using the following four methods. Here, we require that the heavy-flavor hadrons have 1. FlavorCone: The default FlavorCone approach, pT > 16 GeV to roughly select the same phase- where ∆φ is determined from the jet axes (which space regime as the jet-based approaches. are aligned with the flavor-tagged hadrons). We also tested an alternative anti-kt approach where ∆φ 2. FlavorConeJet: An alternative FlavorCone ap- is determined from the flavor-tagged ghosts, but that proach, where ∆φ is determined by the jet mo- has nearly identical behavior as the anti-kt option tested menta (i.e. the vector sum of the constituent mo- above. menta of each jet). All four approaches result in similar behavior at large 13

lead lead LHCb: pT > 20 GeV, zsub > 0.1, η [2.5, 4.5] LHCb: pT > 20 GeV, zsub > 0.1, η [2.5, 4.5] 1.4 ∈ 1.4 ∈ c-hadron tag FlavorCone FlavorCone b-hadron tag 1.2 1.2 Pythia FlavorConeJet FlavorConeJet Pythia ) 1.0 ) 1.0 φ R = 0.5 Anti-kt φ Anti-kt R = 0.5

d ∆ Q-hadron 0.9 d ∆ Q-hadron 0.3 0.8 × 0.8 × σ/ σ/ 0.6 0.6 )(d )(d

/σ 0.4 /σ 0.4 (1 (1

0.2 0.2

0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 ∆φ ∆φ (a) (b)

FIG. 10. Same event selection as Fig.8, but now comparing four different definitions for the ∆ φ observable. Here, the distributions are normalized to the FlavorCone approach in order to highlight the loss in efficiency for the anti-kt method below ∆φ ' R. For Q-hadron, we put an additional offset factor to partially account for the difference in pT acceptance for hadrons versus jets. For FlavorConeJet, the feature at ∆φ ' 0.3 is due to the misalignment of the jet axis and jet momentum for abutting jets.

∆φ, but significant differences are clearly visible in the various exclusive-generalized-kt strategies [108–110], but gluon-splitting regime. The power of the FlavorConeJet none worked as well as the FlavorCone algorithm.18 method degrades at small ∆φ, because the jet momenta While exclusive kt does allow the jet axes to become arbi- recoil against each other as the cones begin to overlap. trarily close in principle, in practice there is still consid- The anti-kt algorithm suffers a precipitous drop in effi- erable heavy-flavor merging in the gluon-splitting regime. ciency in the gluon-splitting regime due to jet merging. Considering all of these results, we advocate the Flavor- Of course, one could reduce the impact of jet merging Cone approach as being best suited to studying heavy- by using a smaller jet radius, at the expense of intro- flavor production in the gluon-splitting regime. ducing more out-of-jet radiation. Finally, because the Q-hadron and FlavorCone distributions are based on the same flavor-tagged hadrons, it is not surprising that they IV. IMPLEMENTATION AT LHCB exhibit the same qualitative features. Without any jet reconstruction, however, the Q-hadron method requires re- The parton-shower studies presented above are very constructing the full pT of the heavy-flavor hadrons, and not just their flight directions. Experimentally, this re- encouraging, but they ignore the realities of detector- quirement inherently leads to a much lower flavor-tagging response effects. Efficient reconstruction of flavor-tagged efficiency. hadrons is the most important ingredient needed to carry out these analyses, and tagging multiple heavy-flavor Beyond Fig. 10, we also considered two additional hadrons in close proximity is a challenge. Furthermore, methods, but neither is as powerful as FlavorCone. In both the SoftDrop and FlavorCone analyses require the the spirit of stable cone algorithms, we studied an iter- flight directions of the flavor-tagged hadrons to be well ated FlavorCone, where the two flavor-tagged hadrons measured. Here, we briefly sketch a possible implemen- provide seed axes that are iteratively adjusted until they tation of these methods at the LHCb detector, which has align with the jet momenta.17 This approach gave similar excellent heavy-flavor reconstruction capabilities. results to the nominal FlavorCone, but occasionally the iteration procedure caused the two heavy-flavor hadrons to merge into the same jet, leading to a loss of performance in the gluon-splitting regime. We also tried 18 Like XCone, recursive exclusive jet algorithms also ensure a fixed number of jets in the final state. In the generalized kt algorithm, 2q the merging and dropping distance measures scale as pT . We considered q = 0.5, 1.0, and 2.0 in exclusive mode with R = 17 Specifically, we start from the original FlavorCone jets. In each 0.5, where the algorithm terminates when there are exactly two iteration step, we define new jet axes based on the jet momenta, undropped jets in the final state. We tried using the resulting and repartition the hadrons to the closest jet axis within the jet exclusive kt jets directly in the analysis, as well as using them radius R. This process is guaranteed to terminate in a finite as jet axes for cone finding and as seeds for iterative stable cone number of steps [24]. finding. 14

