String Formation Beyond Leading Colour
CCOEPP-MN-15-1LU-TP-15-16MCNET-15-09
String Formation Beyond Leading Colour
Jesper R. Christiansen , , Peter Z. Skands , : Department of Astronomy and Theoretical Physics, Lund University, S¨olvegatan 14,Lund, Sweden : Theoretical Physics, CERN, CH-1211, Geneva 23, Switzerland : School of Physics and Astronomy, Monash University, VIC-3800, Australia Abstract
We present a new model for the hadronisation of multi-parton systems, in which colour cor-relations beyond leading N C are allowed to influence the formation of confining potentials(strings). The multiplet structure of SU (3) is combined with a minimisation of the stringpotential energy, to decide between which partons strings should form, allowing also for“baryonic” configurations (e.g., two colours can combine coherently to form an anticolour).In e + e − collisions, modifications to the leading-colour picture are small, suppressed by bothcolour and kinematics factors. But in pp collisions, multi-parton interactions increase the num-ber of possible subleading connections, counteracting their naive /N C suppression. More-over, those that reduce the overall string lengths are kinematically favoured. The model, whichwe have implemented in the PYTHIA 8 generator, is capable of reaching agreement not onlywith the important (cid:104) p ⊥ (cid:105) ( n charged ) distribution but also with measured rates (and ratios) ofkaons and hyperons, in both ee and pp collisions. Nonetheless, the shape of their p ⊥ spectraremains challenging to explain. The description of hadronic final states at high-energy colliders involves a complicated cocktail ofphysics effects, dominated by QCD [1–3]. For the calculation of inclusive hard-scattering cross sec-tions, factorisation allows most of the complicated long-distance physics to be represented in the formof universal parton distribution functions (PDFs) [4], while the short-distance parts can be calculatedperturbatively. Perturbative aspects, such as hard-process matrix elements, parton showers, and decay(chains) of short-lived resonances, are generally coming under increasingly good control, due to acombination of advances: better amplitude calculations (including better automation and better inter-faces [5–12]), better parton-shower algorithms (e.g. ones based on QCD dipoles [13–19]), and bettertechniques for how to combine them (matching and merging, see [3,12,20–22] and references therein).These successes build on an extensive prior experience with perturbative approximations to QCD atboth fixed and infinite order, and the tractable nature of the perturbative expansions themselves.To describe the full ( exclusive ) event structure, however, several additional soft-physics effectsmust be accounted for, such as hadronisation, multiple parton interactions (MPI), Bose-Einstein cor-relation effects, and beam remnants. These are connected with the rich structure of QCD beyond1 a r X i v : . [ h e p - ph ] M a y erturbation theory and are vital, each in their own way, to the understanding of issues such asunderlying-event/pileup effects on isolation and accurate jet calibrations, and the interpretation ofidentified-particle rates and spectra.For these aspects, explicit calculations can only be performed in the context of simplified phe-nomenological models, constructed so as to capture the essential features of full (nonperturbative)QCD. An example relevant to this paper is the Lund string model of hadronisation [23, 24], whosecornerstone is the observation that the static QCD potential between a quark and an antiquark in anoverall colour-singlet state grows linearly with the distance between them, for distances larger thanabout 0.5 fm [25]. This is interpreted as a consequence of the gluon field between the charges forminga high-tension “string” (with tension κ ∼ / fm ), which subsequently fragments into hadrons.While the details of the string-breaking process may be complicated (the Lund model invokesquantum tunnelling to describe this aspect [23]), the first question that any hadronisation model needsto address is therefore simply: between which partons do confining potentials arise? In string-basedmodels, this is equivalent to answering the question between which partons string pieces should beformed. Traditionally, Monte Carlo event generators make use of the leading-colour (LC) approxima-tion to trace the colour flow on an event-by-event basis (see [3, 26]), leading to partonic final states inwhich each quark is colour-connected to a single (unique) other parton in the event (equivalent to aleading-colour QCD dipole [27]). Gluons are represented as carrying both a colour and an anticolourcharge, and are hence each connected to two other partons. At the level of strings, this is interpreted asgluons forming transverse “kinks” on strings whose endpoints are quarks and antiquarks [23]. Studiesat ee colliders show this to be a quite reasonable approximation in that environment, and the traditionalLund string model, implemented in PYTHIA [28–30], is capable of delivering a good description ofthe vast majority of ee collider data (for recent studies, see, e.g., [31–34]).The question of colour reconnections (CR) — broadly, whether other string topologies than theLC one could lead to non-negligible corrections with respect to the LC picture — was studied atLEP [35–42], chiefly in the context of CR uncertainties on W mass determinations in ee → W W [43],with conclusion that excluded the very aggressive models and disfavoured the no CR scenario at 2.8standard deviation [44]. The uncertainty on the W mass from this source ended up at ∆ m W ∼
