Disentangling Heavy Flavor at Colliders

TL;DR

This paper introduces two innovative analysis strategies using jet substructure techniques to study charm and beauty quarks at colliders, aiming to test QCD predictions and improve heavy-flavor modeling.

Contribution

It proposes the SoftDrop declustering and FlavorCone algorithms for detailed heavy-flavor analysis, enhancing understanding of quark kinematics and production mechanisms.

Findings

SoftDrop can test heavy-quark splitting functions

FlavorCone effectively isolates heavy-flavor production

Strategies are well-suited for LHCb, ATLAS, and CMS

Abstract

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 $J / \psi$ or $\Upsilon$, 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.

Tables

Table 1. Table 1: The cross sections for each of the fat-jet flavor-tagged categories determined from Pythia \xspace and Herwig++ \xspace , where the total cross section is normalized to the nominal inelastic cross section of 100 ​ mb 100 mb 100~{}\mathrm{mb} . Because we only consider one flavor label at a time, the sum of the c 𝑐 c -categories equals the sum of the b 𝑏 b -categories. We also show cross sections for quarkonium production in Pythia \xspace .

	$\sigma \left(\text{Pythia}\text{<span>xspace</span>}\right)$ [$\mu$b]	$\sigma \left(\text{Herwig++}\text{<span>xspace</span>}\right)$ [$\mu$b]
${(0,0)}_c$	$9.96\times {10}^2$	$5.28\times {10}^2$
${(0,1)}_c$	$7.56\times {10}^1$	$2.64\times {10}^1$
${(1,1)}_c$	$6.87\times {10}^0$	$2.87\times {10}^0$
${(0,2)}_c$	$1.00\times {10}^1$	$5.64\times {10}^0$
otherc	$8.86\times {10}^{-1}$	$2.47\times {10}^{-1}$
${(0,0)}_b$	$1.07\times {10}^3$	$5.52\times {10}^2$
${(0,1)}_b$	$1.34\times {10}^1$	$9.58\times {10}^0$
${(1,1)}_b$	$8.40\times {10}^{-1}$	$5.03\times {10}^{-1}$
${(0,2)}_b$	$9.50\times {10}^{-1}$	$5.94\times {10}^{-1}$
otherb	$1.13\times {10}^{-2}$	$7.75\times {10}^{-3}$
${(0,1)}_{J/\psi }$	$3.03\times {10}^{-1}$	–
${(0,1)}_{{\rm Y}}$	$1.54\times {10}^{-2}$	–

Table 2. Table 2: The discrimination power between splitting and non-splitting events in Pythia \xspace for c ​ c ¯ 𝑐 ¯ 𝑐 c\bar{c} and b ​ b ¯ 𝑏 ¯ 𝑏 b\bar{b} . For each dijet observable, the values shown correspond to classifier separation δ λ subscript 𝛿 𝜆 \delta_{\lambda} from Eq. ( 12 ), where larger values indicate better performance. Regardless of the choice of discriminant, the FlavorCone \xspace approach outperforms the traditional anti- k t subscript 𝑘 𝑡 k_{t} \xspace approach.

	$c\overline{c}$		$b\overline{b}$
	FlavorCone\xspace	Anti-$k_t$\xspace	FlavorCone\xspace	Anti-$k_t$\xspace
$\Delta y$	0.09	0.03	0.05	0.02
$\Delta \varphi$	0.41	0.30	0.31	0.24
$\Delta R$	0.43	0.31	0.32	0.25
$m_{jj}$	0.42	0.30	0.29	0.22

Table 3. Table 3: The cross section in the LHCb fiducial region, defined with the nominal FlavorCone \xspace algorithm, for c ​ c ¯ 𝑐 ¯ 𝑐 c\bar{c} and b ​ b ¯ 𝑏 ¯ 𝑏 b\bar{b} production from four different predictions. Note that there is a greater disagreement in these rates than there is in the differential shapes in Fig.  9 .

	$\sigma \left(c\overline{c}\right)$ [$\mu$b]	$\sigma \left(b\overline{b}\right)$ [$\mu$b]
Pythia\xspace	2.02	0.97
Vincia\xspace	1.40	0.59
Dire\xspace	2.55	0.93
PowhegBox\xspace	1.27	0.55

Equations

$$\frac{\min (p_{T1},p_{T2})}{p_{T1}+p_{T2}}>z_{\mathrm{cut}}{\left(\frac{R_{12}}{R}\right)}^{\beta },$$

$$z_{\mathrm{tag}}=\frac{p_{\mathrm{T}}^{\mathrm{tag}}}{p_{T1}+p_{T2}},$$

$$z_g\equiv \frac{p_{T1}}{p_{T1}+p_{T2}},$$

$$p_i(z_g)=\frac{{\overline{P}}_i(z_g)}{{\int }_{z_{\mathrm{cut}}}^{1/2}\mathrm{d}{z}^{\prime }{\overline{P}}_i(z^{\prime })}\Theta (z>z_{\mathrm{cut}}),$$

$$\displaystyle\overline{P}_{i}(z)=\sum_{jk}\bigl{(}P_{i\to jk}(z)+P_{i\to jk}(1-z)\bigr{)}.$$

$${\overline{P}}_q(z)\simeq {\overline{P}}_g(z)\simeq \frac{1-z}{z}+\frac{z}{1-z}+\frac{1}{2}.$$

$$P_{Q\to Qg}(z)=\frac{1-z}{z}+\frac{z}{2}-2{\mu }_{Qg}^2.$$

$${\mu }_{Qg}^2=\frac{m_Q^2}{m_{Qg}^2-m_Q^2},$$

$$P_{g\to Q\bar{Q}}(z)=z^2+(1-z{)}^2+{\mu }_{Q\bar{Q}}^2,$$

$${\mu }_{Q\bar{Q}}^2=\frac{2m_Q^2}{m_{Q\bar{Q}}^2},$$

$$z_{\mathrm{sub}}=\frac{p_{\mathrm{T}}^{\mathrm{sub}}}{p_{\mathrm{T}}^{\mathrm{lead}}+p_{\mathrm{T}}^{\mathrm{sub}}}.$$

$${\delta }_{\lambda }=\frac{1}{2}\int \mathrm{d}\lambda \frac{{\left(p_A(\lambda )-p_B(\lambda )\right)}^2}{p_A(\lambda )+p_B(\lambda )}.$$

Full text

††preprint: MIT-CTP/4880

I Introduction

The production of charm and beauty quarks at the Large Hadron Collider (LHC) is studied both as a fundamental 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 to test the properties of the SM Higgs boson, whose largest branching fraction is to a pair of beauty quarks de Florian et al. (2016). Similarly, identifying large-radius jets with double-flavor-tagged substructure enables searching for new physics scenarios involving high-$p_{\rm T}$\xspaceHiggs bosons ATL (2012); CMS (2013, 2016); ATL (2016); Khachatryan et al. (2016); Aaboud et al. (2016a). To address SM backgrounds in both cases, it is essential to understand the mechanisms for heavy-flavor production at the LHC within quantum chromodynamics (QCD). Of particular importance is the process of gluon splitting to heavy-quark pairs $g\to Q\bar{Q}$, where $Q$ denotes a $b$ or $c$ quark, which is challenging to study both theoretically and experimentally.

In this article, we present two analysis strategies aimed at testing key features of heavy-flavor production at the LHC. First, we use a jet-declustering method to study heavy-flavor kinematics within identified jets, with the goal of testing the well-known but as-of-yet-unmeasured massive $1\to 2$ splitting functions of QCD. Second, we 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 contribution from gluon splitting. Both of these analyses can in principle be performed by any of the LHC experiments. Here, we focus on the LHCb detector, which covers the pseudorapidity range $\eta\in[2,5]$, since its excellent heavy-flavor-identification capabilities offer the best short-term prospects. In the appendices, we also present results for ATLAS and CMS, which cover $\eta\in[-2.5,2.5]$. Qualitatively, the results of both proposed analyses are the same for LHCb and for ATLAS/CMS.

