Colour Reconnection from Soft Gluon Evolution
Abstract
We consider soft gluon evolution at the amplitude level to expose the structure of colour reconnection from a perturbative point of view. Considering the cluster hadronization model and an universal Ansatz for the soft anomalous dimension we find strong support for geometric models considered earlier. We also show how reconnection into baryonic systems arises, and how larger cluster systems evolve. Our results provide the dynamic basis for a new class of colour reconnection models for cluster hadronization.
KATP232018
UWTHPH201823
HERWIG201802
MCnet1822
1 Introduction
Multipurpose Monte Carlo event generators (MCEG) Bahr:2008pv (); Sjostrand:2006za (); Sjostrand:2014zea (); Gleisberg:2008ta () play a central role for experimental analyses and phenomenological investigations by simulating particle collisions at a realistic level, including the full complexity and several orders of magnitude difference in relevant energy scales. Starting from a socalled hard scattering at a large momentum transfer, and possibly multiple partonic interactions in hardonic collisions, subsequent radiation in a parton shower is building up the first level of jet substructure and as such provides the first and foremost input to predicting the physical behaviour of observables. These multiple emissions do account for leading logarithmic contributions at all orders of the strong coupling, which can be systematically addressed within analytic approaches to the resummation of the QCD perturbative series either by the direct analysis of scattering amplitudes and cross sections, or by means of effective field theory methods.
As the typical scales or interparton separations reach small scales of the order of , perturbative evolution needs to stop and phenomenological models are used to describe the transition of the partonic ensemble into the observed hadrons. These models take into account nonperturbative corrections, which are included in the analytical approach by means of nonperturbative shape functions, and typically interpret lowmass partonic systems as excited hadronic systems which then break up into stable or unstable hadrons. This applies both to the string hadronization Andersson:1983ia (); Fischer:2016zzs () as well as cluster hadronization paradigms Webber:1983if () employed in LHCage MC event generators. The physical picture behind these models is that at the end of the perturbative evolution colour charge is already confined into small phase space regions such that interpretation of excited hadronic systems does make sense, however in the complex and dense environment of hadronhadron collisions this picture can be spoiled, and is typically invalidated by the presence of multiple partonic scatters.
One therefore expects that further dynamics of exchanging colour charges between the systems are present, which will reduce the relative separations of colour singlet systems. These physics is encoded in colour reconnection models, which are crucial for the description of minimum bias and underlying event data Sjostrand:1993hi (); Lonnblad:1995yk (); Gieseke:2012ft (); Christiansen:2015yca (); Bierlich:2015rha (); Reichelt:2017hts () at hadron colliders. The lack of most models to describe baryon production at hadron colliders has also led to improved models of colour reconnection Christiansen:2015yqa (); Gieseke:2017clv ().
While the spectrum of colour singlet systems is typically already predicted by the large parton shower evolution, the colour reconnection dynamics is a genuine subleading effect which we expect to be composed of by a perturbative as well as a nonperturbative component. Though the effects have yet only been addressed within the context of nonperturbative models, work is ongoing in analysing multiparton emission dynamics beyond the leading approximation Platzer:2013fha (); Martinez:2018ffw () stemming from a detailed analysis of factorisation properties of cross sections, and the resummation of logarithmically enhanced terms. While no consistent connection of such approaches to established hadronization models has been made yet, we take – in this work – the perturbative structure as a starting point to calculate colour reconnecting effects in amplitude level evolution as an input to more constrained, and hence more predictive colour reconnection models.
This paper is organised as follows: In Sec. 2 we review the details of colour preconfinement and cluster hadronization. Following this in Sec. 3 we introduce the concept of perturbative colour evolution of a scattering amplitude, detailing how different colour structures get mixed in the (infrared) renormalised amplitude by means of a renormalization group equation in colour space. Using this as a proxy to design how a perturbatively inspired colour reconnection model would look like, we state a general algorithm in Sec. 4, where we calculate the probabilities to change from an initial colour flow to a final one by evaluating overlaps of the evolved amplitude and a target colour structure, including baryonic configurations. Sec. 5 analyses in very detail the (exactly solvable) evolution of a twocluster system in various kinematic regimes and provides an analytic backup of the dynamics which are typically considered in phenomenological approaches, before we present numerical results for systems of small clusters using a full colour flow evolution in Sec. 6. These results are then considered in Sec. 7 to isolate building blocks for a fullfledged new model of colour reconnection, the implementation of which is subject to ongoing work.
