Colour reconnections in Herwig++
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Abstract
We describe the implementation details of the colour reconnection model in the event generator Herwig++. We study the impact on finalstate observables in detail and confirm the model idea from colour preconfinement on the basis of studies within the cluster hadronization model. Moreover, we show that the description of minimum bias and underlying event data at the LHC is improved with this model and present results of a tune to available data.
Keywords
Large Hadron Collider Parton Shower Minimum Bias Underlying Event Parton Distribution Function1 Introduction
Highenergy hadronic collisions at the Large Hadron Collider (LHC) require a sound understanding of soft aspects of the collisions. All hard collisions are accompanied by the underlying event (UE) which adds hadronic activity in all phase space regions. The physics of the underlying event is similar to the physics in minimum bias (MB) interactions and very important to understand to quantify the impact of pileup in highluminosity runs at the LHC. A wide range of measurements at the Tevatron and the LHC gives us a good picture of MB interactions and the UE [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Data has also shown that a good part of the underlying event is due to hard multiple partonic interactions (MPI). By now, the three major Monte Carlo event generators Herwig [14], Pythia [15, 16] and Sherpa [17] have an MPI model implemented to simulate the underlying event.
Such a model of independent multiple partonic interactions was first implemented in Pythia [18] where its relevance for a description of hadron collider data was immediately shown. On a similar physics basis, but with some differences in the detailed modelling the jimmy addon to the old Herwig program, was introduced [19]. In these models, the average number of additional hard scatters is calculated from a few input parameters and then for each hard event the additional number of hard scatters is sampled. The individual scatters in turn are modelled similarly to the primary hard scatters from QCD 2→2 interactions at leading order, with parton shower and hadronization applied as usual. The current underlying event model in Sherpa [17] is similar but will be replaced by a new approach [20]. The current model in Pythia differs from the original development in some details and follows the idea of interleaved partonic interactions and showering [21, 22].
In the recent releases of Herwig an MPI model is also included [23]. It comes with two main parameters, the minimum transverse momentum \(p_{\perp }^{\min }\) of the additional hard scatters and the parameter μ ^{2}, that can be understood as the typical inverse proton radius squared and appears in the spatial transverse overlap of the incoming hadrons. Good agreement with Tevatron data was found with this model. Soft interactions were added to this model in order to improve consistency with more general theoretical input as the total cross section and the elastic slope parameter in highenergy hadronic collisions [24]. The distribution of transverse momenta in the nonperturbative region below \(p_{\perp }^{\min }\) was modelled similarly to the proposal in [25]. Furthermore, it is assumed that the soft partons are distributed differently from the hard partons inside the hadron. The additional parameters introduced here are fixed by requiring a description of the total cross section and the slope parameter, so we are still left with only two parameters. Once again, a good description of Tevatron data on the UE was found, now also where softer interactions play a role. The model for soft interactions smoothly extrapolates from the perturbative into the nonperturbative region, similar to a model for intrinsic transverse momentum in initialstate radiation [26].
2 Modelling colour reconnections
The cluster hadronization model [36] is based on planar diagram theory [37]: The dominant colour structure of QCD diagrams in the perturbation expansion in 1/N _{c} can be represented in a planar form using colour lines, which is commonly known as the N _{c}→∞ limit. The resulting colour topology in Monte Carlo events with partons in the final state features open colour lines after the parton showers. Following a nonperturbative isotropic decay of any left gluons in the parton jets to light quarkantiquark pairs, the event finally consists of colourconnected partons in colour triplet or antitriplet states. These parton pairs form coloursinglet clusters.
In dijet production via e ^{+} e ^{−} annihilation the invariant mass spectrum of these clusters is independent of the scale of the hard process [36, 38]. The mass distribution peaks at small values, \(\mathcal{O}(1~\mbox{GeV})\), and quickly falls off at higher masses. Descriptively speaking, the cluster constituents tend to be close in momentum space. This property of perturbative QCD is referred to as colour preconfinement, as already stated above. The invariant cluster mass largely consists of the constituent rest masses, which gives rise to a pronounced peak at the parton rest mass threshold. Hence, clusters are interpreted as highly excited prehadronic states. In the cluster hadronization model hadrons normally arise from nonperturbative, isotropic cluster decays. The Herwig implementation of this hadronization model is described in more detail in Ref. [14].