We start with the case of c-hadron tagging. Charm hardest is. This correction can be derived from data us- 0 + + quarks primarily hadronize into a D , D , or Ds me- ing events with (at least) three c-hadron tags. + son, a Λc baryon, or any of their antiparticles. Each of We now turn to the case of b-hadron tagging. Un- these charm hadrons has at least one all-charged-particle fortunately, there are no all-charged-particle b-hadron decay with a sizable branching fraction.19 Therefore, a decays with percent-level branching fractions, making potential strategy at LHCb is to fully reconstruct one c- exclusive reconstruction of b-hadrons inefficient. Inclu- hadron, which would provide excellent momentum reso- sive secondary-vertex-based b-hadron tagging, however, 21 lution of σ(pT)/pT 1% and σ(φ) 2 mrad (see App. A is highly efficient. Backgrounds from c-hadron decays of Ref. [111]). Combining≈ this exclusive≈ tag with the ex- can be highly suppressed by requiring the reconstructed cellent vertex resolution at LHCb permits precise deter- secondary-vertex mass to be greater than the c-hadron mination of the c-hadron impact parameter, making it masses; a similar strategy was employed by CMS in possible to distinguish prompt-charm production from Ref. [92]. The flight direction is measured sufficiently charm produced in b c decays. With full reconstruc- well at LHCb, σ(∆φb¯b) 50 mrad, to employ this in- tion, one could choose→ to use the reconstructed charm clusive strategy in both≈ the SoftDrop and FlavorCone hadron directly in the jet finding, instead of using it only analyses. Events in the Pythia data sample with more as a ghost particle for tagging; this would mitigate track than two b-tagged hadrons are rare enough that improper sharing between (sub)jets. pT-ordering of b-tagged hadrons will have negligible im- In principle, LHCb could also fully reconstruct the pact on the FlavorCone analysis. second c-tagged hadron, but the efficiency of perform- For the relatively low-pT jets studied at LHCb, the ing exclusive reconstruction of both c-hadrons will be distinction between rapidity y and pseudorapidity η is low. For charm tagging (as opposed to exclusive charm non-negligible. In the parton-shower studies above, we reconstruction), LHCb developed a c-jet tagging algo- assumed access to full four-vector information for all par- rithm in Run 1 that is largely based on properties of ticles, including the ghost tags. At LHCb, it is much the c-hadron. This algorithm achieves a c-tagging effi- easier to determine η, though one can reliably estimate y ciency of 20–25%, while also providing excellent c–b dis- since LHCb has good particle identification to infer the crimination [112]. Removing the two (out of ten) fea- appropriate hadron mass value. In practice, we expect tures that depend on properties of the jet (as opposed the distinction between y and η could be implemented as to the c-hadron) used by the LHCb machine-learning- a simple unfolding correction. At worst, one could use a based c-jet-tagging algorithm should make this tagger FlavorCone variant based only on η, and we checked that suitable for use in both the SoftDrop and FlavorCone this does not have a large impact on performance. analyses. That said, most of the discriminating power for Finally, we note that the SoftDrop jet-declustering c-jet tagging comes from a single feature, the corrected analysis can be performed on fat jets clustered using only mass, so a simple analysis based on secondary vertices charged particles. Given that charged-particle recon- is likely sufficient.20 Either way, the expected resolu- struction at LHCb is far superior to that of neutral par- tion on σ(∆φcc¯) is (10 mrad) [111], which is more than ticles, a charged-only strategy may be desirable. From a sufficient for both SoftDropO and FlavorCone, given that theoretical perspective, charged-only distributions can be none of the plots shown above resolve features finer than treated using generalized fragmentation functions called ∆φmin 50 mrad. track functions [113, 114], appropriately adapted to treat ' The only remaining issue is that of pT ordering of the heavy-flavor fragmentation. From an experimental tag- charm hadrons in FlavorCone, since only the two hard- ging perspective, inclusive flavor-tagging of hadrons is est tags are used in our analysis. Only 7% of events in already largely based on charged particle information, so our FlavorCone analysis of Pythia data contain more there is relatively little loss of information in only using than two c-tagged hadrons with pT > 2 GeV, which charged particles for fat jet reconstruction. Note that means that cc¯ events with three (or more) reconstructed while zg itself is a dimensionless observable that is rather c-hadron tags will be rare. Therefore, pT ordering in Fla- insensitive to the charged/neutral ratio, SoftDrop de- vorCone will be less important than the small correction pends on an angular-ordered clustering tree, which ben- required to account for the case where one of the two efits from the improved angular resolution provided by hardest c-hadrons is not reconstructed, but the third- charged particles. Either way, comparing the zg distributions obtained using all particles versus only charged ones will provide a valuable cross check.22