2 Preconfinement and Cluster Hadronization
The cluster hadronization model WEBBER1984492 () is an essential ingredient for Monte Carlo Event Generators such as Herwig Bellm:2015jjp () and Sherpa Gleisberg:2008ta () to convert the partons at scales of the parton shower infrared cutoff of order into observed hadrons at energy scales of order . The cluster model is based on the property of colour preconfinement AMATI197987 () which essentially states that at any scale the colour structure of the parton shower is such that colour singlet combinations of partons can be formed with an asymptotically invariant mass distribution and that this mass distribution is independent of the properties of the hard scattering process or the parton shower itself.
The colour flow of an event is determined through the parton shower, which generally uses the leading colour approximation in order to define the colour flow of a splitting. Since in the large limit HOOFT1974461 () the singlet contribution from emitting gluons is colour suppressed, they can be represented through a colour and an anticolour line, unambiguously determining the colour flow of a splitting in the parton shower evolution of a state. At the end of the parton shower evolution each coloured parton is colour connected to an anticoloured parton forming a colour singlet cluster. The properties of a cluster are uniquely defined by the invariant cluster mass
(1) 
and the kinematics and the flavours of the consitutent quarks (, ) forming the cluster. Since our analysis will primarily concentrate on the cluster level the specific flavours of the quarks are not of interest and can be neglected. Also for our purposes it is sufficient to stay in the massless parton limit in which Eq. 1 simplifies to . Mass effects of light quarks are briefly discussed in Sec. 6.4 but not investigated further.
The assignment of colour connections between quark and antiquark pairs is not without flaws. While at collisions the colour connections emerging from the parton shower do lead to an asymptotically invariant mass distribution of clusters, the situation becomes ill defined when multiple parton interactions, as they appear during hadronic collisions, are taken into account. Since it is unclear how the colour connections between different scattering centres emerges, nonperturbative models are necessary in order to rearrange the colour flow to arrive at a sensible description of data Sjostrand:1987su (). One paradigm which inspired at least the development of one model is the so called notion of a colour preconfined state which states that after the evolution of the parton shower has terminated, the colour connected partons are close in momentum space leading to a distribution of invariant cluster masses which peaks at small values dictated by the parton shower infrared cutoff. Current developments in this direction, regarding spacetime hadronization models are underway and seem to look promising for future studies FerreresSole:2018vgo ().
While the bulk of the developments in recent years has been focused more on the nonperturbative modelling side of event generation there hasn’t been much progress on finding further motivation for colour reconnection or rearrangements of colour flows from the perturbative point of view. In this paper we approach this vast topic with a perturbatively inspired evolution of the colour flow due to softgluon exchanges and analyse the properties of the resulting cluster configurations which are favoured by our perturbative Ansatz.