Our jet-declustering method is based on the SoftDrop\xspacealgorithm Larkoski et al. (2014a) and its precursor, the (modified) MassDrop tagger Butterworth et al. (2008); Dasgupta et al. (2013a, b). 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. As shown in LABEL:Larkoski:2015lea, the kinematics of the two resulting subjets match the famous Altarelli-Parisi $1\to 2$ splitting functions for massless QCD Altarelli and Parisi (1977). SoftDrop\xspacehas been used by CMS Collaboration (2016) and STAR Kauder in the context of heavy-ion collisions, and a related strategy was proposed to test the dead cone effect for boosted top quarks Maltoni et al. (2016). Here, we extend the analysis to QCD with heavy-flavor quarks, exploiting the ability to flavor-tag subjets to test the splitting kinematics of $Q\to Qg$ and $g\to Q\bar{Q}$. Because the SoftDrop\xspacealgorithm works equally well on tagged and untagged jets, we can compare our massive results directly to the massless case. In addition, this method can be applied to quarkonium states like the ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\xspace$ and $\Upsilon$, potentially providing new insights into the puzzle of quarkonium polarization and fragmentation Schub et al. (1995); McGaughey (1996); Sansoni (1996); Abe et al. (1997); Brambilla et al. (2011); Faccioli et al. (2014).

Our gluon-splitting study is based on a new jet algorithm, referred to as FlavorCone\xspace, that identifies conical jets by centering the jet axes along the flight directions of well-identified flavor-tagged hadrons. Unlike standard jet algorithms like anti-$k_{t}$\xspaceCacciari et al. (2008a), the FlavorCone\xspacemethod allows two jet axes to become arbitrarily close. This feature, partially inspired by the XCone jet algorithm Stewart et al. (2015); Thaler and Wilkason (2015), is ideal for studying gluon splitting to heavy flavor, where the two outgoing heavy quarks are often more collimated than the jet radius $R$. In standard jet analyses, overlapping heavy-flavor jets are typically merged, with a precipitous drop in efficiency at angular scales smaller than $R$. In the FlavorCone\xspacemethod, by contrast, heavy-flavor jet axes can be arbitrarily close, with the separate jet constituents determined by nearest-neighbor partitioning. In this way, the FlavorCone\xspacealgorithm enables a full exploration of the heavy-flavor production phase space, interpolating between the traditional regime of well-separated jets to the overlapping regime dominated by gluon splitting.

Like standard approaches to studying high-$p_{\rm T}$\xspaceheavy-flavor production at the LHC, the SoftDrop\xspaceand FlavorCone\xspacestrategies involve tagging (sub)jets that contain heavy-flavor hadrons. As we will see below, however, both the SoftDrop\xspaceand FlavorCone\xspaceanalyses require a definition of flavor tags that is more closely tied to heavy-flavor hadrons than typically required for tagging applications. Specifically, it will be essential to reconstruct the flight directions of heavy-flavor hadrons. For SoftDrop\xspace, these flight directions are used to define flavor-tagged subjet categories. For FlavorCone\xspace, these flight directions directly determine the central jet axes. In this way, the experimental requirements for—and challenges of—performing both analyses are largely shared.

As an alternative to (sub)jet tagging, one could perform exclusive reconstruction of heavy-flavor hadrons. From the experimental perspective, tagging is typically more efficient than reconstruction, since there are relatively few heavy-flavor decay modes that can be fully reconstructed. From the theoretical perspective, analyses based on flavor-tagged jets are less sensitive to nonperturbative physics than those directly based on heavy-flavor hadrons. To the extent that the typical jet scale \mbox{p_{\rm T}}\xspace R is larger than the heavy-flavor-hadron masses, the properties of heavy-flavor jets can be reliably calculated in (resummed) perturbative QCD, without the use of heavy-flavor fragmentation functions. Of course, there are always nonperturbative corrections from hadronization and the underlying event, but jet-level measurements are generally expected to be closer to parton-level perturbative calculations. In any case, jet-based and hadron-based analyses provide complementary information and both should be pursued when studying heavy flavor.

We validate the performance of these methods at the 13 TeV LHC using parton-shower generators. Our primary focus is on Pythia\xspace8.212 Sjostrand et al. (2006, 2008); Sjöstrand et al. (2015), which includes heavy-quark mass effects using matrix-element corrections Norrbin and Sjostrand (2001), and allows a leading-order classification of events into gluon-splitting and non-gluon-splitting topologies. For the SoftDrop\xspacestudy, we compare Pythia\xspaceto Herwig++\xspace 2.7.1 Bahr et al. (2008); Bellm et al. (2016) in order to test the robustness of the $1\to 2$ subjet kinematics to different showering and hadronization models.111Because we are focusing on relatively low-$p_{\rm T}$\xspacejets at LHCb, we generate minimum bias events, which precludes the use of NLO generators. For the FlavorCone\xspacestudy, we also consider alternative perturbative-shower results from Vincia\xspace 2.0.01 Giele et al. (2008); Fischer et al. (2016) and Dire\xspace 0.900 Höche and Prestel (2015), as well as matched next-to-leading-order (NLO) results from PowhegBox\xspace v2 Nason (2004); Frixione et al. (2007); Alioli et al. (2010, 2011), all using Pythia\xspacefor hadronization.222These programs are not compatible with a common underlying event model; therefore, we turn off multiple parton interactions (MPI) in Pythia\xspacefor the FlavorCone\xspacestudy to focus on perturbative physics. In the case of Pythia\xspace, we tested that the addition of MPI does not impact our conclusions. Where needed, we use FastJet 3.1.2 Cacciari et al. (2012a) for jet finding and the RecursiveTools fjcontrib\xspace 1.024 Cacciari et al. (2012a) for SoftDrop\xspace.

The remainder of the article is organized as follows. In Sec. II, we show how SoftDrop\xspacedeclustering can be used to study heavy-flavor kinematics within large-radius jets, including the kinematics of quarkonium production. In Sec. III, we define the FlavorCone\xspacejet algorithm and demonstrate how it can be used to disentangle heavy-flavor production processes in QCD. We do not include detector-response effects on the distributions presented here, though we do discuss the prospects for applying these methods in the realistic LHCb environment in Sec. IV. We conclude in Sec. V, leaving additional plots to the appendices.

II SoftDrop\xspaceJet Declustering to Probe Heavy-Flavor Kinematics

The goal of our jet-declustering analysis is to study the collinear-splitting kernels of QCD appropriate for massive quarks.333For related work, see LABEL:Anderle:2017qwx. These kernels form the basis for parton showers like Pythia\xspace, so we expect jet-declustering measurements will help improve theoretical predictions in the collinear regime. We also present results for quarkonium production within an identified jet. The current Pythia\xspacemodels for ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$\xspaceand $\Upsilon$ production are known to be incomplete, so measurements of the quarkonium-splitting kinematics should provide valuable information. In this section, we use the SoftDrop\xspacealgorithm along with heavy-flavor tagging to reveal the massive-quark splitting kernels.

II.1 Review of SoftDrop\xspace

SoftDrop\xspaceis a jet-grooming technique that removes wide-angle soft radiation from a jet. This algorithm is a generalization of the (modified) MassDrop tagger from Refs. Butterworth et al. (2008); Dasgupta et al. (2013a, b), with an additional angular exponent $\beta$ that controls the degree of grooming. In general, SoftDrop\xspacereduces the dependence of the jet kinematics on other aspects of the full event, such as the underlying event, color correlations to the initial state, and pileup contamination. Here, we will be primarily interested in using SoftDrop\xspaceto define $1\to 2$ splitting kinematics.

SoftDrop\xspacestarts from a jet of radius $R$ that has been clustered with some jet algorithm, typically anti-$k_{t}$\xspace. Regardless of the clustering algorithm used to form the initial jet, one builds a Cambridge-Aachen (C/A) Dokshitzer et al. (1997); Wobisch and Wengler (1998) clustering tree from the jet constituents. Working backwards from the top of the tree, SoftDrop\xspacerecursively checks whether the two branches of the tree satisfy the following condition, set by the the grooming parameters $z_{\rm cut}$ and $\beta$:

[TABLE]

where $p_{{\rm T}i}$ are the transverse momenta of the two branches and $R_{12}$ is their rapidity-azimuth separation. If the condition in Eq. (1) is not satisfied, then the softer of the two branches is dropped and the procedure is repeated on the next node down the C/A tree. The procedure terminates once the SoftDrop\xspacecondition is satisfied, and the two final branches define the two SoftDrop\xspacesubjets.

The SoftDrop\xspacealgorithm has proven to be a valuable tool for the study of jets; see, for example, Refs. Larkoski and Thaler (2014); Dasgupta et al. (2015); Kasieczka et al. (2015); Adams et al. (2015); Luisoni and Marzani (2015); Larkoski and Moult (2016); Aad et al. (2016); Dolen et al. (2016); Frye et al. (2016a, b); Maltoni et al. (2016); Moult et al. (2016); Dasgupta et al. (2016); Chien and Vitev (2016); Salam et al. (2016). As already mentioned, SoftDrop\xspacehas been shown both theoretically Larkoski et al. (2015) and experimentally Collaboration (2016); Kauder to expose the basic splitting functions of massless QCD. Using a parton-shower analysis, we argue below that SoftDrop\xspacecan also be used to directly study the massive QCD splitting functions.

