The Partonic Origin of Multiplicity Scaling in Heavy and Light Flavor Jets

Abstract

Research shows that Koba–Nielsen–Olesen (KNO)-like scaling is fulfilled inside the jets, which indicates that KNO scaling is violated by complex vacuum quantum chromodynamics (QCD) processes outside the jet development, such as single and double parton scattering or softer multiple parton interactions. In the current work, we investigated the scaling properties of heavy-flavor jets using Monte-Carlo simulations. We found that while jets from leading-order flavor-creation processes exhibit flavor-dependent patterns, heavy-flavor jets from production in parton showers follow inclusive-jet patterns. This suggests that KNO-like scaling is driven by initial hard parton production and not by processes in the later stages of the reaction.

1. Introduction

Final-state multiplicities in small colliding systems are known to follow a negative binomial distribution (NBD) regardless of the collision species over several orders of magnitude of energy ranges [1,2,3]. It has also been observed in $e^{+}{e}^{-}$ collisions that the multiplicity distributions at different collision energies collapse into a single distribution when the so-called Koba–Nielsen–Olesen (KNO) scaling [4,5] is applied. However, KNO scaling was found to be violated at higher energies and in more complex, hadronic collision systems [6,7]. The origin of the scaling, and the reason for its breakdown, are still not completely understood, although many explanations have been proposed in the past decades [8,9,10,11,12]. In our earlier study, we found that a KNO-like scaling was fulfilled within the jets. This indicates that KNO scaling is violated by complex quantum chromodynamics (QCD) processes outside the jet development, such as single and double parton scatterings or softer multiple parton interactions (MPI) [13].

In our manuscript, we investigated the scaling properties of heavy-flavor (HF) jets in comparison to an inclusive jet sample. The heavy flavor is mostly produced in hard (large momentum-exchange) processes, in the early stages of the collision. The most relevant perturbative QCD processes that contribute to the production cross-section are leading-order (LO) flavor creation, next-to-leading order (NLO) gluon-splitting, as well as flavor excitation [14]. The parton shower and fragmentation of heavy-flavor jets are different from light-flavor jets due to two main reasons: the color charge effect; that is, heavy flavor jets are initiated by quarks as opposed to light-flavor jets that are mostly gluon-initiated [15]; and the dead-cone effect, meaning that small-angle gluon radiations off a massive parton are forbidden in QCD, and as a consequence, heavy-flavor fragmentation is harder and results in different jet substructures [16,17,18,19].

In the current work, we modeled both light and heavy-flavor jets using the PYTHIA 8 Monte-Carlo event generator [20], and we differentiate the samples according to the process the jets are created in. Whether the KNO-like scaling observed for inclusive jets is retained or violated in heavy-flavor jets can shed light on the origin of the scaling itself, and also on possible mechanisms that are responsible for the violation of the scaling in heavy-flavor jets. The methods we present can further be used to gain insight into the flavor-dependent evolution of the jets. Future measurements targeted on the scaling of light and heavy-flavor jets can also serve as validation tools for heavy-flavor production and fragmentation models.

2. Analysis Method

We simulated proton–proton collisions at $\sqrt{s}$$=7$ TeV center-of-mass energy utilizing the PYTHIA 8 (version 8.226) Monte-Carlo event generator with the Monash tune and HardQCD settings [20,21]. PYTHIA 8 is tuned to describe both the fundamental physical observables of the leading hard process and the underlying event, and it is known to reproduce final-state multiplicities well [22,23]. In PYTHIA 8, the hard parton scatterings and decays are simulated using LO matrix elements (ME). These are amended by initial and final state radiations, which create the parton shower (PS) in perturbative QCD calculations based on Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) splitting kernels [20], as well as soft and hard multiple parton interactions, integrated into a single framework [24]. The hadronic final state is then produced with the Lund string fragmentation model [25].