3 Perturbative Colour Evolution
QCD scattering amplitudes are vectors in both colour and spin space, and can be decomposed in a basis of contributing colour structures,
(2) 
where we have suppressed the helicity degrees of freedom as we are mainly interested in the colour structure. These bases are typically overcomplete, and nonorthogonal. This poses a computational, but not a conceptual constraint, and in this work we consider the colour flow basis, which at first sight has the worst scaling behaviour in terms of overcompleteness, however also exhibits the closest link to the actual flow of colour charge through a scattering amplitude, and provides us with a convenient connection to the parton shower evolution and (pre)confinement properties. We stress that this basis is not limited to considerations in the large limit, and has indeed been shown to be a convenient tool for organising fullcolour evolution at the amplitude level Platzer:2013fha (); Martinez:2018ffw (), besides its earlier use in the efficient recursive computation of treelevel amplitudes, see e.g. Maltoni:2002mq (). Specifically, colour structures in the colour flow basis can be labelled by permutations which describe how colour charge is flowing from one leg to another,
(3) 
Virtual corrections are in general both, ultraviolet and infrared divergent. These divergencies are regulated within dimensional regularization, and absorbed into renormalizing bare quantities at a given scale . While ultraviolet divergences in this way relate to the running of the strong coupling, the infrared singularities drive the evolution of a scattering amplitude and the renormalization program in this case can be used to sum large logarithmic contributions of infrared origin to all orders in perturbation theory. To be precise, we can relate the bare amplitude to the renormalized amplitude as
(4) 
where is the set of outgoing momenta, is the dimensional regularization parameter in dimensions, and is the scale at which the (infrared) renormalization has been performed. The renormalization constant is an operator in the space of colour structures and sums the infrared divergences to all orders, resulting in a finite renormalized amplitude.
By taking a logarithmic derivative of the bare amplitude with respect to we obtain an evolution equation^{1}^{1}1We consider the evolution at most to oneloop level such that no implicit dependence arises through the running of .
(5) 
where the soft anomalous dimension matrix
(6) 
encodes the residues of the divergencies contained in . At one loop, the soft anomalous dimension reads
(7) 
where for two outgoing or two incoming massless partons and and for incoming/outgoing or outgoing/incoming pairs, and in lowest order. The solutions to the evolution equation take the form
(8) 
where represents the hard scattering amplitude before the evolution.
The evolution operator, neglecting the noncusp terms as they are diagonal in colour space, and assuming that only final state, massless partons are present, is
(9) 
Here we have not chosen a fixed scale to provide the initial condition for the evolution but rather have chosen an upper limit on the integration per pair of partons, assuming that is always less than the scales , which amounts to a specific choice of hard amplitude which is already encoding logarithms of the universal hard scale and the specific mass parameters which we consider here. The reason for this splitting will soon become clear, however the main point to make here is that the evolution operator is a matrix exponential in colour space and is describing iterated soft gluon exchanges between any two legs to all orders in the strong coupling. We have illustrated this in Fig. 1.
In terms of the colour flow basis introduced earlier, the action of the evolution operator can be summarised in iterating socalled colour reconnectors Platzer:2013fha () which, once per action, will swap two indices of the permutation labelling the specific colour flow, and introduce longer sequences of transpositions when exponentiated. If the colour flows in a basis tensor can be considered to indeed represent physical colour singlet systems, than the evolution operator can be expected to be the basic object describing the physics of colour reconnection at the amplitude level. We shall use this observation as a starting point for our model investigation.
4 The General Algorithm and Baryonic Reconnections
The specific configuration we obtain from a preconfining parton evolution as discussed in Sec. 2, with a universal cluster mass spectrum, can be seen as driven by a cross section resulting from an amplitude which has been dominated by a colour structure corresponding to the assignment of clusters identified in the final state,
(10) 
In this case we assume that the logarithms of have been summed by the parton shower evolution with , effectively corresponding to a veto of radiation off dipoles with masses around the shower infrared cutoff and we view the initial step of colour reconnection as an evolution in colour space to scales of order below the initial cluster masses and the parton shower infrared cutoff. To this extend we identify , and we use
(11) 
as an Ansatz for the evolution. The starting point for colour reconnection of a cluster configuration represented through a colour structure is then to consider the overlap between the evolved amplitude and a new colour structure to constitute a reconnection amplitude,
(12) 
Here we have removed the partial amplitude for the colour flow we start to evolve, as it will only an overall normalisation which is irrelevant for the reconnection probability, which we now take to be
(13) 
where runs over all possible colour flows.