II.2 Event Selection and Flavor Classification

Because we want to compare the splitting kinematics for jets that contain different numbers of heavy-flavor-tagged hadrons, we define an event selection that is independent of the heavy-flavor content. We start from large-radius merged jets without applying any flavor-tagged hadron requirements, and then use the following analysis workflow.

• •

We identify all flavor-tagged hadrons with \mbox{p_{\rm T}}\xspace>2\mathrm{\>Ge\kern-1.00006ptV}\xspace and treat their flight directions as ghost particles Cacciari et al. (2008b) for the purposes of jet clustering.444A ghost is a particle with infinitesimal energy, but well-defined direction, that is clustered for the purpose of (sub)jet heavy-flavor tagging. See Sec. IV for a discussion of the experimental aspects of flavor-tagged hadrons. For the case of charm tagging, we require that the $c$-hadron does not come from a $b$ decay.

• •

We cluster the hadrons and the ghosts into anti-$k_{t}$\xspace fat jets with $R=1.0$.

• •

The hardest jet is required to have $\eta\in[3,4]$, so that the full nominal jet cone is within LHCb acceptance, and \mbox{p_{\rm T}}\xspace>20\mathrm{\>Ge\kern-1.00006ptV}\xspace, which is a typical jet scale in LHCb. In App. C, we show results relevant for ATLAS and CMS using a larger $p_{\rm T}$ threshold.

• •

We apply the SoftDrop\xspacejet-declustering algorithm to the hardest jet, taking the SoftDrop\xspaceparameters to be $\beta=0$ and $z_{\rm cut}=0.1$. Note that with this choice of parameters SoftDrop\xspaceacts identically to the modified MassDrop tagger with $\mu=1$ Dasgupta et al. (2013b).

• •

For each flavor-tagged hadron that is kept after SoftDrop\xspace, we calculate

[TABLE]

where $p_{\rm T1}$ and $p_{\rm T2}$ are the transverse momenta of the two SoftDrop\xspacesubjets. To count as a flavor tag in the classification scheme below, we require $z_{\rm tag}>0.05$.

The resulting SoftDrop\xspacesubjets, and their flavor labels, form the basic objects of interest for subsequent analysis. We choose $z_{\rm tag}$ to be half the value of $z_{\rm cut}$ in order to reduce kinematic dependence on the tagging condition, though one could further optimize the relationship between $z_{\rm tag}$ and $z_{\rm cut}$ to balance perturbative control against dependence on heavy-flavor fragmentation. Experimentally, the method used to tag the hadron flavor must provide a measurement of the hadron flight direction, using, for example, the vector formed by connecting the $pp$-collision point to the hadron-decay vertex.

These selection requirements are loose, and require some care to implement properly in event generators. Within the LHCb acceptance, we often find that the fat jet comes not from the hard-scattering process but from underlying event activity. For this reason, we only test the Pythia\xspaceand Herwig++\xspaceevent generators, since they have full implementations of the underlying event including MPI.

To study the splitting kinematics, we use the flavor-tagged ghosts to label the SoftDrop\xspacesubjets. Because the ghosts participate in the original anti-$k_{t}$\xspaceclustering and subsequent SoftDrop\xspacedeclustering, they can become constituents of the subjets. By counting the number of ghosts within each subjet that satisfy $z_{\rm tag}>0.05$, we associate flavor labels and interpretations to the fat jets. In cases where the ghost particles are removed by the SoftDrop\xspaceprocedure, we simply ignore them when assigning flavor labels.555The dropped ghost tags are interesting in their own right, since they can be used to diagnose counterintuitive kinematical scenarios. The main reason not to consider them for this study is to avoid a proliferation of curves on the following plots.

Specifically, we label our jets as $(n_{1},n_{2})$, where $n_{1}$ is the number of heavy-flavor hadrons tagged in the first subjet and $n_{2}$ the number tagged in the second subjet, defined such that $n_{1}\leq n_{2}$. The cases we focus on are:

• •

No tagged subjets (0,0): Labeling light quarks generically as $q$, this category comes from $g\to gg$, $q\to qg$, and $g\to q\bar{q}$.

• •

One single tag (0,1): This most likely arises from a heavy quark emitting a gluon, $Q\to Qg$.

• •

Two single tags (1,1): Here, both subjets are associated with heavy flavor, which usually arises from gluon splitting, $g\to Q\bar{Q}$.

• •

One double tag (0,2): Double-tagged subjets come from cascaded splittings such as $g\to gg$ followed by $g\to Q\bar{Q}$, making their interpretation in terms of splitting functions more complicated.

We also rarely find jets with $n_{1}+n_{2}>2$, which are ignored in the analysis presented below. For simplicity, we only treat one flavor label at a time, such that the $c$-tagged categories do not include an explicit veto on $b$-tagged objects, and vice versa. In Fig. 2, we show some example SoftDrop\xspaceevent displays.

II.3 Splitting Function Interpretation

Because the above flavor-tagged categories are based 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\xspaceprocedure or fails the $z_{\rm tag}$ condition, and this has to be accounted for when interpreting the observed distributions. For example, a (0,2) jet with one dropped tag becomes a (0,1) jet. In addition, soft $g\to Q\bar{Q}$ splittings can contaminate the flavor labels, though this effect is highly suppressed by the $z_{\rm tag}$ condition. Being mindful of migration, it is instructive to discuss the expected kinematical distributions for each of the flavor-tagged categories.

We are specifically interested in the momentum sharing $z_{g}$ between the SoftDrop\xspacesubjets, and adopt a modified definition of $z_{g}$ compared to the literature

[TABLE]

where the $1$ and $2$ subjet labels are derived from the $(n_{1},n_{2})$ flavor-tagged label, instead of being ordered by $p_{\rm T}$\xspace, such that $z_{g}\in[z_{\rm cut},1-z_{\rm cut}]$. In cases where $n_{1}=n_{2}$, we randomize the ordering of the subjets resulting in a $z_{g}$ distribution that is symmetric about $z_{g}=1/2$.

For the (0,0) case, which has no flavor tags, this is essentially massless QCD with $N_{f}=3$. As shown in LABEL:Larkoski:2015lea and experimentally measured by CMS Collaboration (2016) and STAR Kauder , the $z_{g}$ distribution is closely related to the massless-QCD splitting kernels. Specifically, for $\beta=0$ and to lowest order in $\alpha_{s}$, the probability distribution for $z_{g}$ is given by

[TABLE]

where $i$ labels the initiating parton for the jet. Here, $\overline{P}_{i}(z)$ are symmetrized versions of the QCD splitting functions for parton $i$ summed over all final state partons,

[TABLE]

Because we are not distinguishing between quark and gluon (sub)jets in this analysis, the measured $p(z_{g})$ distribution probes a combination of all massless splittings: $g\to gg$, $q\to qg$, and $g\to q\bar{q}$. For $N_{f}=3$, the symmetrized splitting functions for quarks and gluons are identical to this order,

[TABLE]

Note that $\overline{P}$ does not include the Casimir factor ($C_{q}=4/3$ and $C_{g}=3$), which drops out from the $p(z_{g})$ distributions at lowest order in $\alpha_{s}$.

For the (0,1) case of one flavor tag, the dominant contribution comes from $Q\to Qg$. In this case, the $z_{g}$ distribution depends on the quasi-collinear splitting function Catani et al. (2001), which is not symmetrized over the two subjets:666Note that we are using the reversed convention of $z$ versus $1-z$ in the splitting function in order to match the definition of $z_{g}$.

[TABLE]

Here, the mass ratio is

[TABLE]

and $m_{Qg}$ is the invariant mass of the heavy quark plus gluon system. Taking the $m_{Q}\to 0$ limit and symmetrizing $z\to 1-z$, one recovers Eq. (6) as expected. By comparing the (0,1) and (0,0) distributions, it is possible to test the splitting-function form in Eq. (7).777Pythia\xspaceimplements the heavy-flavor splitting functions using a matrix-element correction Norrbin and Sjostrand (2001) instead of the $-2\mu^{2}_{Qg}$ term in Eq. (7).

For the (1,1) category with one flavor tag in each subjet, the dominant process is $g\to Q\bar{Q}$. The quasi-collinear splitting function for this case is Catani et al. (2001)

[TABLE]

the mass ratio is

[TABLE]

and $m_{Q\bar{Q}}$ is the invariant mass of the heavy-quark pair.888In the Pythia\xspaceimplementation, the $\mu^{2}_{Q\bar{Q}}$ term is multiplied by an additional factor of $4z(1-z)$ Sjöstrand et al. (2015). This explains why the $(1,1)$ category in Fig. 3 exhibits a downturn towards $z\to 0$ and $z\to 1$. As one goes to higher jet $p_{\rm T}$\xspace, this additional factor is less important, and one recovers the expected upturn from Eq. (9); see Figs. 13 and 13. Note the absence of any singular behavior in the $z\to 0$ or $z\to 1$ limits, as expected since this process does not have a soft singularity.