Using the possibility in PYTHIA 8 to restrict event generation to certain hard processes, we created four different sets of data. As the baseline for our study, we used an inclusive-jet sample, where any hard QCD scattering process was allowed above an appropriately selected value of the minimal momentum transfer in the hardest process ($\widehat{p_{\mathrm{T}}}$) depending on the jet transverse momentum ($p_{\mathrm{T}}^{\mathrm{jet}}$), as detailed in [26]. Next, we used samples with ME flavor creation, where hard $2\to 2$ parton scatterings were allowed only with heavy-flavor outgoing partons: $gg\to b\overline{b}\left(c\overline{c}\right)$ and $q\overline{q}\to b\overline{b}\left(c\overline{c}\right)$. This provided wide-angle heavy-flavor jets created directly in the leading process of the event. Finally, we created a sample that is dominated by b-jets from the PS, by allowing only those $2\to 2$ processes that do not directly create heavy flavor: $gg\to gg$, $gg\to q\overline{q}$, $qg\to qg$, $q\overline{q}\to gg$, $q\overline{q}\to q^{\prime }\overline{q^{\prime }}$ (where incoming HF is allowed, but only light flavor exits), and finally three more processes: $q\overline{q^{\prime }}\to q{q}^{\prime }$, $q\overline{q^{\prime }}\to q\overline{q^{\prime }}$ and $\overline{q}\overline{q^{\prime }}\to \overline{q}\overline{q^{\prime }}$ (where outgoing and incoming flavors are the same and q and $q^{\prime }$ may be of the same flavor) [27]. In this case, the heavy quark pair is produced in a later step, e.g., in a $g\to b\overline{b}$ gluon-splitting process, typically with smaller opening angles. The heavy quarks then often manifest in the final state as secondary jets (besides the leading jet), or may even end up in the same jet.

In all cases, charged-particle jets were clusterized from final-state charged pions, kaons, and (anti)protons with $p_{\mathrm{T}}$$>0.15$ GeV using the anti-$k_{\mathrm{T}}$ jet-clustering algorithm [28] with a resolution parameter of $R=0.7$ in the mid-rapidity range $\left|\eta \right|<1$ and full azimuth coverage. The reconstructed jets were categorized in 20 different $p_{\mathrm{T}}^{\mathrm{jet}}$ ranges, from 15 to 400 $\mathrm{GeV}$. Regarding the charm and beauty jet samples, the corresponding heavy quark was required to fall within the cone of the selected jet, similarly to jet-tagging methods that are utilized in the experiment [29,30].

3. Results and Discussion

In Figure 1, we plot the mean and the root mean square (RMS) values of the event multiplicity (N) distributions at central pseudorapidity ($\left|\eta \right|<1$), in the function of $p_{\mathrm{T}}^{\mathrm{jet}}$, separately for inclusive jets, b-jets, and c-jets from ME flavor creation, as well as for b-jets from parton shower processes. As one expects, events having jets with a higher $p_{\mathrm{T}}^{\mathrm{jet}}$ contain more final-state hadrons on the average, and the distribution also becomes broader toward higher $p_{\mathrm{T}}^{\mathrm{jet}}$. Heavy-flavor jets from ME flavor creation correspond to a lower average multiplicity at a given $p_{\mathrm{T}}^{\mathrm{jet}}$, while heavy-flavor from the parton shower follows the trend of inclusive jets. The difference is especially prominent for higher $p_{\mathrm{T}}^{\mathrm{jet}}$.

As a next step, we fitted the multiplicity distributions with a negative binomial distribution function in each of the jet transverse momentum ranges,

$$P_N=\frac{\Gamma (Nk+a)}{\Gamma \left(a\right)\Gamma (Nk+1)}{p}^{Nk}{(1-p)}^a,$$

(1)

where a, k, and p are parameters related to the mean and dispersion of the distribution of the multiplicity N. In Figure 2, we show the multiplicity distributions after all the $p_{\mathrm{T}}^{\mathrm{jet}}$ ranges have been scaled on top of each other using the NBD fits. The scaling approximately holds for all four jet samples we investigated. However, for jets containing charm or beauty from flavor creation, the data show minor departures from the NBD fits: the distribution is wider for larger $p_{\mathrm{T}}^{\mathrm{jet}}$, while narrower for smaller $p_{\mathrm{T}}^{\mathrm{jet}}$ values.