4.1 Baryonic Colour Reconnections
In Gieseke:2017clv () the concept of colour reconnection to baryonic clusters has been investigated and proven to be central to the description of baryon production at hadron colliders. In the framework of our perturbatively inspired colour reconnection we can also accommodate for such reconnections provided that there are at least three clusters, or colour flows, to be considered. It is then possible to associate a baryon/antibaryon pair to a colour structure which has been suitably antisymmetrized in three fundamental and three antifundamental indices
(14) 
The normalisation constant is taken to reproduce the normalisation of a single mesonic configuration,
(15) 
This allows us to define a baryonic reconnection amplitude
(16) 
where denotes the permutation with the colour and anticolour indices corresponding to the baryonic system removed,
(17) 
The generalised reconnection probability is then
(18) 
with
(19) 
We also consider the possibility of evolving an already existing Baryon, for which we introduce ’unbaryonizing’ reconnection amplitudes
(20) 
These allow us to quantify how relevant such an evolution step would be for a highmass baryonic system, which would not have entered the reconnection dynamics any more in the case of the models considered before.
5 A TwoCluster Sandbox
The goal of this Section is to gain an analytical insight into colour reconnection from soft gluon evolution. In order to do so we study the simplest possible situation of the evolution of a two cluster system. In this case there are just two possible colour flows and according to Eq. 11 the evolution of the single state can be expressed in the following way:
(21) 
In that case we can provide an explicit expression for the exponent of the evolution operator,
(22) 
where . However, as will be discussed at the end of Section 6.4 the numerical investigation showed that the Coulomb term in has negligible effect on the massless cluster evolution, which we do not consider in this section for simplicity. We obtain the compact form for in terms of the cluster masses or the partons fourmomenta:
(23) 
The evolution is governed by the exponential of the matrix which has the structure
(24) 
where we introduced new variables: , , and . Let us now determine a reconnection probability when the initial colour flow is :
(25) 
We remind the reader that we work in the basis in which the scalar products are nonorthogonal states and therefore we have and . Hence can be expressed in terms of evolution matrix in the following way:
(26) 
where are matrix elements of . We will now explore properties of . The minimal and maximal mixing is obtained at (with ) or () respectively, where most of the reconnection values are encountered for extremal cluster mass configurations: In the case when the condition can be translated to the kinematical situation when masses of final cluster and fulfil the following equality:
(27) 
On the other hand in the case the value of is obtained when
In the limit when is large^{2}^{2}2Already for the value of and for it is equal to the condition above is much simpler:
Since we assume that both and are not equal to at the same time, the condition above is only fulfilled when
which is also consistent with the numerical results presented in Fig. 15. In general the reconnection probability is bigger than the probability that the system does not change when:
Finally, let us cast some light on the rapidity dependence of the result. In order to do it we will work in the frame when particles with fourmomenta and are backtoback, i.e. and , then for we express the fourmomenta in the following way: . Then the scalar products have the simple form:
and the condition for the minimal mixing from Eq. 27 can be rewritten as
(28) 
such that, when and , meaning that the initial partons in clusters have similar transverse momenta and are close in the rapidity, the reconnection mixing is minimal. Therefore, more likely will be reconnections when the values for the quark antiquark pairs of the original clusters are big, which is also confirmed numerically Fig. 5. It is also interesting to see the transverse momentum dependence of the result from Eq. 28 which we plan to investigate in the future while constructing a phenomenological model based on the current studies.
6 Numerical Results
A significant difference towards other colour reconnection models Gieseke:2012ft (); Gieseke:2017clv () is that we do not directly compare clusters and then choose a configuration that would leave us with prespecified properties such as a lower invariant cluster mass. Since we calculate the probabilities to evolve into different colour flows we first show that our approach leads to reasonable results compatible with the effects of conventional colour reconnection algorithms. In order to analyse the effect of the colour reconnection we mostly compare kinematic variables associated to the clusters before and after reconnection. We first consider ‘mesonic‘ reconnections and later proceed to include ‘baryonic’ reconnections as outlined in Sec. 4.1. We generate initial cluster configurations using the RAMBO method Kleiss:1985gy (), and a variation of the Jadach algorithm JADACH1975297 (), which was used for the UA5 model ALNER1987445 (). While RAMBO is performing a flat phase space population, including a cluster configuration which would not be expected from a preconfining shower evolution, the UA5 algorithm provides us already with a very physical mass spectrum.