Finally, the (0,2) category, where one subjet has two flavor tags, does not have a simple interpretation in terms of $1\to 2$ splitting functions. In a parton shower, this configuration can be obtained from $g\to gg$ followed by $g\to Q\bar{Q}$. More intuitively, one can think of the double-tagged subjet as being a color-octet configuration that radiates soft gluons via $(Q\bar{Q})_{8}\to(Q\bar{Q})_{8}g$. In this color-octet interpretation, the (0,2) distribution is expected to look like the (0,1) case with the replacement $m_{Q}\to 2m_{Q}$, since the different Casimir factors do not appear in $P(z)$ at lowest order. It is of particular interest to compare the (0,2) category to the quarkonium case studied in Sec. II.5.

In addition to $z_{g}$, the other natural kinematic observable for SoftDrop\xspacejets is $R_{g}$, the opening angle between the two subjets. For massless partons, the $R_{g}$ distribution was calculated to next-to-leading-logarithmic accuracy in LABEL:Larkoski:2014wba. The $R_{g}$ distribution for massive partons has not been calculated in the literature, though LABEL:Maltoni:2016ays used a variant of $R_{g}$ to test the dead cone effect for boosted top quarks. We do not show the perturbative predictions here, since for the jet $p_{\rm T}$\xspacerange of interest for LHCb, the $R_{g}$ distribution is dominated by nonperturbative physics and is relatively insensitive to the flavor content of the jets. For completeness, we show the $R_{g}$ distributions in App. A.

In the analysis below, we treat the jet fragmentation process as if it were rotationally symmetric about the jet axis. As recently discussed in LABEL:Anderle:2017qwx, though, it is interesting to study the angle between the jet production plane and the subjet decay plane. For the case of $g\to Q\bar{Q}$, this angle is sensitive to gluon polarization, motivating future multi-differential studies of the full SoftDrop\xspacesubjet decay phase space.

II.4 Results: Heavy-Quark Splittings

Using this SoftDrop\xspacejet-declustering strategy, we first consider the inclusive cross section for each of the flavor-tagged categories in Table 1. Quantitatively, the predictions obtained from Pythia\xspaceand Herwig++\xspacedo not agree: both the absolute and relative cross sections show sizable discrepancies. There is qualitative agreement, however, as both generators predict that the $(0,0)$ category with no flavor tags dominates the total rate, followed by the $(0,1)$ category which is largely due to $Q\to Qg$. The $(1,1)$ and $(0,2)$ categories, which arise from $g\to Q\bar{Q}$ and cascaded splittings, respectively, are predicted to have similar rates, while events with $n_{1}+n_{2}>2$ are rare as expected.

We next show SoftDrop\xspacedistributions for $z_{g}$, as defined in Eq. (3), for both $c$-tagged and $b$-tagged fat jets. We focus on the (0,0), (0,1), and (1,1) categories here, leaving the (0,2) category to Sec. II.5. In Fig. 3, we show results from Pythia\xspace. The (0,0) distribution, which has no flavor tags, agrees with those already found in LABEL:Larkoski:2015lea, with the important caveat that $z_{g}$ is defined here to be symmetric about $z_{g}=1/2$ (instead of stopping at $z_{g}=1/2$). For the (0,1) category, the $z_{g}$ distribution agrees qualitatively with the $Q\to Qg$ splitting function, peaking towards $z\to 0$ as expected from Eq. (7). The (1,1) category, which is largely due to $g\to Q\bar{Q}$, has no singular structures near $z\to 0$ or $z\to 1$. Figure 4 shows that the analogous distributions obtained using Herwig++\xspaceare similar to those obtained from Pythia\xspace, with some small differences observed near the endpoints.

All three categories exhibit distinct behavior that qualitatively agrees with predictions from perturbative QCD. While the $1\to 2$ splitting functions of massive QCD are well known theoretically, they have never been probed in such a direct manner experimentally. By exploiting the ability to flavor-tag SoftDrop\xspacesubjets, our approach provides the opportunity to directly probe the splitting kinematics of $Q\to Qg$ and $g\to Q\bar{Q}$. Having confirmed that our parton-shower results agree qualitatively with the expected theoretical predictions in Sec. II.3, we look forward to tests of these $z_{g}$ distributions in data.

II.5 Results: Quarkonium Production

The SoftDrop\xspacejet-declustering strategy is also applicable to fat jets containing identified quarkonium states. In terms of flavor content, jets with a quarkonium-tagged subjet are similar to the (0,2) category defined above, so it is interesting to compare their $z_{g}$ distributions to see whether the underlying physics is similar or not. Traditionally, quarkonium production within a jet is studied using fragmentation functions, which describe the momentum fraction carried by a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$\xspaceor $\Upsilon$ hadron within a resolved jet. Here, we pursue a complementary approach using $z_{g}$, which gives the momentum fraction carried by a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$\xspace-tagged or $\Upsilon$-tagged subjet within a SoftDrop\xspacefat jet.

As a preamble, it is worth emphasizing that there are considerable theoretical uncertainties in both the production and splittings associated with the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$\xspaceand $\Upsilon$. A standard approach to study these quarkonium states is to use nonrelativistic QCD (NRQCD) Bodwin et al. (1995); Cho and Leibovich (1996a, b), though there is a long-standing quarkonium-polarization puzzle associated with this approach; see, for example, LABEL:Brambilla:2010cs and references therein. More recently, Refs. Baumgart et al. (2014); Bain et al. (2016) used the method of fragmenting jet functions (FJF) Procura and Stewart (2010) as an alternate method to calculate ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$\xspaceproduction. Specifically, LABEL:Bain:2016clc considered the kinematics of ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$\xspaceproduction within a resolved jet, finding that their theoretical predictions for the showering, and hence splitting functions, associated with the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$\xspacedisagreed with those implemented in Pythia\xspace. In Pythia\xspace, a ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$\xspaceproduced in the color-octet channel is treated as a loosely bound $c\bar{c}$ state, and its total showering is the sum of the showers originating from either quark. By contrast, in the FJF approach, a produced ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$\xspaceis showered through DGLAP evolution of the splitting kernels from $2m_{c}$ up to the jet energy scale. There have been other treatments of quarkonia showers discussed in the literature; see, for example, LABEL:Ernstrom:1996am. Given these theoretical uncertainties, we expect measurements of $z_{g}$ to help clarify the mechanism of quarkonium production within jets. Furthermore, LHCb recently published a study of prompt ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$\xspaceproduction in jets Aaij et al. (2017). Their results are consistent with the predictions of the FJF-based approach, and in stark disagreement with Pythia\xspace, providing additional motivation to measure $z_{g}$ for quarkonium production.

In Table 1 we give the predicted rates for ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$\xspace-tagged and $\Upsilon$-tagged jet production. These rates are more than an order of magnitude smaller than those of the double-flavor-tagged $(0,2)$ category with the same quark flavor. In Fig. 5, we compare the $z_{g}$ distributions for quarkonium-tagged jets to the $(0,1)$ and $(0,2)$ categories. In the context of Pythia\xspace, soft-gluon radiation from heavy-flavor quarks primarily differs from soft-gluon radiation from quarkonium only in the overall color factor. The quarkonium distributions are not included in the equivalent Herwig++\xspaceplot shown in Fig. 6, since quarkonium production is not available in the version of Herwig++\xspaceused in these studies. Since the $z_{g}$ observable is insensitive to Casimir factors at lowest order, the distributions in Fig. 5 are all similar. Given the calculation in LABEL:Bain:2016clc, and the LHCb results in LABEL:LHCb-PAPER-2016-064, it will be fascinating if this prediction is borne out in data.

III FlavorCone\xspaceJets to Disentangle Heavy-Flavor Production

We now transition from studying heavy-flavor within a single jet to heavy-flavor production in the event as a whole. There are multiple production channels for heavy 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, with the caveats that the $\alpha_{s}$ order at which these diagrams appear depends on the PDF scheme employed, and that at higher orders there is no gauge-invariant definition of these categories. Still, making a gluon-splitting versus non-gluon-splitting distinction provides helpful intuition about the relevant phase-space regions populated by these diagrams. The main challenge of using traditional jet algorithms in the gluon-splitting regime is jet merging. In this section, we first review the jet-merging issue, and then introduce our FlavorCone\xspacejet algorithm designed to resolve overlapping heavy-flavor jets.