35 MeV , corresponding to about 0.05%.There are strong physical reasons to think that CR effects should be highly suppressed at LEP,however. Firstly, there is a “trivial” parametric suppression of beyond-LC effects of order /N C ∼ . Secondly, the two W decay systems are separate colour-singlet systems, with a space-timeseparation of order of the inverse W width, Γ W ∼ . This separation implies that interferenceeffects between the two systems should be highly suppressed for wavelengths shorter than / Γ W , i.e.,there can be essentially no perturbative cross-talk between them. This line of argument motivatedthe phrasing of CR models that operate only at the non-perturbative level as the most physicallyreasonable [43], an observation that we shall also adhere to in the present work. Thirdly, the QCDcoherence of perturbative parton cascades implies that, inside each W (or Z ) decay system, angles ofsuccessive QCD emissions tend to be ordered from large to small [45], so that there is very little space-time overlap between the QCD dipoles inside each system. This means that, even if one were to allowto set up confining potentials between non-LC-connected partons, these would tend to correspond to larger opening angles and therefore they would have a higher total potential energy (longer strings)than the equivalent LC ones. The LC topology should therefore also be dynamically favoured overany possible non-LC ones. All these factors contribute to an expectation of quite small effects, at leastin the context of e + e − collisions.Moving to pp collisions (and using pp as a shorthand to for any generic hadron-hadron collision,including in particular also p ¯ p ones), the situation changes dramatically. Trivially, one must now in-2 h N æ T p Æ CDFPythia 8 (Def)Pythia 8 (no CR)
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Figure 1: Measurements of (cid:104) p ⊥ (cid:105) ( n Ch ) in minimum-bias events at 630 GeV [50] (a), 1960 GeV [51](b), and 7000 GeV [53] (c), compared to PYTHIA 8.175 [29] (tune 4C [56]), with and without colourreconnections switched on. (Plots from mcplots.cern.ch [55].)clude coloured initial-state partons, with associated coloured beam remnants. But more importantly,the modern understanding of the underlying event (UE) and of soft-inclusive (minimum-bias/pileup)physics in general, especially at high particle multiplicities, is that they are dominated by contributionsfrom multiple parton interactions (MPI) [46]. In a pp event that contains several MPI systems, there isa non-negligible possibility of phase-space overlaps between final states from different MPI systems.Moreover, since the MPI scattering centres must all reside within the proton radius, which is of thesame order as the transverse size of QCD strings, the initial-state (beam) jets will all “sit” right on topof each other, a situation which should affect the fragmentation especially at high rapidities. Finally,unlike the case for angular-ordered partons inside a jet, there is no perturbative principle that predis-poses colour-connected partons from different MPI or beam-remnant systems to have small openingangles; indeed a recent study [47] found that such “inter-MPI/remnant” invariant masses (denoted i -type and n -type in [47]) tend to be among the largest in the events, corresponding to a high potentialenergy in a string context, and hence with the most to gain from potential reconnections. For thesereasons, we expect qualitatively larger effects in pp collisions.There are also tantalising hints from hadron-collider data that nontrivial physics effects are presentat the hadronisation stage in pp collisions. The most important such clue is furnished by the depen-dence of the average (charged) particle p ⊥ on the particle multiplicity, (cid:104) p ⊥ (cid:105) ( n ch ) . Measurements ofthis quantity in minimum-bias events, first made at the ISR [48] and since by UA1 [49], CDF [50, 51]and the LHC experiments [52–54], reveal that (cid:104) p ⊥ (cid:105) grows with n ch , as can be seen in the plots infig. 1 (from mcplots.cern.ch [55]). This cannot be accounted for by independently hadronising MPIsystems, for which the expectation would be that (cid:104) p ⊥ (cid:105) ( n ch ) should be almost flat, as is also illustratedby the “no CR” curves in fig. 1. (If each MPI hadronises independently, then per-particle quantitiessuch as (cid:104) p ⊥ (cid:105) should be independent of the number of MPI, which is correlated with n Ch [46].) Theobservation that (cid:104) p ⊥ (cid:105) increases with n Ch therefore strongly suggests that some form of collectivehadronisation phenomenon is at play, correlating partons from different MPI systems.3iven these arguments, and the realisation [57] that precision kinematic extractions of the topquark mass at hadron colliders (see e.g., [58–64] for experimental methods and [65] for a recent phe-nomenology review) can be significantly affected by colour reconnections , several toy models haveappeared [47, 57, 68–71], relying mainly on potential-energy minimisation arguments to reconfigurethe partonic colour connections for hadronisation. Although these models have had some successin describing the (cid:104) p ⊥ (cid:105) ( n Ch ) distribution (as e.g., in fig. 1), the lack of rigorous underpinnings haveimplied that large uncertainties remain, which still contribute about a 500 MeV uncertainty on thehadronic top mass extraction [62, 65, 71]. In this paper, we take a first step towards creating a morerealistic model, combining the earlier string-length minimisation arguments with selection rules basedon the colour algebra of SU (3) .An alternative line of argument, pursued in particular in the EPOS model [72], invokes the notionof hydrodynamic collective flow to explain the (cid:104) p ⊥ (cid:105) ( n Ch ) distribution (as well as the so-called CMS“ridge effect” [73, 74] and a host of other pp observables [72]). Certainly, the presence of hydroeffects in pp is a hypothesis that, if confirmed, would have far-reaching consequences, and it will be animportant task for future experimental and phenomenological studies to find