19 For example, the folowing decays can each be cleanly and ef- ficiently reconstructed at LHCb: B(D0 → K−π+) ≈ 4%, 21 Another potential strategy is to reconstruct one b-tag using a dis- + − + + + − + + + − B(D → K π π ) ≈ 9%, B(Ds → K K π ) ≈ 5%, and placed J/ψ → µ µ decay, and the other one using an inclusive + − + B(Λc →pK π ) ≈ 6%. secondary-vertex b-tag. 20 The corrected mass takes the reconstructed secondary-vertex 22 As shown in Ref. [40], there can be significant event-by-event mass msv, and adds a correction based on the momentum trans- fluctuations in the momentum fraction z when going from all p 2 2 verse to the direction of flight msv + (psv sin δθ) +|psv sin δθ|, particles to charged particles. But these fluctuations often have where δθ is the angle between ~psv and the flight vector. a mild impact on the final distribution, see Ref. [114]. 15

V. CONCLUSION cision QCD calculations of heavy-flavor production, and lead to improved parton-shower modeling of heavy-flavor In this article, we outlined two analysis strategies de- physics. signed to gain a better understanding of the origin and kinematics of heavy flavor at colliders. First, we showed how SoftDrop jet declustering of a fat jet can expose VI. ACKNOWLEDGEMENTS the well-known but as-of-yet-unmeasured splitting kernels for heavy-flavor quarks and quarkonia in QCD. Sec- We thank Vava Gligorov, Ben Nachman, Frank ond, we showed how the FlavorCone jet algorithm can Petriello, Ira Rothstein, Gavin Salam, and Peter Skands help separate the gluon-splitting and non-gluon-splitting for helpful discussions. Feynman diagrams were drawn heavy-flavor production mechanisms. Our parton-shower using Ref. [115]. The work of NLR and JT is sup- studies were focused on the LHCb experiment because ported by the U.S. Department of Energy (DOE) under of its superior ability to identify and reconstruct heavy- grant contract numbers DE-SC-00012567 and DE-SC- flavor hadrons, offering the best short-term prospects for 00015476. NLR is also supported in part by the Ameri- carrying out these measurements. We also expect that can Australian Association’s ConocoPhillips Fellowship. similar techniques can be pursued by ATLAS and CMS, PI and MW are supported by the U.S. National Science especially in the higher-pT jet range. Foundation grant PHY-1607225. A key theme from this study is the value of tagging heavy-flavor hadrons, as opposed to the more traditional strategy of tagging heavy-flavor jets. This theme has al- Appendix A: Additional SoftDrop Distributions ready been emphasized in jet substructure studies based on subjet flavor-tagging [4,5], which use ghost associa- The two natural observables to describe SoftDrop sub- tion in a similar way as our (n1, n2) flavor-categorization jets are the momentum sharing zg, defined in Eq. (3), and scheme. The FlavorCone algorithm goes one step fur- the opening angle Rg. In Sec.II, we showed the zg distri- ther, using flavor-tagged hadrons to define jets, in con- butions for our various tagging categories in Figs.3 and trast to the typical approach of first finding jets and then 5. In this appendix, we show the analogous distributions flavor-tagging them. Experimentally, tagging hadrons is for Rg in Fig. 11. far more challenging than tagging jets, but we think the For the case of β = 0, the differential Rg cross sec- physics opportunities