III.1 Traditional Approach to Heavy-Flavor Rates

At the LHC, it is well-known that the gluon-splitting and flavor-excitation processes can dominate the total rate over flavor creation; see, for example, Refs. Zanderighi (2007); Aad et al. (2013a); Voutilainen (2015). These two dominant processes are challenging to calculate since, as emphasized by their $2\to 3$ representations in Figs. 1(a) and 1(b), they involve multiple emission scales. The cross sections for these channels have been calculated at NLO in perturbative QCD; see, for example, Refs. Nason et al. (1988, 1989); Beenakker et al. (1991); Olness et al. (1999); Frixione et al. (1998); Cacciari et al. (1998); Nason et al. (1999); Tung et al. (2002); Czakon et al. (2008) and references therein. Fixed-order results have been interfaced with parton-shower programs to provide predicted rates for the LHC Norrbin and Sjostrand (2000); Banfi et al. (2007); Cacciari et al. (2012b). Perturbative QCD predictions for heavy-flavor production have been tested experimentally for $b$ quarks—at the Tevatron Abe et al. (1993a, b); Abbott et al. (2000a, b), and at the LHC by ATLAS Aad et al. (2011); Aaboud et al. (2016b); Aad et al. (2012a, 2013b, b, 2014), CMS Khachatryan et al. (2011); Chatrchyan et al. (2012a, 2014a, b, 2013, 2014b), and LHCb Aaij et al. (2014)—and for a combination of flavors at ATLAS Aad et al. (2013a). Many of these studies have noted a tension between the theoretical predictions and the experimental data, especially in the gluon-splitting regime.

The phase-space region where the disagreement is largest is also where analysis strategies based on traditional jet algorithms break down. In gluon-splitting events, the heavy quarks tend to have small angular separation due to the gluon propagator. Standard jet algorithms, such as anti-$k_{t}$\xspace, have difficulty in this situation, since events where two heavy quarks fall into a single jet are typically cut from the analysis.999Alternatively, one could separately perform a subjet analysis on anti-$k_{t}$\xspacejets with multiple flavor tags ATL (2012); CMS (2013, 2016); ATL (2016). This strategy, however, does not result in a smooth transition between the gluon-splitting and non-gluon-splitting regimes. This limitation is unavoidable for anti-$k_{t}$\xspace, since the jet axes cannot get any closer than the jet radius $R$, usually taken to be $R=0.4$ or $0.5$.101010Other cone-like algorithms can even lead to jet merging below $\sqrt{2}R$ or $2R$; see discussion in LABEL:Thaler:2015uja. Despite this, almost all of the LHC analyses referenced above use anti-$k_{t}$\xspace, and suffer the associated loss of performance.

It is worth mentioning an alternative strategy for studying heavy-flavor production based on flavored jet algorithms Banfi et al. (2006, 2007). These algorithms, which work equally well on heavy-flavor quarks or hadrons, attempt to neutralize gluon-splitting topologies through a recursive clustering strategy. In this way, events with $g\to Q\bar{Q}$ are not even categorized as heavy-flavor production, giving complementary information to FlavorCone\xspacejets.

III.2 A New Approach: FlavorCone\xspace

We now introduce a simple jet algorithm aimed at reconstructing the gluon-splitting phase space. In an event with $n$ heavy-flavor-tagged hadrons, we define $n$ FlavorCone\xspacejets of radius $R$ as follows.

The flight direction of each flavor-tagged hadron defines a separate jet axis. 2. 2.

Any particle that is further away than $R$ from a jet axis is left unclustered. 3. 3.

Each remaining particle is clustered to the nearest jet axis. 4. 4.

The momentum of each jet is then determined by the summed momenta of its constituents.

As is appropriate for $pp$ collisions, we use the rapidity-azimuth distance $\Delta R=\sqrt{(\Delta y)^{2}+(\Delta\phi)^{2}}$ to determine the unclustered region and the nearest jet axis.111111In an experimental analysis, it may be preferable to use pseudorapidity $\eta$ instead of rapidity $y$ to avoid complications of assigning masses to reconstructed particles. See Sec. IV for further discussion. Example events clustered with the FlavorCone\xspacealgorithm are shown in Fig. 7.

By construction, the FlavorCone\xspacealgorithm does not include a merging step, so there are always exactly $n$ jets in the event, regardless of whether the $n$ flavor-tagged hadrons are widely separated or almost collinear. In the well-separated regime, the resulting jets are perfect cones centered on the flight directions of the flavor-tagged hadrons, yielding results that are similar to anti-$k_{t}$\xspace. Crucially, there is no limitation on the jet axes getting closer than $R$, and abutting jet regions are determined by nearest-neighbor partitioning. Of course, this feature relies fundamentally on the ability of experiments like LHCb to accurately reconstruct the flight direction of heavy-flavor hadrons, even when the hadrons are almost collinear, as discussed further in Sec. IV.

The partitioning scheme used for FlavorCone\xspaceis motivated by the XCone jet algorithm Stewart et al. (2015); Thaler and Wilkason (2015), which is also designed to yield a fixed, predetermined number of jets. Beyond just the simplicity of the FlavorCone\xspacealgorithm, its main advantage over XCone is that flavor-tagged hadrons always end up in separate jets, whereas XCone can still allow merging. The XCone algorithm is infrared and collinear (IRC) safe, since it starts from a set of IRC-safe seed-jet axes and then applies an iterative procedure to find the jet regions that (locally) minimize an $N$-jettiness measure Stewart et al. (2010). The FlavorCone\xspacealgorithm is also IRC safe, since additional soft particles or collinear splittings cannot impact the flight direction of a flavor-tagged hadron, to the extent that $m_{b,c}\gg\Lambda_{\rm QCD}$.121212Similar to XCone, FlavorCone\xspacecan be sensitive to the IRC regime, since the algorithm will identify $n$ jets even if the $n$ flavor-tagged hadrons of interest have very low $p_{T}$. For this reason, one may wish to impose additional requirements on the FlavorCone\xspacejets to ensure that one is in the perturbative regime. Here, we set a minimum $p_{\rm T}$\xspace on the flavor tag and then place an additional cut on the overall jet $p_{\rm T}$\xspace, which introduces some mild dependence on the $b$-quark fragmentation function. Alternatively, one could place a cut on the relative $p_{\rm T}$\xspace of the tag and the jet, in the same spirit as the $z_{\rm tag}$ condition in Eq. (2).

In general, the FlavorCone\xspacejet axis and the FlavorCone\xspacejet momentum are not aligned. In this respect, the FlavorCone\xspaceaxes are similar in spirit to the winner-take-all axes Larkoski et al. (2014b); Bertolini et al. (2014); Salam , especially since flavor-tagged hadrons often carry a large fraction of the jet momentum in the non-gluon-splitting regime. If desired, one could apply an iterative procedure to find $n$ mutually stable cones using the $n$ flavor-tagged-hadron directions as seed axes. As mentioned in Sec. III.5, we find better performance by simply using flavor-tagged-hadron axes directly, since this avoids the issue of abutting jets repelling (or merging into) each other after iteration, which tends to wash out the gluon-splitting phase space.

III.3 Event Selection and Classification

Using this FlavorCone\xspacealgorithm, we now outline an analysis strategy to disentangle the mechanisms for heavy-flavor production. Our focus here is on the LHCb experiment, though we show related distributions for ATLAS and CMS in App. C. Our analysis workflow is as follows.

• •

We select events that have at least two heavy-flavor hadrons with $\eta\in[2.5,4.5]$, so that the full FlavorCone\xspace(with $R=0.5$) is within LHCb acceptance. For concreteness in the plots below, we require these hadrons to have \mbox{p_{\rm T}}\xspace>2\mathrm{\>Ge\kern-1.00006ptV}\xspace, though the specific selection criteria would depend on the implementation details; see Sec. IV.

• •

These heavy-flavor hadrons are ordered in $p_{\rm T}$\xspace, and the two hardest ones are used to form FlavorCone\xspacejets with $R=0.5$.131313Although reconstructing the full four-momenta of the hadrons is challenging at the LHC, ordering them by $p_{\rm T}$\xspaceis more feasible. That said, events with more than two reconstructed flavor-tagged hadrons are expected to be rare; see Sec. IV.

• •

The leading FlavorCone\xspacejet is required to have \mbox{p_{\rm T}}\xspace>20\mathrm{\>Ge\kern-1.00006ptV}\xspace, though larger $p_{\rm T}$\xspacethresholds would likely be needed at ATLAS or CMS.

