Kinematic strangeness production in cluster hadronization
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Abstract
We present a modification to the nonperturbative strangeness production mechanisms in the MonteCarlo event generator Herwig in order to make the processes more dynamic and collective. We compare the model to a series of observables for soft physics at both LEP and LHC.
1 Introduction
The nonperturbative elements of simulating LHC events remain an active area of research in light of recent ALICE and CMS data [1, 2]. Signs of strangeness enhancement and collective effects in high multiplicity events respectively have inspired several phenomenological models, ranging from interacting strings [3, 4], to relativistic hydrodynamics [5], to tweaks to the existing multiple parton interaction mechanisms [6] and colour reconnection [7, 8] models. Monte Carlo event generators [5, 9, 10, 11] provide a useful testing ground for these models.
Arguably the most successful models of hadronization which try to reproduce strangeness enhancement in highmultiplicity events are rooted in the physics of collectivity, where the dense environment of high multiplicity events leads to more complicated systems which interact with one another. Heavy ion event generators typically prefer a hydrodynamic viewpoint, where the quarkgluon plasma acts as a perfect fluid, changing the dynamics of hadronization. Highenergy pp event generators tend to use sophisticated iterations of the more conventional proton collision techniques, such as the DIPSY rope model where several overlapping Lund strings [12] combine into a higherrepresentation colour field, which then may enhance strangeness production and may also shove each other transversely outwards, mimicking the fluid behaviour of quarkgluon plasma. Another model [13] has attempted to use a thermodynamics inspired route to string fragmentation and was able to explain a harder transverse momentum spectrum for heavier particles.
Herwig [9] has recently developed a new model for colour reconnection, where baryonic clusters were allowed to be produced in a geometric fashion [8], in an attempt to explain the results of [1]. The model was able to create heavier hadrons, and in particular more baryons, but in order to better describe the data, the nonperturbative gluon splitting mechanism was allowed to produce \(s\bar{s}\) pairs as well as the default lighter species. However, the production weight was simply set to a flat number, tuned to Minimum Bias events at the LHC. In this paper, we will mainly focus on the fundamental mechanisms of strangeness production in cluster hadronization, namely the production rate of \(s\bar{s}\) pairs during nonperturbative gluon splitting, cluster fission, and cluster decay. In doing so, we are taking the first steps to a rework of strangeness production in the Herwig hadronization phase. A full model would also need to consider colour reconnection, since this rearranges the colour topology and thus the mass distribution inside an event, affecting the scaling that we are interested in studying.
In this study, we aim to introduce a simple dynamic model of strangeness production in Herwig, in which each nonperturbative production stage uses the kinematic information of the relevant surrounding coloursinglet system. After reviewing the current mechanisms of hadronization in Sect. 2, we perform two separate tunes to a number of light strange meson observables for LEP and LHC Minimum Bias events in Sect. 3. We show that the tuned current strangeness production parameters are drastically different between the two collider types, and propose a massbased scaling for the relevant production weights in Sect. 4, comparing two different masslike measures to scale the probability. In Sect. 5, we tune our new model and compare the results with the old model in Herwig, as well as perform a comparison to the default Lund string model in Pythia [10] with the Monash tune [14]. We briefly summarize the work and possible future avenues for research in Sect. 6.
2 The Herwig Hadronization Model
Figure 1 sketches a schematic event, focusing on the final state. After generating a hard matrix element for the event, Herwig performs a parton shower, producing a number of soft and collinear partons. After the parton shower reaches \(\mathcal {O}(1)\) GeV, the hadronization phase of simulation occurs. In Herwig, the hadronization model is the cluster model [15], based on the colour preconfinement [16] property from the angularordered parton shower. A cluster can be considered to be a highly primordial, excited coloursinglet \(q{\bar{q}}\) pair.