6.1 Mesonic reconnections
A known issue which concerns the modelling of LHC events is that it is a priori not clear how the colour connection between different scattering centres of multi parton interactions looks like. The clusters emerging from these interactions are in general too heavy meaning that they consist of quark antiquark pairs which are not close in momentum space, in terms of a small invariant mass of the pairs. In this case colour reconnection models are used to restore the notion of a colour preconfined state leading to a shift towards lower invariant cluster masses. In Fig. 2 we show the invariant mass distribution for five clusters before and after colour reconnection where the phase space was sampled with the two phase space algorithms mentioned above, RAMBO and UA5type.
While for the RAMBO kinematics the invariant mass distribution gets shifted towards lower values the clusters generated with the UA5 model already consist of quark antiquark pairs close in momentum space, which leads to clusters with the smallest invariant mass possible. Considering different colour flows will eventually connect quark antiquark pairs well separated in rapidity leading to heavier clusters as seen in the right plot of Fig. 2. When sampling the cluster kinematics with the UA5 model but enforcing random colour connections between the quarks and anti quarks for the initial configuration the resulting clusters are large and (in rapidity span) overlapping. A sketch of this configuration is shown in Fig. 3 due to comprehensibility for the simple case of two cluster evolution. For this configuration our Ansatz for colour reconnection again chooses colour flows leading to a shift towards lower values in terms of invariant cluster mass.
The effect on the invariant mass spectrum can clearly be seen in Fig. 4, where we plotted the logarithm of the invariant cluster masses.
We conclude that the approach followed in this paper naturally prefers colour flows leading to a configuration with lower invariant cluster masses. We also stress here that the algorithm does not veto any colour flows which would lead to an increase in terms of invariant cluster mass. To get an intuitive picture of what happens on the quark level the rapidity difference, , between the quark antiquark pairs which were participating in the reconnection process is shown in Fig. 5 for the RAMBO phase space and for the UA5 model with random initial colour connections.
In both figures we see that colour flows resulting in clusters consisting of quark antiquark pairs which are closer in rapidity are clearly preferred. Again the effect is more pronounced for the UA5 model with random initial colour connections.
6.2 Baryonic reconnections
Within the context of our model a baryonic colour flow can be introduced as explained in Sec. 4.1. In Fig. 6 the average baryonic reconnection probability for both phase space algorithms is shown.
Depending on the phase space algorithm we employ to sample the initial configurations, the average baryonic reconnection probability ranges between and . The first striking observation is that the probability rises with the number of clusters considered. The more clusters in an event, the more likely it is to find a candidate for baryonic reconnection. For RAMBO kinematics the initial colour configuration has no effect on the average reconnection probability. For the UA5 phase space it strongly depends on the initial colour configuration. Since the original UA5 cluster configuration already is in a state where the quarks are colour connected to their closest neighbours in phase space the probability for reconnection into a different mesonic state is suppressed, which raises the probability to end up in a baryonic state. If the quarks are randomly connected the average baryonic reconnection probability is suppressed since the probability for mesonic reconnection is high. Also the more clusters we consider, the higher the probability is to find a candidate for baryonic reconnection. Now we proceed to study the three, four and five cluster evolution with the RAMBO phase space in detail. The distribution of the reconnection probabilities is shown in Fig. 7.
The baryonic reconnection probability tends to prefer lower values with a pronounced peak at zero. The tail towards higher values in the distribution might indicate some preferred kinematic configurations for the evolution into a baryonic state. With only one possible baryonic configuration, the three cluster evolution is convenient to analyse and to extract a kinematic dependence.
In Fig. 8 the probability to evolve into a baryonic state with respect to the sum of average values of the quarks and antiquarks that would constitute a baryonic cluster, is shown, where is defined as the distance between the constituent quarks in the plane
(29) 
and we define as
(30) 
where the subscripts , , denote the quarks(antiquarks) inside the baryonic(antibaryonic) clusters.