• •

The subleading FlavorCone\xspacejet is required to have $z_{\rm sub}>0.1$, where we define

[TABLE]

This cut avoids highly asymmetric events that are more difficult to predict from perturbation theory.

We perform this analysis separately for $b\bar{b}$ and $c\bar{c}$ final states. In principle, one could also look at mixed $bc$ events, but we do not pursue that here.

For comparison, we also consider events clustered using anti-$k_{t}$\xspacewith $R=0.5$, where flavor tagging is performed by treating the flight directions of the hardest two heavy-flavor hadrons as ghost particles in the clustering. The two resulting anti-$k_{t}$\xspaceflavor-tagged jets are then treated in the same way as the FlavorCone\xspacejets, with the same cut on the leading jet $p_{\rm T}$\xspaceand subleading $z_{\rm sub}$. In keeping with the traditional strategy, anti-$k_{t}$\xspacejets that contain two flavor-tagged hadrons are rejected from the analysis.

When using the Pythia\xspaceparton shower, it is possible to classify generated events as being either splitting or non-splitting using the event record. If the two flavor-tagged hadrons can be traced back to a common gluon parent from the shower, then the event is labelled as splitting. Every other event is classified as non-splitting, which includes prompt production via flavor creation and excitation, as well as gluon-splitting-like events where the two heavy-flavor hadrons do not come from a common gluon parent from the shower.141414We also tested alternative classification schemes based on the hard-production vertex. We found cases, however, where the hard production is labeled as flavor creation or excitation, but the actual heavy-flavor hadrons within the LHCb acceptance come from a subsequent gluon splitting. Of course, this splitting/non-splitting distinction is not physical, as it cannot be implemented experimentally and is ambiguous beyond the strongly-ordered parton-shower limit. Nonetheless, it allows us to isolate the way that Pythia\xspacetreats the gluon-splitting regime and test whether the FlavorCone\xspacealgorithm is sufficiently sensitive to the underlying physics.

III.4 Results: Clear Separation of Gluon Splitting

The strategy above selects events with two flavor-tagged jets. To probe the physics of heavy-flavor production, we consider the cross section differential in four dijet observables: $\Delta y$, $\Delta\phi$, $\Delta R$, each determined using the jet axes; and the invariant mass of the dijet system $m_{jj}$. For anti-$k_{t}$\xspace, the jet axis and momentum are aligned, whereas for FlavorCone\xspace, the jet axis is aligned with the flight direction of the flavor-tagged hadron.

In order to determine the potential separation power between splitting and non-splitting events, we first calculate the classifier separation as implemented in TMVA Hocker et al. (2007).151515An alternative classifier metric is mutual information Larkoski et al. (2014c), which is closely related to classifier separation Andersen et al. (2016). Given two event categories $A$ and $B$, and probability distributions $p_{A}(\lambda)$ and $p_{B}(\lambda)$ for the observable $\lambda$, the classifier separation $\delta_{\lambda}$ is defined as

[TABLE]

The value of $\delta_{\lambda}$ always lies within $[0,1]$, where $\delta_{\lambda}=0$ means that $\lambda$ has no discriminating power and $\delta_{\lambda}=1$ indicates ideal separation.

In Table 2, we show the $\delta$ values for each of the dijet observables as computed from Pythia\xspace. As expected, $\Delta y$ is not a good discriminant, since back-to-back jets from flavor creation can also have a small rapidity separation. The remaining three observables show good performance in separating the splitting and non-splitting categories. These observables are, of course, strongly correlated, and we checked that combining them in a multivariate analysis provides little improvement. Table 2 clearly shows that the FlavorCone\xspaceapproach outperforms anti-$k_{t}$\xspace, yielding a 30–40% gain in performance as measured by $\delta_{\lambda}$. This is largely due to heavy-flavor merging by the anti-$k_{t}$ algorithm.161616In the spirit of footnote 9, one could consider double-tagged anti-$k_{t}$ jets as a separate event category to be included in the calculation of $\delta_{\lambda}$. Depending on the precise double-tagged selection criteria one uses, the resulting performance can be comparable to FlavorCone\xspace. Alternatively, one could use a smaller jet radius to reduce the impact of jet merging.

To highlight the separation power achievable using the FlavorCone\xspacealgorithm, we show the full distribution for $\Delta\phi$ in Fig. 8, with the other three observables given in App. B. We present both $b\bar{b}$ and $c\bar{c}$ events, broken down into the splitting and non-splitting categories. In order to isolate the phase-space region dominated by gluon splitting, one can select events with \Delta\phi\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 0.5pt\sim}\hss}\raise 1.0pt\hbox{<}}1. Note that the $\Delta\phi$ distribution smoothly approaches zero, with no change of behavior as the angle approaches the jet radius $R=0.5$. It is in this region that the FlavorCone\xspacealgorithm performs particularly well, while traditional jet algorithms result in jet merging (see Sec. III.5).

In addition to Pythia\xspace, we apply this analysis to three alternative predictions—Vincia\xspace, Dire\xspace, and PowhegBox\xspace—all interfaced to Pythia\xspacefor hadronization. Already from the total cross sections in Table 3, one can see substantial differences between the generators, but the origin of the disagreement is not clear from the rates alone. In the normalized distributions in Fig. 9, we can see in more detail the different predictions for the splitting-enriched and non-splitting-enriched regions of phase space. While there is qualitative agreement about the peaks at $\Delta\phi=0$ and $\pi$, quantitatively the predictions are sufficiently different that direct comparison to LHC data should result in improved modeling of heavy-flavor production in parton-shower generators.

III.5 Alternative Approaches

To better understand the behavior of FlavorCone\xspacejets, it is instructive to make a comparison to alternative reconstruction strategies. In Fig. 10, we compare the $\Delta\phi$ distributions obtained using the following four methods.

FlavorCone\xspace: The default FlavorCone\xspaceapproach, where $\Delta\phi$ is determined from the jet axes (which are aligned with the flavor-tagged hadrons). 2. 2.

FlavorConeJet: An alternative FlavorCone\xspaceapproach, where $\Delta\phi$ is determined by the jet momenta (i.e. the vector sum of the constituent momenta of each jet). 3. 3.

Anti-$k_{t}$\xspace: The default anti-$k_{t}$\xspaceapproach using $E$-scheme recombination, where the jet axes and jet momenta align. 4. 4.

$Q$-hadron: Omitting any jet clustering and determining $\Delta\phi$ from the flavor-tagged hadrons alone. Here, we require that the heavy-flavor hadrons have \mbox{p_{\rm T}}\xspace>16\mathrm{\>Ge\kern-1.00006ptV}\xspace to roughly select the same phase-space regime as the jet-based approaches.

We also tested an alternative anti-$k_{t}$\xspaceapproach where $\Delta\phi$ is determined from the flavor-tagged ghosts, but that has nearly identical behavior as the anti-$k_{t}$\xspaceoption tested above.

All four approaches result in similar behavior at large $\Delta\phi$, but significant differences are clearly visible in the gluon-splitting regime. The power of the FlavorConeJet method degrades at small $\Delta\phi$, because the jet momenta recoil against each other as the cones begin to overlap. The anti-$k_{t}$\xspacealgorithm suffers a precipitous drop in efficiency in the gluon-splitting regime due to jet merging. Of course, one could reduce the impact of jet merging by using a smaller jet radius, at the expense of introducing more out-of-jet radiation. Finally, because the $Q$-hadron and FlavorCone\xspacedistributions are based on the same flavor-tagged hadrons, it is not surprising that they exhibit the same qualitative features. Without any jet reconstruction, however, the $Q$-hadron method requires reconstructing the full $p_{\rm T}$\xspaceof the heavy-flavor hadrons, and not just their flight directions. Experimentally, this requirement inherently leads to a much lower flavor-tagging efficiency.