We face a similar situation if we consider multiple parton interactions in single hadron collisions. The MPI model in Herwig equips the event with a number of further QCD parton scatters, in addition to the primary partonic subprocess. For each of these subprocesses a pair of gluons, initiating the scatter, is extracted from the colliding hadrons. The chosen colour topology for this extraction corresponds to the N _{c}→∞ limit. As stated above, this limit is justified in perturbative branchings. In nonperturbative regimes, however, it is rather a QCDmotivated model than an assessable approximation.
2.1 Plain colour reconnection
 1.
Create a list of all quarks in the event, in random order. Perform the subsequent steps exactly once for every quark in this list.
 2.
The current quark is part of a cluster. Label this cluster A.
 3.Consider a colour reconnection with all other clusters that exist at that time. Label the potential reconnection partner B. For the possible new clusters C and D, which would emerge when A and B are reconnected (cf. Fig. 4), the following conditions must be satisfied:
 The new clusters are lighter,where m _{ i } denotes the invariant mass of cluster i.$$ m_C+m_D < m_A+m_B , $$(1)

C and D are no colour octets.

 4.
If at least one reconnection possibility could be found in step 3, select the one which results in the smallest sum of cluster masses, m _{ C }+m _{ D }. Accept this colour reconnection with an adjustable probability p _{reco}. In this case replace the clusters A and B by the newly formed clusters C and D.
 5.
Continue with the next quark in step 2.
2.2 Statistical colour reconnection
Clearly, it is impossible to locate the global minimum of λ, in general, since an event with 100 parton pairs, for instance, implies about 100!≈10^{158} possible cluster configurations to be tested. The Simulated Annealing algorithm from Ref. [40], however, has proven useful in solving optimisation problems like this approximately. The SCR model is an application of this algorithm with λ as the objective function to be minimised.
We would like to stress that the annealing model is used only as a numerical tool to minimize the colour length introduced above and hence give no physical interpretation to the model parameters themselves. We argue later, that merely the idea of minimizing the colour length is indeed meaningful and physical.
3 Characteristics of colour reconnection
In this section we want to study hadronizationrelated quantities which allow us to understand colour reconnection from an event generator–internal point of view. Here, a set of typical values for c, α, f and N _{steps} in the SCR model, as well as for p _{reco} in the PCR model, was used, which was obtained from tunes to experimental data, as described below in Sect. 4.
3.1 Colour length drop
With soft inclusive hadronhadron generator settings there are, generally speaking, two important classes of events. One of the two are events where there is no notable change in the sum of squared cluster masses, λ. In another large fraction of events, however, colour reconnection causes an extreme drop in λ. An obvious interpretation for this drop is that the colour reconnection procedure replaces disproportionally heavy clusters by way lighter ones.
In Fig. 5(b) we show the colour length drop in hard dijet events in pp collisions. We observe a notable decrease of large colour length drops, Δ_{if}=1, with increasing cut on the jet transverse momentum at parton level. The reason for this decrease is that higher momentum fractions are required for the hard dijet subprocess, whereas in soft events the remaining momentum fraction of the proton remnants is higher. Hence clusters containing a proton remnant are less massive in hard events, which implies less need for colour reconnection.
The distribution of the colour length drop in e ^{+} e ^{−} annihilation events looks completely different, as shown in Fig. 5(c). We find that colour reconnection has no impact on the colour length in the bulk of dijet events. We show only the Δ_{if} distribution from the SCR model here, but the PCR model yields similar results. These results confirm that due to colour preconfinement partons nearby in momentum space in most cases are combined to colour singlets already. In events with hadronic W pair decays, however, hadrons emerge from two separate colour singlets. If there is a phase space overlap of the two parton jet pairs, the production of hadrons is expected to be sensitive to colour reconnection. We address this question later on in Sect. 4.1. Here we want to remark that the fraction of WW events with nonvanishing colour length drop is slightly higher than for the dijet case. Nevertheless, the vast majority of WW events is not affected by colour reconnection, too.