The median shows a rising probability with lower values which indicates that the formation into a baryonic cluster is preferred if the three quarks and the three antiquarks are close together in space. We note that the baryonic and the antibaryonic cluster can still be overlapping since we do not take the distance between them into account.
6.3 Unbaryonization
Colour reconnection algorithms that allow reconnection into a baryonic state are structured in a way that once a baryonic cluster is formed, it is not considered for any further modifications. This clearly biases the reconnection procedure but has been necessary in order to cope with the rising complexity of many cluster systems. In principle a system could evolve into a baryonic state and then evolve again into a mesonic state which in turn lowers the amount of baryonic clusters occurring in an event. In the context of our model we can study the evolution back into a mesonic state by considering unbaryonization where we start the evolution with a baryonic configuration as the initial state and calculate the probability to evolve into a mesonic cluster configuration. In Fig. 9 we show the average probabilities for unbaryonization in terms of and , where . This further adds to the intuitive picture that the probability for forming a baryonic cluster is high if the three quarks and the three antiquarks are close in momentum space. This also suggests that the colour field between quarks is enhanced if they do fly in the same direction.
In principle this could be used in a model such as Gieseke:2017clv () to decide whether a baryonic cluster should be kept or not which allows for more flexibility and may unbias reconnection algorithms.
6.4 Parameter variations and general findings
The Ansatz for the colour flow evolution, Eq. 11 depends on the two parameters and . Also the socalled Coulomb term proportional to affects the weights of the different colour flows. The reconnection probability is a dynamic quantity as it strongly depends on the kinematics of the cluster constituents before and after reconnection and the parameter which can be viewed as a cutoff parameter of the colour flow evolution in Eq. 9. In Fig. 10 we show the distribution of invariant cluster masses for four cluster evolution with two different values of and the corresponding colour length drop Gieseke:2012ft () which is defined as
(31) 
where and denote the colour length before and after colour reconnection in an event which is defined as the sum of squared invariant cluster masses
(32) 
If there is no colour reconnection and approximately vanishes. If there was quite a significant change in which indicates a big effect due to colour reconnection. The kinematics of the four clusters were sampled with the RAMBO method. In order to have more physical cluster masses we sample them with a centreofmass energy of which is closer to the cluster mass spectrum at the end of a typical shower evolution.
Comparing the four cluster evolution with the different values for we see that the lower , the more likely it is to pick a colour flow which results in a reduction of invariant cluster masses. For all values the distribution of peaks at zero and is then distributed towards the positive and negative region where the majority of the values are in the positive region indicating a reduction in terms of invariant cluster masses. Negative values of are also possible since we do not veto any colour flows which would result in higher invariant cluster masses. For the colour reconnection algorithm has the highest impact, severely shifting the distribution of invariant cluster masses towards smaller values which can also be seen for . Because can be interpreted as the cutoff parameter of the colour flow evolution if the colour flow evolves into a state of preferably low invariant cluster masses.
In Sec 6.1 we showed the effect of the algorithm on a relatively unphysical distribution of cluster masses as a simple proof of concept, that our Ansatz and our algorithm indeed produce reasonable results. In this section we study the evolution of colour flow and the behaviour of the model at different centreofmass energies with the RAMBO method where we compare with a more physical centreofmass energy of , probing a spectrum of smaller clusters. We show the reconnection probability for the case of two cluster evolution for the two different centreofmass energies with different values in Fig. 11.
While at the reconnection probability covers the whole range from zero to one, at and they are narrowly distributed around . With for small centreofmass energies the values are closer together which leads to similar reconnection probabilities in each event. This can be countered by reducing the parameter such that the ratio in the logarithm is the same as for higher energies which is also shown in Fig. 11 for the two cluster evolution with and .
With a smaller parameter the reconnection probabilities start to cover the whole range, and we observe the same behaviour for baryonic reconnections. The smaller the value of the amplitude in colour space will continue to evolve down to a much smaller scale which, in the end will result in a colour flow with preferably small invariant cluster masses. In a full model the parameter could be tuned to data in order to define the cutoff at which the evolution is bound to stop and to verify if the amplitude does indeed favour a state of small invariant cluster masses or a state which allows for more fluctuations in terms of cluster size.