justify investing in the development tion was first calculated in Ref. [10], where it was shown of hadron-based tagging strategies. Theoretically, it will that the Rg distribution exhibits single-logarithmic be- be interesting to study the behavior of fixed-order QCD havior for high-pT jets. For our LHCb study, however, calculations when using these heavy-flavor categories. we are working with rather low pT jets with pT > 20 GeV, Beyond just the excellent tagging performance of where the distributions are largely controlled by nonper- LHCb, the heavy-flavor strategies presented here are mo- turbative effects. Indeed, the SoftDrop algorithm often tivated by the need for a more robust theoretical defi- terminates at the first stage of declustering, such that nition of heavy flavor. In many collider-based studies, Rg R, as reflected by peaks shown in Fig. 11. Given ' a jet with g QQ¯ will be tagged as a heavy-flavor these Rg distributions, it is in some sense surprising that jet, yet the physics→ of g QQ¯ is very different from the zg distributions in Figs.3 and5 show no indication that of Q Qg, with different→ underlying production for large nonperturbative corrections. mechanisms→ and different final-state kinematics. A similar point was emphasized in the context of flavored jet algorithms [80, 100]. By individually identifying heavy- Appendix B: Additional FlavorCone Distributions flavor hadrons, one can more easily separate double-tag versus single-tag (sub)jets, mitigating the confusions that In Sec.III, we demonstrated the separation achievable arise from gluon splitting to heavy flavor. We expect that between splitting and non-splitting events in Pythia us- the SoftDrop zg distributions will be helpful in validating ing the FlavorCone algorithm. The ∆φ distribution was new flavor-tagging methods, and we hope that this study shown in Fig.8, and for completeness we show the ∆ y, inspires more sophisticated heavy-flavor categorization. ∆R, and mjj observables in Fig. 12. As demonstrated Finally, heavy-flavor production is important for in TableIII, ∆ R and mjj are effective discriminants be- stress-testing event generators. Even though all of the tween gluon splitting and non-splitting events, similar to generators tested here are based (in principle) on the ∆φ, though all three variables are highly correlated. By same underlying QCD splitting kernels, the differences contrast, ∆y is not an effective discriminant because of in their distributions are substantial, especially in the the prevalence of back-to-back dijets from flavor creation. gluon-splitting regime. It would be particularly interesting to see how heavy-flavor production evolves as a Appendix C: Implementation at ATLAS and CMS function of pT, since the relative importance of each contribution varies as function of jet kinematics. The measurements proposed here should inspire further pre- While the focus of our study has been on the LHCb 16

LHCb: p > 20 GeV, z > 0.05, η [3, 4] LHCb: p > 20 GeV, z > 0.05, η [3, 4] T tag ∈ T tag ∈

5 (0, 0)c c-hadron tag 5 (0, 0)b b-hadron tag

(0, 1)c Pythia (0, 1)b Pythia

) 4 ) 4 g (1, 1)c R = 1.0 g (1, 1)b R = 1.0 R R d d 3 3 σ/ σ/ )(d )(d 2 2 /σ /σ (1 (1 1 1

0 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Rg Rg (a) (b)

LHCb: p > 20 GeV, z > 0.05, η [3, 4] LHCb: p > 20 GeV, z > 0.05, η [3, 4] T tag ∈ T tag ∈