3.2 Classification of clusters

The first class are the clusters consisting of partons emitted perturbatively in the same partonic subprocess. We call them htype (hard) clusters.

The second class of clusters are the subprocessesinterconnecting clusters, which combine partons generated perturbatively in different partonic subprocesses. They are labelled as itype (interconnecting) clusters.

The remaining clusters, which can occur in hadron collision events, are composed of at least one parton created nonperturbatively, i.e. during the extraction of partons from the hadrons or in soft scatters. In what follows, these clusters are called ntype (nonperturbative) clusters.
3.3 Resulting physics implications
The characteristics of clusters that have been studied in this section clearly confirm the physical picture we have started out with. The colour reconnection model in fact reduces the invariant masses of clusters that are mostly of nonperturbative origin. These arise as an artefact of the way we colourconnect additional hard scatters in the MPI model with the rest of the event.
At this nonperturbative level we have no handle on the colour information from theory, hence we have modelled it. First in a very naïve way when we extract the ‘first’ parton from the proton, but only to account for a more physical picture later, where we use colour preconfinement as a guiding principle. We therefore conclude that our ansatz to model colour reconnections in the way we have done it reproduces a meaningful physical picture.
4 Tuning and comparison of the model results with data
In this section we address the question of whether the MPI model in Herwig, equipped with the new CR model, can improve the description of the ATLAS MB and UE data, see Fig. 2. To that end we need to find values of free parameters (tune parameters) of the MPI model with CR that allow to get the best possible description of the experimental data. Since both CR models can be regarded as an extension of the cluster model [36], which is used for hadronization in Herwig, the tune of Herwig with CR models may require a simultaneous retuning of the hadronization model parameters to a wide range of experimental data, primarily from LEP (see Appendix D from Ref. [14]). Therefore, we start this section by examining whether the description of LEP data is sensitive to CR parameters.
4.1 Validation against e^{+}e^{−} LEP data
As the W bosons are produced on shell and significantly boosted at \(\sqrt{s}=189~\mbox{GeV}\), the finite W width can cause the two W bosons to travel long distances before decaying. In the limit of a very small W width, large reconnection effects between the two W systems should thus be suppressed in the model. The moderate sensitivity of the particle flow to colour reconnections implies, however, that colour reconnection effects are small in WW events. Note that also the largely vanishing colour length drop in WW events, cf. Fig. 5(c) and the discussion in Sect. 3.1, supports this conclusion. Hence we retain the described generic reconnection models also for WW events and do not introduce an extra suppression mechanism.
4.2 Tuning to data from hadron colliders
Now that we have validated the CR models by comparison against LEP data, we are ready to tune their parameters to data provided by hadron colliders. Before LHC data was available, the MPI model in Herwig [24] was tuned by subdividing the twodimensional parameter space, spanned by the model’s main parameters, the inverse proton radius squared μ ^{2} and the minimum transverse momentum \(p_{\perp }^{\min }\), into a grid. For each of the parameter points on this grid, the total χ ^{2} against the Tevatron underlyingevent data [1, 57] was calculated. A region in the parameter plane was found, where similarly good values for the overall χ ^{2} could be obtained.
While tuning the MPI models including colour reconnection we are dealing with a larger number N of tunable parameters p _{ i }, where N=4 in case of the PCR (p _{disrupt}, p _{reco}, \(p_{\perp }^{\min }\) and μ ^{2}) and N=7 in case of the SCR model (p _{disrupt}, \(p_{\perp }^{\min }\), μ ^{2}, α, c, f and N _{steps}). Hence the simple tuning strategy from above is ineffective. A comprehensive scan of 7 parameters, with 10 divisions in each parameter would require too much CPU time.
Instead, we use a parametrizationbased tune method which is much more efficient for our case. The starting point for this tuning procedure is the selection of a range \([p_{i}^{\rm min},\, p_{i}^{\rm max}]\) for each of the N tuning parameters p _{ i }. Event samples are generated for random points of this Ndimensional hypercube in the parameter space. The number of different points depends on the number of input parameters to ensure a well converging behaviour of the final tune. Each generated event is directly handed over to the Rivet package [58] to analyse the generated events. This allows the computation of observables for each parameter point, which construct the input for the tuning process. The obtained distributions of observables for each parameter variation are the starting point for the main part of the tune, which is achieved using the Professor framework [59]. Professor parametrizes the generator response to the probed parameter points. In that way it finds the set of parameters, which fits the selected observables best. The user is able to affect the tuning by applying a weight for each observable, which specifies the impact of the variable for the tuning process.