Another interesting topic is the mass distributions for different centre of mass energies. In Fig. 12 we show the mass distributions of four clusters for and divided by the centreofmass energy with the possibility to produce baryonic clusters.
The distributions of the reconnected clusters are roughly the same although for there is much less room to evolve into a state of smaller cluster masses since we used . The mass distribution of baryonic clusters is also shifted between the two centre of mass energies. Since the soft anomalous dimension matrix only depends on the ratio of the invariant cluster masses and the cut off parameter , it is possible to find an energy independent prescription which should lead to the same distribution of cluster masses.
A continuation of the arguments of the logarithms towards very small cluster masses, has been considered but did not show any change in our findings and can as such be used to prevent numerical instabilities should the relevant small masses be encountered in a full model. The strong coupling parameter is clearly a direct measure of the overall reconnection strength and while we have chosen it to reflect the strong coupling at a lowish scale, , it should, in practice also be considered a tunable parameter of the model.
Until now we neglected quark masses completely, which lead to the simplified form of the invariant cluster mass. Assuming the same quark (constituent) masses for light quarks ( GeV for up and down quarks) , which are used in the cluster hadronization model, only small effects were found in terms of the mass distribution and no sizeable effects on the reconnection probabilities. For heavy quarks () more severe effects are expected but we leave this topic for a detailed study in the scope of a fullfledged model implementation. Switching off the Coulomb term does not change our findings for the highmass systems, while we see some effects for smallmass systems.
7 Towards a Full Model
Due to its complexity, the approach followed in this paper is limited to a small number of clusters and should be seen as a theoretical consideration to constrain the structure of an improved colour reconnection model. It clearly makes a full colour flow evolution unfeasible to implement on the typically large systems of clusters encountered in a typical high energy collisions. We will mainly use the insights we gained from looking at evolution of small systems to extrapolate a simplified model which could be suitable for implementation. A crucial aspect is to identify independently evolving subsystems in the cases with the highest number of clusters we did consider here. This will allow us to formulate ‘microscopic’ input to a model iterating over small numbers of clusters within a large ensemble.
The cluster configurations, being essentially colour structures in the colour flow basis, can be labelled by permutations and an important question to ask is what the minimum number of transpositions one requires to transform the initial configuration into a final, reconnected configuration. This number is directly related to the power of the number of colours when evaluating the overlap between the two colour structures, with more transpositions leading to a higher suppression. In Fig. 13 we show the number of transpositions for four and five cluster evolution where the phase space was populated with the RAMBO method.
For both cases we note the peak at one transposition, i.e. a reconnection within a two cluster system, between the initial and final state. This indicates the existence of independently evolving subsystems where the contributions from the remainder of the event are suppressed. Crucially, this indicates that colour reconnection is not simply a suppressed effect which would have indicated a much higher rate of nonreconnected systems, as well as a much steeper drop of the other reconnection dynamics with the number of transpositions. Our finding is then also indicative of a choice of evolving small subsystem out of a larger configuration as depicted in Fig. 14, where mixing with well separated clusters can actually be neglected.
Colour reconnection models implemented in event generators often rely on very simplified models in order to handle the complex structure of hadronic collisions. In an old model in the Herwig event generator, for example, reconnections would only be accepted if they allowed for a smaller mass configuration, and with a fixed probability which essentially was inferred by tuning to underlying event data. While this approach has benefits in terms of efficiency and simplicity, and has shown to provide a reasonable description of data Gieseke:2012ft () it does not take into account the full kinematic dynamics and complexity of a hadronic event.
In order to make contact with this simple model, and to highlight the fact that geometric models as well as the nontrivial kinematic dependence of our Ansatz provide a much more dynamic model, we consider the reconnection probability projected to the variable of the old model, which essentially is the ratio of the sum of cluster masses before and after reconnection. Having generated kinematics of two clusters with the RAMBO algorithm with , , and for comparison with , , to visualise the energy dependence, and plot the median of the reconnection probabilities over the ratio of the sum of invariant cluster masses, which would result from the two different possible colour flows, see Fig. 15.