Nonperturbative gluon splitting,

Colour reconnection,

Cluster fission,

Cluster decay to hadron pairs,

Unstable hadron decays.
The three other listed stages in hadronization are each allowed to contribute to the overall strangeness in the event, since they each produce new \(q{\bar{q}}\) pairs. We briefly recall the details of each step as presented in depth in [9].
2.1 Nonperturbative gluon splitting
Once the parton shower ends, all gluons undergo a nonperturbative splitting into \(q{\bar{q}}\) pairs. The species of the pair is determined by a given weight, e.g. in the tune from [8] the weights of up, down, and strange are 2:2:1. The default version of Herwig does not allow for strangeness production at this step, only \(u{\bar{u}}\) and \(d{\bar{d}}\) pairs. The only constraint on the gluon splitting is that the gluon mass is at least twice the constituent mass of the species in question, and the gluons are split isotropically.
After all the gluons in an event have been split, nearest neighbours in momentum space are most likely to be nearest neighbours in colour space [16], and clusters are formed from the momentumspace neighbouring \(q{\bar{q}}\) pairs, with a mass distribution decoupled from the hard scattering process that created them.
2.2 Cluster fission
2.3 Cluster decay
2.4 Herwig strangeness parameters
The Herwig parameters that control nonperturbative
strangeness production are the gluon splitting weight  SplitPwtSquark, and the cluster fission & decay weight  PwtSquark. In the original model, cluster fissioning and cluster decaying are controlled by the same parameter. The first step in our understanding of the different contributions is to disentangle cluster fission from cluster decay and introduce one additional parameter which controls the production of a \(s\bar{s}\) pair during cluster fission  FissionPwtSquark. The decay parameter remains the same.
3 Tuning of the existing model
In this section we tune the parameters for strangeness production of the existing model first to LEP and then to LHC data. Hadronization models are typically tuned to LEP data if they do not rely on ppspecific event topology, e.g. multiple parton interactions and their effects on colour reconnection, since LEP provides a clean QCD final state environment which imposes relatively strict constraints on what one’s hadronization model is allowed to do.
The tuning is achieved by using the Rivet and Professor frameworks for Monte Carlo event generators [17, 18]. In order to understand the overall effects of strangeness production on different stages of the event generation, we keep all other hadronization parameters that were previously tuned to LEP data at their default values [9, 19]. In the first tune (TUNE1), we only consider the effects of the parameters that are directly responsible for strangeness production as explained in Sect. 2.
In a second tuning attempt (TUNE2), we introduce the new parameter for the cluster fission stage. Tuning these 3 different parameters will allow us to study the phases of strangeness production during event generation and will shed light on the differences between LEP and LHC.
We note that this section is an extended part of the introduction to visualize and highlight the effects of the aforementioned different parameters and to see at which stage nonperturbative strangeness production is preferred.
3.1 LEP tuning

Mean charged multiplicities for rapidities \(y<1.0, y<1.5\) and \(y<2.0\)

\(K^0\) spectrum

Mean \(\pi ^0\) multiplicty

Mean \(K_S+K_L\) multiplicity

Mean \(K^0\) multiplicity

Mean \(\pi ^+/\pi ^\) multiplicty

Mean \(K^+K^\) multiplicity

Ratio (w.r.t \(\pi ^{\pm }\)) of mean \(K^{\pm }\) multiplicity

Ratio (w.r.t \(\pi ^{\pm }\)) of mean \(K^{0}\) multiplicity

\(K^{\pm }\) scaled momentum
Results of the parameter values for strangeness production at the different stages of the event generation (LEP). In both default Herwig and TUNE1, cluster fission and decay have the same parameter. In TUNE2, they are allowed to be different, but the tuning procedure returned equal values. In default Herwig, there is no \(g\rightarrow s\bar{s}\) option
3.2 LHC tuning

\(K^++K^\) yield in INEL pp collisions at \(\sqrt{s}=7\,\mathrm {TeV}\) in \(y<0.5\)

\(K/\pi \) in INEL pp collisions at \(\sqrt{s}=7\,\mathrm {TeV}\) in \(y<0.5\)

\(K_S^0\) rapidity distribution at \(\sqrt{s}=7\,\mathrm {TeV}\)