5 (0, 1)c c-hadron tag 5 (0, 1)b b-hadron tag

(0, 2)c Pythia (0, 2)b Pythia

) 4 ) 4 g g (0, 1)J/ψ R = 1.0 (0, 1)Υ R = 1.0 R R d d 3 3 σ/ σ/ )(d )(d 2 2 /σ /σ (1 (1 1 1

0 0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Rg Rg (c) (d)

FIG. 11. The distribution for SoftDrop Rg in (left) c-tagged and (right) b-tagged events. The top row (a,b) shows the same ﬂavor categories as Fig.3, and the bottom row (c,d) shows the same ﬂavor categories as Fig.5.

experiment, the SoftDrop and FlavorCone strategies can the pT threshold for identifying heavy-flavor hadrons to also be applied at ATLAS and CMS, which we refer to be 20 GeV (instead of 2 GeV) to account for the difficulty as general purpose detectors (GPDs). The primary ex- of GPDs to resolve low pT tracks. perimental challenge at GPDs is achieving the required heavy-flavor-hadron tagging efficiencies, as well as asso- The SoftDrop zg results for the GPD workflow are shown in Fig. 13, to be compared to the LHCb results in ciated pT and flight-direction measurements. As emphasized in Sec.IV, it is known how these challenges can be Figs.3 and5. As one goes to the higher- pT regime of the overcome at LHCb in the near term. Nevertheless, CMS GPDs, there are a number of important differences. At has already probed small angular separations between the perturbative level, one expects relatively little change b-hadrons in Ref. [92], so in the future, we expect the in the zg distributions as a function of pT [12]. That said, stringent heavy-flavor-tagging requirements for SoftDrop the nonperturbative effect of underlying event is very and FlavorCone can be met at a GPD. relevant at low jet pT values, where it tends to make zg more flatly distributed. Thus, going to higher pT in To show the expected performance of our methods at a Fig. 13, the zg distributions exhibit stronger singularities GPD, we repeat the main results from our parton-shower towards z 0 and z 1 as expected. Another impor- studies, albeit with three changes to the analysis work- tant difference→ is related→ to category migration. Because flow. First, we require the entire jet to be contained we are taking the ztag requirement to be half of the zcut within the GPD acceptance of η [ 2.5, 2.5] (instead requirement, there is phase space for subsequent g QQ¯ ∈ − → of η [2, 5]). Second, we increase the pT threshold to emissions to cause noticeable migration from the (0, 0) to 200 GeV∈ (instead of 20 GeV) to account for the typical the (0, 1) category. This was not as much of an issue at jet scales used for triggering at a GPD. Third, we change low jet pT, since the phase space for additional gluon ra- 17

lead lead LHCb: pT > 20 GeV, zsub > 0.1, η [2.5, 4.5] LHCb: pT > 20 GeV, zsub > 0.1, η [2.5, 4.5] 1.0 ∈ 1.0 ∈ gluon splitting c-hadron tag gluon splitting b-hadron tag 0.8 non-splitting Pythia 0.8 non-splitting Pythia ) )

y total R = 0.5 y total R = 0.5

d ∆ 0.6 d ∆ 0.6 σ/ σ/

)(d 0.4 )(d 0.4 /σ /σ (1 (1 0.2 0.2

0.0 0.0 3 2 1 0 1 2 3 3 2 1 0 1 2 3 − − − − − − ∆y ∆y (a) (b)

lead lead LHCb: pT > 20 GeV, zsub > 0.1, η [2.5, 4.5] LHCb: pT > 20 GeV, zsub > 0.1, η [2.5, 4.5] 0.8 ∈ 0.8 ∈

0.7 gluon splitting c-hadron tag 0.7 gluon splitting b-hadron tag non-splitting Pythia non-splitting Pythia