Beyond Fig. 10, we also considered two additional methods, but neither is as powerful as FlavorCone\xspace. In the spirit of stable cone algorithms, we studied an iterated FlavorCone\xspace, where the two flavor-tagged hadrons provide seed axes that are iteratively adjusted until they align with the jet momenta.171717Specifically, we start from the original FlavorCone\xspacejets. In each iteration step, we define new jet axes based on the jet momenta, and repartition the hadrons to the closest jet axis within the jet radius $R$. This process is guaranteed to terminate in a finite number of steps Stewart et al. (2015). This approach gave similar results to the nominal FlavorCone\xspace, 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 various exclusive-generalized-$k_{t}$ strategies Catani et al. (1993); Ellis and Soper (1993); Catani et al. (1991), but none worked as well as the FlavorCone\xspacealgorithm.181818Like XCone, recursive exclusive jet algorithms also ensure a fixed number of jets in the final state. In the generalized $k_{t}$ algorithm, the merging and dropping distance measures scale as $p_{\rm T}^{2q}$. We considered $q=0.5$, $1.0$, and $2.0$ in exclusive mode with $R=0.5$, where the algorithm terminates when there are exactly two undropped jets in the final state. We tried using the resulting exclusive $k_{t}$ jets directly in the analysis, as well as using them as jet axes for cone finding and as seeds for iterative stable cone finding. While exclusive $k_{t}$ does allow the jet axes to become arbitrarily close in principle, in practice there is still considerable heavy-flavor merging in the gluon-splitting regime. Considering all of these results, we advocate the FlavorCone\xspaceapproach as being best suited to studying heavy-flavor production in the gluon-splitting regime.

IV Implementation at LHCb

The parton-shower studies presented above are very encouraging, but they ignore the realities of detector-response effects. Efficient reconstruction of flavor-tagged hadrons is the most important ingredient needed to carry out these analyses, and tagging multiple heavy-flavor hadrons in close proximity is a challenge. Furthermore, both the SoftDrop\xspaceand FlavorCone\xspaceanalyses require the flight directions of the flavor-tagged hadrons to be well measured. Here, we briefly sketch a possible implementation of these methods at the LHCb detector, which has excellent heavy-flavor reconstruction capabilities.

We start with the case of $c$-hadron tagging. Charm quarks primarily hadronize into a $D^{0}$, $D^{+}$, or $D_{s}^{+}$ meson, a $\Lambda_{c}^{+}$ baryon, or any of their antiparticles. Each of these charm hadrons has at least one all-charged-particle decay with a sizable branching fraction.191919For example, the folowing decays can each be cleanly and efficiently reconstructed at LHCb: $\mathcal{B}(D^{0}\!\to\!K^{-}\pi^{+})\approx 4\%$, $\mathcal{B}(D^{+}\!\to\!K^{-}\pi^{+}\pi^{+})\approx 9\%$, $\mathcal{B}(D_{s}^{+}\!\to\!K^{-}K^{+}\pi^{+})\approx 5\%$, and $\mathcal{B}(\Lambda_{c}^{+}\!\to\!pK^{-}\pi^{+})\approx 6\%$. Therefore, a potential strategy at LHCb is to fully reconstruct one $c$-hadron, which would provide excellent momentum resolution of \sigma(\mbox{p_{\rm T}}\xspace)/\mbox{p_{\rm T}}\xspace\approx 1\% and $\sigma(\phi)\approx 2$ mrad (see App. A of LABEL:Ilten:2015hya). Combining this exclusive tag with the excellent vertex resolution at LHCb permits precise determination of the $c$-hadron impact parameter, making it possible to distinguish prompt-charm production from charm produced in $b\to c$ decays. With full reconstruction, one could choose to use the reconstructed charm hadron directly in the jet finding, instead of using it only as a ghost particle for tagging; this would mitigate track sharing between (sub)jets.

In principle, LHCb could also fully reconstruct the second $c$-tagged hadron, but the efficiency of performing exclusive reconstruction of both $c$-hadrons will be low. For charm tagging (as opposed to exclusive charm reconstruction), LHCb developed a $c$-jet tagging algorithm in Run 1 that is largely based on properties of the $c$-hadron. This algorithm achieves a $c$-tagging efficiency of 20–25%, while also providing excellent $c$–$b$ discrimination Aaij et al. (2015). Removing the two (out of ten) features that depend on properties of the jet (as opposed to the $c$-hadron) used by the LHCb machine-learning-based $c$-jet-tagging algorithm should make this tagger suitable for use in both the SoftDrop\xspaceand FlavorCone\xspaceanalyses. That said, most of the discriminating power for $c$-jet tagging comes from a single feature, the corrected mass, so a simple analysis based on secondary vertices is likely sufficient.202020The corrected mass takes the reconstructed secondary-vertex mass $m_{\rm sv}$, and adds a correction based on the momentum transverse to the direction of flight $\sqrt{m^{2}_{\rm sv}+(p_{\rm sv}\sin{\delta\theta})^{2}}+|p_{\rm sv}\sin{\delta\theta}|$, where $\delta\theta$ is the angle between $\vec{p}_{\rm sv}$ and the flight vector. Either way, the expected resolution on $\sigma(\Delta\phi_{c\bar{c}})$ is $\mathcal{O}(10\,{\rm mrad})$ Ilten et al. (2015), which is more than sufficient for both SoftDrop\xspaceand FlavorCone\xspace, given that none of the plots shown above resolve features finer than $\Delta\phi_{\rm min}\simeq 50\,{\rm mrad}$.

The only remaining issue is that of $p_{\rm T}$\xspaceordering of the charm hadrons in FlavorCone\xspace, since only the two hardest tags are used in our analysis. Only 7% of events in our FlavorCone\xspaceanalysis of Pythia\xspacedata contain more than two $c$-tagged hadrons with \mbox{p_{\rm T}}\xspace>2\mathrm{\>Ge\kern-1.00006ptV}\xspace, which means that $c\bar{c}$ events with three (or more) reconstructed $c$-hadron tags will be rare. Therefore, $p_{\rm T}$\xspaceordering in FlavorCone\xspacewill be less important than the small correction required to account for the case where one of the two hardest $c$-hadrons is not reconstructed, but the third-hardest is. This correction can be derived from data using events with (at least) three $c$-hadron tags.

We now turn to the case of $b$-hadron tagging. Unfortunately, there are no all-charged-particle $b$-hadron decays with percent-level branching fractions, making exclusive reconstruction of $b$-hadrons inefficient. Inclusive secondary-vertex-based $b$-hadron tagging, however, is highly efficient.212121Another potential strategy is to reconstruct one $b$-tag using a displaced ${{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}}\xspace\to\mu^{+}\mu^{-}$ decay, and the other one using an inclusive secondary-vertex $b$-tag. Backgrounds from $c$-hadron decays can be highly suppressed by requiring the reconstructed secondary-vertex mass to be greater than the $c$-hadron masses; a similar strategy was employed by CMS in LABEL:Khachatryan:2011wq. The flight direction is measured sufficiently well at LHCb, $\sigma(\Delta\phi_{b\bar{b}})\approx 50$ mrad, to employ this inclusive strategy in both the SoftDrop\xspaceand FlavorCone\xspaceanalyses. Events in the Pythia\xspacedata sample with more than two $b$-tagged hadrons are rare enough that improper $p_{\rm T}$\xspace-ordering of $b$-tagged hadrons will have negligible impact on the FlavorCone\xspaceanalysis.

For the relatively low-$p_{\rm T}$\xspacejets studied at LHCb, the distinction between rapidity $y$ and pseudorapidity $\eta$ is non-negligible. In the parton-shower studies above, we assumed access to full four-vector information for all particles, including the ghost tags. At LHCb, it is much easier to determine $\eta$, though one can reliably estimate $y$ since LHCb has good particle identification to infer the appropriate hadron mass value. In practice, we expect the distinction between $y$ and $\eta$ could be implemented as a simple unfolding correction. At worst, one could use a FlavorCone variant based only on $\eta$, and we checked that this does not have a large impact on performance.

Finally, we note that the SoftDrop\xspacejet-declustering analysis can be performed on fat jets clustered using only charged particles. Given that charged-particle reconstruction at LHCb is far superior to that of neutral particles, a charged-only strategy may be desirable. From a theoretical perspective, charged-only distributions can be treated using generalized fragmentation functions called track functions Chang et al. (2013a, b), appropriately adapted to treat heavy-flavor fragmentation. From an experimental tagging perspective, inclusive flavor-tagging of hadrons is already largely based on charged particle information, so there is relatively little loss of information in only using charged particles for fat jet reconstruction. Note that while $z_{g}$ itself is a dimensionless observable that is rather insensitive to the charged/neutral ratio, SoftDrop\xspacedepends on an angular-ordered clustering tree, which benefits from the improved angular resolution provided by charged particles. Either way, comparing the $z_{g}$ distributions obtained using all particles versus only charged ones will provide a valuable cross check.222222As shown in LABEL:Anderle:2017qwx, there can be significant event-by-event fluctuations in the momentum fraction $z$ when going from all particles to charged particles. But these fluctuations often have a mild impact on the final distribution, see LABEL:Chang:2013iba.