\(K_S^0\) transverse momentum distribution at \(\sqrt{s}=7\,\mathrm {TeV}\)
Results of the parameter values for strangeness production at the different stages of the event generation (LHC). In both default Herwig and TUNE1, cluster fission and decay have the same parameter, while in TUNE2 they are allowed to be different. In default Herwig, there is no \(g\rightarrow s\bar{s}\) option
3.3 Summary
The general approach in tuning a hadronization model is to tune the parameters to LEP data and then assume it is able to describe LHC observables as well since hadronization is assumed to factorize and should not depend on the process involved.
The main difference between LEP and LHC is the denser hadronic environment one encounters due to multiple parton interactions and therefore also the enhanced effect of colour reconnections on the distribution of final state particles. Be that as it may, we believe that the probability to produce strangeness e.g at the stage of nonperturbative gluon splitting should be a universal parameter and be independent of the process in question.
Since the data shows that clearly different parameter values are preferred at LHC and LEP the approach to have a single valued probability is not suited for the description of both LHC and LEP observables. It may capture the average effect but it does not allow for fluctuations on an eventbyevent basis. We tackle this problem by assuming that the rate at which strangeness is produced depends on the hadronic density of the immediate environment, which will be discussed in the next section.
4 Kinematic strangeness production
As mentioned above, the various splitting probabilities and weights are flat numbers tuned to data, without any considerations for the topology of a given event. In order to have a more dynamic picture, where the splitting probabilities depend on the environment, we choose to scale the weights with respect to coloursinglet masses. The mass of a coloursinglet system at a given phase of hadronization scales the probability for strangeness production up or down, depending on a characteristic mass scale for each step.
The weights in Eq. 4 are only for strangeness production, and they are relative to the production weights of up and down quarks. In the limit of a very heavy coloursinglet, the rate of producing strangeness will be the same as that of the lighter quarks, while in the lowmass limit, only the lighter quarks will be allowed to be produced. The appeal of an exponential scaling is that this model only introduces one extra parameter to the default model of hadronization in Herwig, and indeed, it does not introduce any extra parameters if one splits the fission and decay parameters. Thus we avoid a proliferation of parameters in our model, and we still have a natural mechanism to allow for eventbyevent fluctuations in strangeness production.
4.1 Nonperturbative gluon splitting
At the end of the shower, instead of immediately splitting the gluons into \(q{\bar{q}}\) pairs with the species determined by their given weights, we instead collect the various coloursinglet systems in the event, what we call preclusters. While colour preconfinement dictates that the mass distribution of clusters is independent of the hard energy scale, there are no such constraints on the masses of the coloursinglet preclusters. As shown schematically in Fig. 6, a parton shower can produce gluons and quarkantiquark pairs at a perturbative level, separating the event into a number of different preclusters with a variety of masses.
The characteristic mass scale for preclusters will unfortunately depend on the type of collider one uses. As shown in Fig. 4, there is a very broad tail for the proton colliders due to the number of preclusters that one can produce. This is a byproduct of the type of dense and complicated final state environment of high energy hadron colliders. At LEP, there are two peaks for the precluster mass distribution, one at close to 91.2 GeV, corresponding to events where there are only gluon emissions from the outgoing \(q{\bar{q}}\) legs from the hard scattering process, and very few coloursinglets fall between the two peaks, due to the simple fact that perturbative gluon splitting is suppressed compared to perturbative gluon emission.
4.2 Cluster fission and decay
At the cluster fission and cluster decay level, the coloursinglet is the cluster itself. We allow the characteristic mass scale and characteristic production probability to be different for the two phases. As shown in Fig. 5, the typical cluster masses at the cluster fission and cluster decay stages are roughly similar for both LEP and LHC, which we hope to reflect in the characteristic mass scales for the two tunes. We note that Figs. 4 and 5 are plotted without turning on the exponential scaling, which would change the mass distribution slightly, but the figures are benchmarks of the typical coloursinglet total invariant masses.
4.3 Coloursinglet masses
Results for the tuned characteristic mass scales \(m_0\), in units of GeV, of our new model using the total invariant mass of a coloursinglet object for LEP and LHC tunes respectively
Invariant mass  LEP  LHC 