To quantify the deviations from the scaling behavior and to mitigate the effects of fluctuations, we calculated the higher moments of the multiplicity distributions in a similar manner to [13]. Here, the qth moment in a given $p_{\mathrm{T}}^{\mathrm{jet}}$ window is defined as

$$\langle N^q\rangle =\sum _{N=1}^{+\infty }{P}_N{N}^q,$$

(2)

where $P_N$ is the probability distribution corresponding to the event multiplicity N. If the scaling is fulfilled and the mean of the distribution scales with a factor $\lambda$, then it is expected that the $q^{th}$ moment scales with ${\lambda }^q$ as

$$\langle N^q\left(p_{\mathrm{T}}^{\mathrm{jet}}\right)\rangle ={\lambda }^q\left(p_{\mathrm{T}}^{\mathrm{jet}}\right)\langle N^q\left(p_0\right)\rangle ,$$

(3)

where $p_0$ is chosen so that the scaling factor is $\lambda \left(p_0\right)=1$. In Figure 3, we show the first nine moments of the multiplicity distributions divided by their order q in the function of the mean charged-hadron multiplicity $\langle N\rangle$ at $\left|\eta \right|<1$, on a log–log scale. The four panels correspond to the four jet categories. The linear fits show a similar trend for all four cases, which means that the scaling is also present for heavy-flavor jets.

Figure 4 summarizes the slopes of the fits for the first nine statistical moments, as well as the goodness-of-fit parameter ${\chi }^2/NDF$. All slopes are around unity within ≈5%. As expected, the goodness of linear fit is worse for higher moments. The b-jets coming from parton shower processes follow the same trend as the inclusive jets. On the other hand, heavy-flavor jets from matrix element production in the simulations correspond to slope parameters that are slightly but significantly different from unity: in the case of charm, slopes of the fits for moments $2\le q\le 6$ tend to be lower than unity, while in the beauty case the slopes for moments $q\ge 7$ are larger than unity. Furthermore, the goodness of fit for HF ME production tends to be worse than for inclusive jets, ${\chi }^2/NDF\ge 10$ for any $q\ge 5$. This suggests that the KNO-like scaling originates from the hard parton production, and it is less influenced by the parton shower. The similar patterns of the inclusive jets and the b-jets from PS also indicate that event multiplicities are not driven by flavor-dependent jet fragmentation processes.

4. Conclusions

In this manuscript, we summarize our results on the scaling properties of heavy-flavor jets from different production processes, and compare them to those on inclusive jets. We used PYTHIA 8 simulations to evaluate the charged-hadron event multiplicities at central pseudorapidity, in the function of the charged-particle jet transverse momentum within the range $15<p_{\mathrm{T}}^{\mathrm{jet}}<400$ GeV/c. We found that the multiplicity distributions satisfied a KNO-like scaling with $p_{\mathrm{T}}^{\mathrm{jet}}$ for charm and beauty jets similar to what has been observed for inclusive jets. However, we note that multiplicity distributions in events with jets initiated by charm and beauty directly from the leading hard process show some departure from the negative binomial shape, depending on the $p_{\mathrm{T}}^{\mathrm{jet}}$. Further analysis of the statistical moments of the multiplicity distributions shows that the scaling is fulfilled within ≈5% throughout the full $p_{\mathrm{T}}^{\mathrm{jet}}$ range, but the deviations are more significant for leading-order heavy flavor creation, especially in the case of beauty. On the other hand, beauty production from the parton shower tends to deviate less from scaling expectations and follows the inclusive-jet trend within uncertainties. Therefore, we conclude that the KNO-like scaling originates from the parton level of the early stages of the collision, and not from the later stages of the parton shower or jet fragmentation.

A good description of hadron multiplicity distributions is a basic requirement for models and it is generally fulfilled by the most widely used event generators. However, multiplicities in the function of the jet momentum for jets tagged with different flavors can provide means to further validate heavy-flavor production and fragmentation models. Moreover, while event multiplicity is a good proxy for jet multiplicity in the case of jets coming from the leading hard process, this is not necessarily the case for jets that come from secondary hard processes or gluon radiation. Therefore, an interesting extension of the current work in this direction could be to evaluate the scaling in terms of the jet multiplicity instead of event multiplicity, and to see whether (in that case) the scaling of heavy flavor jets from the parton shower follows light or heavy jets.