4.2.1 Tuning to minimumbias data
4.2.2 Tuning to underlyingevent data
The next important question was whether the new model is able to describe the UE data collected by ATLAS at 7 TeV [4]. The measurements are made relative to a leading object (the hardest charged track in this case). Then, the transverse plane is subdivided in azimuthal angle ϕ relative to this leading object at ϕ=0. The region around the leading object, ϕ<π/3, is called the “towards” region. The opposite region, where we usually find a recoiling hard jet, ϕ>2π/3, is called “away” region, while the remaining region, transverse to the leading object and its recoil, where the underlying event is expected to be least ‘contaminated’ by activity from the hard subprocess, is called “transverse” region. Again, we only focus on the tuning of the PCR model here. For the underlyingevent tune two observables were used: The mean number of stable charged particles per unit of η–ϕ, 〈d^{2} N _{ch}/dη dϕ〉, and the mean scalar p _{⊥} sum of stable particles per unit of η–ϕ, 〈d^{2}∑p _{ t }/dη dϕ〉, both as a function of \(p_{\perp }^{\mathrm {lead}}\), with charged particles in the kinematic range p _{⊥}>500 MeV and η<2.5.
4.2.3 Centreofmass energy dependence of UE tunes
To study the energy dependence of the parameters properly, we examine a set of observables at different collider energies, whose description is sensitive to the MPI model parameters. The experimental data should be measured at all energies in similar phasespace regions and under not too different trigger conditions. These conditions were met by two UE observables: 〈d^{2} N _{ch}/dη dϕ〉 and 〈d^{2}∑p _{ t }/dη dϕ〉, both measured as a function of \(p_{\perp }^{\mathrm {lead}}\) (with \(p_{\perp }^{\mathrm {lead}}< 20~\mbox{GeV}\)) by ATLAS at 900 and 7000 GeV (with p _{⊥}>500 MeV) and by CDF at 1800 GeV. Let us first focus on the PCR model. In this case we have four free model parameters, p _{disrupt}, p _{reco}, \(p_{\perp }^{\min }\) and μ ^{2}. For each hadronic centreofmass energy we performed independent fourdimensional tunings. Note that \(p_{\perp }^{\mathrm {lead}}\) denotes the transverse momentum of the hardest track in the case of ATLAS, whereas the CDF underlyingevent analysis uses the p _{⊥} of the leading jet, which we call \(p_{\perp }^{\mathrm {lead}}\) here, as well.
Tune values for \(p_{\perp }^{\min }\). All other model parameters, which do not depend on the c.m. energy, are summarized in Table 2
\(p_{\perp }^{\min }\) [GeV]  

\(\sqrt{s}\) [GeV]  900  1800  7000 
ueee3  1.55  2.26  2.75 
ueee3cteq6l1  1.86  2.55  3.06 
ueeescrcteq6l1  1.58  2.14  2.60 
Parameters of the energyextrapolating underlyingevent tunes. The last two parameters describe the running of \(p_{\perp }^{\min }\) according to Eq. (7)
ueee3  ueee3cteq6l1  ueeescrcteq6l1  

μ ^{2} [GeV^{2}]  1.11  1.35  1.5 
p _{disrupt}  0.80  0.75  0.8 
p _{reco}  0.54  0.61  – 
c  –  –  0.01 
f  –  –  0.21 
N _{steps}  –  –  10 
α  –  –  0.66 
\(p_{\perp ,0}^{\min }\) [GeV]  3.11  2.81  2.64 
b  0.21  0.24  0.21 
Since by construction the MPI model depends on the PDF set, we created two separate energyextrapolated tunes for the CTEQ6L1 and MRST LO** PDFs. In general, both tunes yield similar and satisfactory descriptions of experimental data.^{2} As an example see Fig. 16, in which we compare the ueee3 and ueee3cteq6l1 tunes to ATLAS UE observables, measured in all three regions (toward, transverse and away).