V Conclusion

In this article, we outlined two analysis strategies designed to gain a better understanding of the origin and kinematics of heavy flavor at colliders. First, we showed how SoftDrop\xspacejet declustering of a fat jet can expose the well-known but as-of-yet-unmeasured splitting kernels for heavy-flavor quarks and quarkonia in QCD. Second, we showed how the FlavorCone\xspacejet algorithm can help separate the gluon-splitting and non-gluon-splitting heavy-flavor production mechanisms. Our parton-shower studies were focused on the LHCb experiment because of its superior ability to identify and reconstruct heavy-flavor hadrons, offering the best short-term prospects for carrying out these measurements. We also expect that similar techniques can be pursued by ATLAS and CMS, especially in the higher-$p_{\rm T}$\xspacejet range.

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 already been emphasized in jet substructure studies based on subjet flavor-tagging CMS (2016); ATL (2016), which use ghost association in a similar way as our $(n_{1},n_{2})$ flavor-categorization scheme. The FlavorCone\xspacealgorithm goes one step further, using flavor-tagged hadrons to define jets, in contrast to the typical approach of first finding jets and then flavor-tagging them. Experimentally, tagging hadrons is far more challenging than tagging jets, but we think the physics opportunities justify investing in the development of hadron-based tagging strategies. Theoretically, it will be interesting to study the behavior of fixed-order QCD calculations when using these heavy-flavor categories.

Beyond just the excellent tagging performance of LHCb, the heavy-flavor strategies presented here are motivated by the need for a more robust theoretical definition of heavy flavor. In many collider-based studies, a jet with $g\to Q\bar{Q}$ will be tagged as a heavy-flavor jet, yet the physics of $g\to Q\bar{Q}$ is very different from that of $Q\to Qg$, with different underlying production mechanisms and different final-state kinematics. A similar point was emphasized in the context of flavored jet algorithms Banfi et al. (2006, 2007). By individually identifying heavy-flavor hadrons, one can more easily separate double-tag versus single-tag (sub)jets, mitigating the confusions that arise from gluon splitting to heavy flavor. We expect that the SoftDrop\xspace$z_{g}$ distributions will be helpful in validating new flavor-tagging methods, and we hope that this study inspires more sophisticated heavy-flavor categorization.

Finally, heavy-flavor production is important for stress-testing event generators. Even though all of the generators tested here are based (in principle) on the same underlying QCD splitting kernels, the differences in their distributions are substantial, especially in the gluon-splitting regime. It would be particularly interesting to see how heavy-flavor production evolves as a function of $p_{\rm T}$\xspace, since the relative importance of each contribution varies as function of jet kinematics. The measurements proposed here should inspire further precision QCD calculations of heavy-flavor production, and lead to improved parton-shower modeling of heavy-flavor physics.

VI Acknowledgements

We thank Vava Gligorov, Ben Nachman, Frank Petriello, Ira Rothstein, Gavin Salam, and Peter Skands for helpful discussions. Feynman diagrams were drawn using LABEL:Ellis:2016jkw. The work of NLR and JT is supported by the U.S. Department of Energy (DOE) under grant contract numbers DE-SC-00012567 and DE-SC-00015476. NLR is also supported in part by the American Australian Association’s ConocoPhillips Fellowship. PI and MW are supported by the U.S. National Science Foundation grant PHY-1607225.

Appendix A Additional SoftDrop\xspaceDistributions

The two natural observables to describe SoftDrop\xspacesubjets are the momentum sharing $z_{g}$, defined in Eq. (3), and the opening angle $R_{g}$. In Sec. II, we showed the $z_{g}$ distributions for our various tagging categories in Figs. 3 and 5. In this appendix, we show the analogous distributions for $R_{g}$ in Fig. 11.

For the case of $\beta=0$, the differential $R_{g}$ cross section was first calculated in LABEL:Dasgupta:2013via, where it was shown that the $R_{g}$ distribution exhibits single-logarithmic behavior for high-$p_{\rm T}$\xspace jets. For our LHCb study, however, we are working with rather low $p_{\rm T}$\xspace jets with \mbox{p_{\rm T}}\xspace>20\mathrm{\>Ge\kern-1.00006ptV}\xspace, where the distributions are largely controlled by nonperturbative effects. Indeed, the SoftDrop\xspacealgorithm often terminates at the first stage of declustering, such that $R_{g}\simeq R$, as reflected by peaks shown in Fig. 11. Given these $R_{g}$ distributions, it is in some sense surprising that the $z_{g}$ distributions in Figs. 3 and 5 show no indication for large nonperturbative corrections.

Appendix B Additional FlavorCone\xspaceDistributions

In Sec. III, we demonstrated the separation achievable between splitting and non-splitting events in Pythia\xspaceusing the FlavorCone\xspacealgorithm. The $\Delta\phi$ distribution was shown in Fig. 8, and for completeness we show the $\Delta y$, $\Delta R$, and $m_{jj}$ observables in Fig. 12. As demonstrated in Table 3, $\Delta R$ and $m_{jj}$ are effective discriminants between gluon splitting and non-splitting events, similar to $\Delta\phi$, though all three variables are highly correlated. By contrast, $\Delta y$ is not an effective discriminant because of the prevalence of back-to-back dijets from flavor creation.

Appendix C Implementation at ATLAS and CMS

While the focus of our study has been on the LHCb experiment, the SoftDrop\xspaceand FlavorCone\xspacestrategies can also be applied at ATLAS and CMS, which we refer to as general purpose detectors (GPDs). The primary experimental challenge at GPDs is achieving the required heavy-flavor-hadron tagging efficiencies, as well as associated $p_{\rm T}$\xspace and flight-direction measurements. As emphasized in Sec. IV, it is known how these challenges can be overcome at LHCb in the near term. Nevertheless, CMS has already probed small angular separations between $b$-hadrons in LABEL:Khachatryan:2011wq, so in the future, we expect the stringent heavy-flavor-tagging requirements for SoftDrop\xspaceand FlavorCone\xspacecan be met at a GPD.

To show the expected performance of our methods at a GPD, we repeat the main results from our parton-shower studies, albeit with three changes to the analysis workflow. First, we require the entire jet to be contained within the GPD acceptance of $\eta\in[-2.5,2.5]$ (instead of $\eta\in[2,5]$). Second, we increase the $p_{\rm T}$\xspacethreshold to $200\mathrm{\>Ge\kern-1.00006ptV}\xspace$ (instead of $20\mathrm{\>Ge\kern-1.00006ptV}\xspace$) to account for the typical jet scales used for triggering at a GPD. Third, we change the $p_{\rm T}$\xspace threshold for identifying heavy-flavor hadrons to be $20\mathrm{\>Ge\kern-1.00006ptV}\xspace$ (instead of $2\mathrm{\>Ge\kern-1.00006ptV}\xspace$) to account for the difficulty of GPDs to resolve low $p_{\rm T}$\xspace tracks.

The SoftDrop\xspace$z_{g}$ results for the GPD workflow are shown in Fig. 13, to be compared to the LHCb results in Figs. 3 and 5. As one goes to the higher-$p_{\rm T}$\xspace regime of the GPDs, there are a number of important differences. At the perturbative level, one expects relatively little change in the $z_{g}$ distributions as a function of $p_{\rm T}$\xspace Larkoski et al. (2015). That said, the nonperturbative effect of underlying event is very relevant at low jet $p_{\rm T}$\xspace values, where it tends to make $z_{g}$ more flatly distributed. Thus, going to higher $p_{\rm T}$\xspace in Fig. 13, the $z_{g}$ distributions exhibit stronger singularities towards $z\to 0$ and $z\to 1$ as expected. Another important difference is related to category migration. Because we are taking the $z_{\rm tag}$ requirement to be half of the $z_{\rm cut}$ requirement, there is phase space for subsequent $g\to Q\bar{Q}$ emissions to cause noticeable migration from the $(0,0)$ to the $(0,1)$ category. This was not as much of an issue at low jet $p_{\rm T}$\xspace, since the phase space for additional gluon radiation was more restricted. As one goes to higher jet $p_{\rm T}$\xspace, one could mitigate this category migration by choosing a higher $z_{\rm tag}$ requirement, at the expense of introducing shoulders in the $z_{g}$ distribution. In practice, one would probably want to make measurements of $z_{g}$ with multiple $z_{\rm tag}$ values, to test the stability of the distributions to the tagging conditions.

The FlavorCone\xspace$\Delta\phi$ distributions for the GPD workflow are shown in Fig. 14, to be compared to the LHCb results in Fig. 8. Here, the relative size of the splitting and non-splitting categories are different, with FlavorCone\xspaceproviding even greater separation than achieved at lower $p_{\rm T}$\xspace. We conclude that the GPDs should be able to successfully use these analysis strategies to distill the kinematics and rates associated with heavy flavor, probing a complementary phase space compared to LHCb.