Gluon Splitting  97  48 
Cluster Fission  3  22 
Cluster Decay  23  4 
5 Analysis
We first tune the 3 parameters of our massbased scaling model to the same identified strange particle yields at LEP and LHC as in Sect. 3. The new tunable parameters are MassScale (for gluon splitting), FissionMassScale, and DecayMassScale, which are defined by Eq. 4. The outcome of the tuning procedure for the relevant parameter values is shown in Tables 3 and 4 for LEP and LHC Minimum Bias, for both the total invariant mass measure and the \(\lambda \) measure.
Results for the tuned characteristic mass scales \(m_0\), in units of GeV, of our new model using our \(\lambda \) measure (defined in Eq. 5) of a coloursinglet object for LEP and LHC tunes respectively
\({\lambda }\) measure  LEP  LHC 

Gluon Splitting  72  37 
Cluster Fission  4  20 
Cluster Decay  16  10 
5.1 Discussion
The default version of Herwig did not allow for strange production during the gluon splitting stage. By allowing this process, improvements can be seen in all the considered observables. With our new model, there is a more physically motivated dynamic strangeness production mechanism at all stages of the hadronization.
The multiple parton interaction model in Herwig involves two types of subprocesses, hard and soft. Hard processes are allowed to shower and emit quarks and gluons, while soft ones produce only gluons which may not shower. These soft gluons are all colourconnected to each other and the beam remnants, resulting in a single precluster when undergoing nonperturbative gluon splitting. This type of precluster typically has a large invariant mass due to the large number of soft gluons and the isotropic nature of their momentum distribution, resulting in a high strangeness production weight for this subsystem. The resulting produced strange particles coming from these soft interactions are distributed uniformly in rapidity.
There are three key differences between the LEP and LHC environments during hadronization. Firstly, LEP has a much lower energy scale than the LHC, naturally limiting the possible distribution of coloursinglet masses at the stage of nonperturbative gluon splittings. As a result, a direct comparison between LEP and LHC in our model is not straightforward.
Secondly, while LEP and LHC simulations may have very similar cluster mass distributions, the number of clusters is far higher for the latter. Similarly, at the precluster level, LEP prefers coloursinglets that span the entire final state, as shown in Fig. 4, i.e. no perturbative gluon splittings during the parton shower. This results in the majority of events either having enhanced strangeness production or none at all, at the gluon splitting level, meaning that a flat weight at this level in hadronization can be justified for LEP runs.
Expected value of strangeness production weight of our new model in LEP events at 91.2 GeV, comparing the total invariant mass results with the \(\lambda \) measure results
\({E(w_s)}\) at LEP  Mass  \(\lambda \) 

Gluon Splitting  0.096  0.164 
Cluster Fission  0.297  0.166 
Cluster Decay  0.009  0.016 
Expected value of strangeness production weight of our new model in LHC Minimum Bias events at 7 TeV, comparing the total invariant mass results with the \(\lambda \) measure results
\({E(w_s)}\) at LHC  Mass  \(\lambda \) 