We repeated this procedure also for the SCR model. However, since in this case the tuning procedure was more complicated, as explained below, we concentrated on one PDF set only, namely CTEQ6L1. The first obvious complication was the larger number of parameters to tune. The second complication was associated with the fact that one of the tuning parameters, N _{steps}, is an integer number. The current version of Professor, however, does not provide such an option, instead it treats all parameters as real numbers. Therefore, we decided to carry out fifty separate tunes for different fixed values of N _{steps}, starting from 1 to 50. The last problem that we encountered, which is probably associated with the two previously mentioned problems, was that for some parameter values the predictions from Professor were significantly different from the results we received directly from Herwig++ runs. Initially, we increased the order of the interpolating polynomials from second to fourth, which should improve Professor’s predictions, but this did not improve the situation. Therefore, we first identified regions of the parameter space where this problem appeared most frequently and then excluded these from the tuning procedure. As a result, we obtained an energyextrapolated underlyingevent tune for the SCR model, which we call ueeescrcteq6l1.
In Figs. 13, 14 and 15 we show a comparison of the PCR and SCR energyextrapolated (CTEQ6L1) tunes and the ue72 tune against 〈d^{2} N _{ch}/dη dϕ〉 and 〈d^{2}∑p _{ t }/dη dϕ〉 as a function of \(p_{\perp }^{\mathrm {lead}}\) for p _{⊥}>500 MeV in all three regions (toward, transverse and away) and at three different collider energies. We can see that the quality of the data description is high and at the same level for all tunes. Nevertheless, we favour the SCR model as here we have a clearer physics picture and a more flexible model.
5 Conclusions
We have introduced two different models for nonperturbative colour reconnections in Herwig. The models are of slightly different computational complexity but give very similar results. The tuning results have shown that the SCR is preferred to have parameters that force a quick ‘cooling’ of the system and therefore results in a very similar model evolution as in the simpler PCR model. We therefore consider the PCR as a special case of the SCR model for quick cooling and keep the SCR as the more flexible model for future versions of Herwig++. As a consequence, we understand that the data demands a final state that does not obey a perfectly minimized colour length. We interpret this as a model limitation. At some point the picture of colour lines breaks down. Colour lines themselves are only a valid prescription up to leading order in the N _{ C }→∞ limit. Furthermore, the mechanism addresses the nonperturbative regime where the picture of the colour triplet charges themselves is already a model by itself and possibly completely washed out.
We have studied the mechanism of colour reconnection in detail and found that in fact the nonperturbative parts of the simulation demand the colour reconnection mechanism in order to repair the lack of information on the colour flow. The intuitive picture we have based our model on could be verified. The idea of colour preconfinement is meaningful in the context of the hadronization model and has to be rectified when a model of multiple partonic interactions is applied without further information on the colour structure in between the multiple scatters.
Furthermore, we have shown that by tuning the MPI model with CR we can obtain a proper description of nondiffractive MB ATLAS observables. We present the energyextrapolated tune ueee3, which is an important step towards the understanding of the energy dependence of the model. Finally, we have unified the different tunes of the MPI model in Herwig++ into a simple parametrization of the \(p_{\perp }^{\min }\) dependence in a way that allows us to describe data at different energies with only one set of parameters. News concerning Herwig tunes are available on the tune wiki page [50].
Footnotes
 1.
Apparently, f _{ a }(m _{cut}) is only welldefined for m _{cut} less than the maximum cluster mass. On this interval, the series (f _{ a,n }), with n the number of events taken into account, converges pointwise to the function f _{ a }. This is a more formal definition of the cluster fraction functions.
 2.
The only difference is that the CTEQ6L1 gives more flexibility in the choice of the model parameters.
Notes
Acknowledgements
We are grateful to the other members of the Herwig collaboration for critical discussions and support. We acknowledge financial support from the Helmholtz Alliance “Physics at the Terascale”. This work was funded in part (AS) by the LancasterManchesterSheffield Consortium for Fundamental Physics under STFC grant ST/J000418/1.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
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