The old model would here only have put a step function in place, with no further kinematic dependence present. We also stress the fact that the reconnection probability does not vanish, but saturates at if reconnection would result in a lower sum of invariant cluster masses and mounts up at if or . These limits can already be obtained from our analytic studies in Sec. 5. For both energies it is clear that the dependence of the reconnection probability on the ratio of cluster masses is a more dynamical one than a simple step function.
We therefore suggest to indeed use our findings as an input to more sophisticated reconnection models along the lines of Gieseke:2017clv ():

The analytically known reconnection probability from the evolution of two cluster systems can, with the finding of mostly independently evolving two cluster systems, be used directly to improve the model assumption of these type of reconnections.

The fact that also three cluster systems seem to be rather detached in a large ensemble can be used to supplement the baryon production mechanism in Gieseke:2017clv () with a more dynamic reconnection probability, possibly based on approximating this evolution to the first few orders in a expansion Platzer:2013fha ().

The baryonic reconnection mechanism, which has so far not considered the possibility of unconnecting baryonic clusters in an evolution picture or statistical model Gieseke:2012ft ().
All of these mechanisms are contained within our approach and that such a model will be highly predictive in the sense that with the analogue of the strong coupling and the soft scale it contains effectively two, possibly three parameters if one wants to include the number of colours , as well. We finally note that, though we have essentially been considering the cluster hadronization model in our considerations, a similar dynamics could be implemented in a string picture.
8 Conclusions and Outlook
We have studied to what extent the structure of perturbative colour evolution can be used as an input to improve or constrain existing colour reconnection models. In particular, we have analytically solved the evolution of a twocluster system, as well as numerically studied the evolution of larger systems of up to five clusters. We have found that there is indeed a highly dynamic and nontrivial rearrangement of colour structures already from a simple Ansatz using a oneloop soft anomalous dimension, which confirms earlier work on geometrically inspired reconnection models Gieseke:2017clv ().
The full evolution in colour space is, however, not feasible in a realistic model which needs to cope with several tens to hundreds of clusters. However we have found evidence that in the evolution of larger systems the bulk of the reconnection effects is isolated in small subsystems of two to three clusters which allow to build an iterative model, which can also be based on approximations of the evolution operator Platzer:2013fha (). Our framework also allows to include a probability of reconnecting baryonic clusters into mesonic systems, an important aspect which has not been considered in models of baryonic reconnection so far, but could result in a detailed balance mechanism to establish a realistic fraction of baryons for given partonic constituent dynamics.
We stress the fact that our approach should not only be considered as a motivation for improved models of nonperturbative colour reconnection but does highlight that the perturbative mixing of colour structures, mediated through virtual soft gluon exchanges, should be considered an important ingredient in new approaches to improving parton shower algorithms beyond the leading level Martinez:2018ffw (), and that these effects are in general not mediated in a probabilistic manner, or through treelevel amplitudes. However until such algorithms are fully available, and the dynamics of hadronization is understood in this context, we postpone further aspects to future work and use the findings obtained here as an improved input to existing colour reconnection models.
Acknowledgments
This work has been supported in part by the BMBF under grant number 05H15VKCCA and 05H18VKCC1. This work was also supported by the MCnetITN3 H2020 Marie Curie Initial Training Network, contract number 722104, as well as the European Union’s Horizon 2020 research and innovation programme (grant agreement No 668679), and the COST action (“Unraveling new physics at the LHC through the precision frontier”) No. CA16201. S.P. is grateful to KIT, CERN and MITP for their kind hospitality, and P.K. is grateful to Universität Wien for their hospitality, while several aspects of the present work have been addressed. P.K. also acknowledges the support received from the Karlsruhe House of Young Scientists. A.S. acknowledges support from the National Science Centre, Poland Grant No. 2016/23/D/ST2/02605 and the grant 1807846Y of the Czech Science Foundation (GACR).
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