) 0.6 ) 0.6

R total R = 0.5 R total R = 0.5 0.5 0.5 d ∆ d ∆

σ/ 0.4 σ/ 0.4

)(d 0.3 )(d 0.3 /σ /σ

(1 0.2 (1 0.2

0.1 0.1

0.0 0.0 0 1 2 3 4 5 0 1 2 3 4 5 ∆R ∆R (c) (d)

lead lead LHCb: pT > 20 GeV, zsub > 0.1, η [2.5, 4.5] LHCb: pT > 20 GeV, zsub > 0.1, η [2.5, 4.5] 0.05 ∈ 0.05 ∈ c-hadron tag b-hadron tag 0.04 Pythia 0.04 Pythia

[GeV]) R = 0.5 [GeV]) R = 0.5

jj 0.03 gluon splitting jj 0.03 gluon splitting m m

d non-splitting d non-splitting

σ/ 0.02 total σ/ 0.02 total )(d )(d

/σ 0.01 /σ 0.01 (1 (1

0.00 0.00 0 20 40 60 80 100 0 20 40 60 80 100

mjj [GeV] mjj [GeV] (e) (f)

FIG. 12. Separation observable from the FlavorCone algorithm for (left) c-tagged and (right) b-tagged hadrons. The event selection is the same as Fig.8, but now showing (top row) ∆ y, (middle row) ∆R, and (bottom row) mjj . 18

GPD: pT > 200 GeV, ztag > 0.05, η [ 1.5, 1.5] GPD: pT > 200 GeV, ztag > 0.05, η [ 1.5, 1.5] 6 ∈ − 6 ∈ −

c-hadron tag (0, 0)c b-hadron tag (0, 0)b 5 5 Pythia (0, 1)c Pythia (0, 1)b ) ) g R = 1.0 (1, 1) g R = 1.0 (1, 1) z 4 c z 4 b d d

σ/ 3 σ/ 3 )(d )(d

/σ 2 /σ 2 (1 (1

1 1

0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

zg zg (a) (b)

GPD: pT > 200 GeV, ztag > 0.05, η [ 1.5, 1.5] GPD: pT > 200 GeV, ztag > 0.05, η [ 1.5, 1.5] 8 ∈ − 8 ∈ −

7 c-hadron tag (0, 1)c 7 b-hadron tag (0, 1)b Pythia (0, 2) Pythia (0, 2) 6 c 6 b ) ) g R = 1.0 g R = 1.0 (0, 1) z (0, 1)J/ψ z Υ 5 5 d d

σ/ 4 σ/ 4 )(d )(d 3 3 /σ /σ

(1 2 (1 2

1 1

0 0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

zg zg (c) (d)

FIG. 13. The distribution at a GPD for SoftDrop zg in (left) c-tagged and (right) b-tagged events. The top row (a,b) is analogous to the LHCb results in Fig.3 and the bottom row (c,d) corresponds to Fig.5.

diation was more restricted. As one goes to higher jet pT, flow are shown in Fig. 14, to be compared to the LHCb one could mitigate this category migration by choosing results in Fig.8. Here, the relative size of the splitting a higher ztag requirement, at the expense of introducing and non-splitting categories are different, with Flavor- shoulders in the zg distribution. In practice, one would Cone providing even greater separation than achieved at probably want to make measurements of zg with multi- lower pT. We conclude that the GPDs should be able ple ztag values, to test the stability of the distributions to successfully use these analysis strategies to distill the to the tagging conditions. kinematics and rates associated with heavy flavor, prob- The FlavorCone ∆φ distributions for the GPD working a complementary phase space compared to LHCb.

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lead lead GPD: pT > 200 GeV, zsub > 0.1, η [ 2, 2] GPD: pT > 200 GeV, zsub > 0.1, η [ 2, 2] 3.5 ∈ − 3.5 ∈ − c-hadron tag b-hadron tag 3.0 3.0 Pythia Pythia ) 2.5 ) 2.5 φ R = 0.5 φ R = 0.5

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0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 ∆φ ∆φ (a) (b)

FIG. 14. The distribution at a GPD for the FlavorCone ∆φ observable in (a) c-tagged and (b) b-tagged events. These plots are analogous to the LHCb results in Fig.8.

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