Gluon Splitting  0.555  0.571 
Cluster Fission  0.018  0.020 
Cluster Decay  0.153  0.041 
Taking the characteristic mass scales from Tables 3 and 4, we have translated these into an effective expected value for the weights for the two mass measures. For LEP events, as shown in Table 5, the total invariant mass approach prefers cluster fissioning, while for the \(\lambda \) measure, nonperturbative gluon splitting and cluster fissioning are approximately the same. It should be noted that aside from the gluon splitting weights, there is no direct translation between the kinematic picture and the old model of strangeness production, but these expected values give an idea of the average weights. For gluon splitting at LEP, the weight simply varies between 0 and the maximal value, since preclusters are predominately situated around two peaks, as shown in Fig. 4, and the value shown in Table 5 is simply half the maximal value of 0.192 in the invariant mass case, and 0.328 for the \(\lambda \) measure.
For LHC Minimum Bias events, the expected value for the weights are shown in Table 6. There is very little difference between using the two mass measures at the gluon splitting and cluster fission stages, while cluster decay is significantly suppressed when using the \(\lambda \) measure. The enormous suppression of strangeness production during the later stages of hadronization compared to the gluon splitting is almost certainly a hint that colour reconnection plays a nontrivial role in producing strange hadrons. Our new kinematic model uses a massbased scaling, but colour reconnection aims to lower the cluster masses to some local minimum, meaning that it is in direct conflict with our considerations. For LEP simulations, colour reconnection has a small effect, while in LHC simulations, colour reconnection is a vital phenomenon. Future work will study the correlations between the role colour reconnection plays and our model, in particular, varying the amount of colour reconnection that takes place in an event, and allowing baryonic clusters to form.
Our studies showed that there is virtually no quantitative difference between using the tuned invariant mass parameters and the tuned \(\lambda \) measure parameters. However, the results in Tables 5 and 6 suggest that the \(\lambda \) measure bridges the divide between the two types of collision better.
We can see that our model performs marginally better than Pythia, and significantly better than default Herwig, when trying to describe the \(K^{\pm }\) and drastically better on both counts for the \(K/\pi \) ratio yields, as shown in Fig. 10. However, in the low\(p_{\perp }\) region, both Pythia and our model overestimate the data. When using LHC Minimum Bias tuned parameters for LEP simulations, our model outperforms the default Herwig model, but Pythia describes the data better, as shown in Fig. 11.
We expect that changing nonperturbative strangeness production scaling should not change the overall eventshape observables, such as the Sphericity, and total jet broadening. We have included several of these observables from ALEPH data [20, 21] in Fig. 12, to confirm that there are only minor statistical differences between default Herwig 7 and our new scaling when one is concerned with nonspecies specific observables.
While we have not fully solved the discrepancy between the weights for LEP and LHC strangeness production, we have achieved two results: firstly, we have narrowed the gap between the weights of the two types of collision, and in particular, our model can be used with LHC Minimum Bias tuned parameters to better describe LEP data. Secondly, we have made the first steps to a more sophisticated treatment of hadronization and pair production at the lowenergy scale in Herwig.
6 Conclusion and outlook
We have introduced a threepart model that scales the probability for strangeness production during the hadronization phase of event generation in Herwig. The scaling is directly controlled by the mass of the corresponding event coloursinglet subsystem at each step. With this mechanism, we allow for greater fluctuations in the production of strange pairs on an eventbyevent basis.
We have studied the mechanism for nonperturbative strangeness production in detail and found that the current flat probability model is irreconcilable with both LEP and LHC data. A hadronization model should be able to have minimal effects on LEP simulations, but produce significant effects for LHC simulations.
After allowing a massbased scaling, and tuning the parameters to LEP and LHC data, we find that we are able to narrow the gap between the two collider types, and able to describe some observables better than the Lund string model in Pythia with the Monash tune. We also provide expected values for nonperturbative strangeness production, which capture the average values for eventbyevent fluctuations.
It should be noted that we have not considered heavier hyperons, the production of which has been shown to be increased by creating baryonic clusters at the colour reconnection stage [8]. Baryonic clusters, which are heavier by nature, would modify our model’s strangeness production rates. Understanding the interplay between our new model and colour reconnection will be left for future work.
There is still much left to understand in soft physics, but understanding the correlations created between the various models in hadronization are imperative to having more precise and useful Monte Carlo event generators.
Notes
Acknowledgements
We would like to thank Stefan Gieseke and Peter Skands for their comments on the manuscript. We would also like to thank Stefan Prestel for his helpful comments on the work and Peter Christiansen for his invaluable help understanding experimental analyses. This work has received funding from the European Union’s Horizon 2020 research and innovation programme as part of the Marie SklodowskaCurie Innovative Training Network MCnetITN3 (grant agreement no. 722104). This work has been supported by the BMBF under Grant number 05H18VKCC1.
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