ways of disentangling CReffects from hydro ones. In this context, our paper should therefore also be viewed as an attempt to seehow far one can get without postulating genuine (pressure-driven) collective-flow effects in pp . Withinthis context, it is important to note that CR can mimic flow effects to some extent, via the creation ofboosted strings [75]. Alternatively, it is possible that the effective string tension could be rising, as inthe idea of colour ropes [76], with recent work along these lines reported on by the Lund group [77].Finally, we note that non-hydro rescattering has also been proposed [78] as a potential mechanismcontributing to the rise of (cid:104) p ⊥ (cid:105) ( n Ch ) , though the explicit model of parton-parton rescattering effectspresented in [78] found only very small effects. The possibility of Boltzmann-like elastic (or eveninelastic) final-state hadron-hadron rescattering is still open. As usual, nature’s solution is likely toinvolve an interplay of effects at different levels. Nevertheless, before exploring further effects at thehadron level, we believe it makes good sense to first examine the hadronisation process itself, whichis the topic of this work.Finally, we note that colour flows beyond LC have also been invoked in the context of J/ψ for-mation [79–82], and as a potential mechanism to generate diffractive topologies in ep and pp colli-sions [83, 84].In section 2, we briefly recapitulate the treatment of colour space for the existing MPI modelsin PYTHIA, and present the new model that we have developed, combining the minimisation of thestring potential with the multiplet structure of QCD. In section 3, we constrain the resulting freemodel parameters on a selection of both ee and pp data, discussing the physics consequences of thenew colour-space treatment as we go along. In section 4, we consider implications for precisionextractions of the top quark mass at hadron colliders. Finally, in section 5, we summarise and give anoutlook. In this section, we present the colour-space model that we have developed, which allows strings toform not only between LC-connected partons, but also between specific non-LC-connected ones,following combination rules that approximate the multiplet structure of full-colour QCD. We beginwith a brief summary of the current modelling, in section 2.1. We then turn to a general discussion of For completeness we note that, similarly to above, much smaller effects are expected in e + e − environments [66, 67]. (a) Hadron RemnantMPI 1 MPI 2 MPI 3 (b)
Figure 2: (a): in a strict LC picture, each MPI initiator gluon increases the “colour charge” of thebeam remnant by two units. (b): allowing different MPI initiators to be connected in a colour chainreduces the total colour charge of the remnants. Here for example, no strings will be stretched directlybetween the shown remnant and the final states of MPI 2. (Note that the colour assignments shownare for illustration only, and would be represented by Les Houches Colour Tags [5, 6] in a real eventgenerator.)coherence effects beyond leading N C in section 2.2. Finally, in section 2.3, we present the detailedimplementation of the new model.We emphasise that there is a conceptual difference between colour-space ambiguities , such asthose explored in this work, and physical colour reconnections . The subleading-colour effects we dis-cuss here arise naturally in “full-colour” SU (3) and do not involve any physical exchange of colour ormomentum (although explicit algorithms may of course still employ an iterative-reconnection schemeto find the potential-energy minimum). Strictly speaking, the term colour reconnections should be re-served to describe effects related to dynamical reconfigurations of the colour/string space that involveexplicit exchange of colours and momentum, via perturbative gluon exchanges or non-perturbativestring interactions. Effects of this type are not explored directly in this work, instead we refer theinterested reader to the SK string-interaction models presented in [43, 66, 67]. Somewhat sloppily, wefollow the entrenched convention in the field and use the acronym “CR” for effects of either kind here. In a naive LC picture, each MPI scattering system is viewed as separate and distinct from all othersystems in colour space. The very simplest colour-space options in the old PYTHIA 6 MPI model [46]and the first HERWIG (and HERWIG++) MPI models [85, 86] go a step further, representing eachMPI final state as two quarks (or gluons), colour-connected directly to each other, i.e., treating eachMPI system as a separate hadronising colour-singlet system. However, this ignores that the incomingpartons are coloured, and hence that the total colour charge of each MPI scattering system is in generalnon-zero. These particular models therefore violate colour conservation and are unphysical.To be LC-correct one must take into account that each MPI-initiator parton should cause one ortwo strings to be stretched to its remnant (one for quarks, two for gluons). This conserves colour,but still has the implication that no strings would be stretched between different MPI systems. Thissituation is illustrated in fig. 2a. Physically, this can lead to arbitrarily many strings being stretchedacross the central rapidity region, one or two for each MPI (corresponding to adding their total colourchargers together as scalar quantities, rather than as SU (3) vectors).However, already in the context of earlier works [46,87], it was noted that even this picture cannotbe quite physically correct. Since all the MPI initiators on each side are extracted from one and thesame (colour-singlet) beam particle, and since they are extracted at a rather low scale of order theperturbative evolution cutoff p ⊥ ∼ one to a few GeV, there is presumably some overlap and accom-5anying saturation effects, implying that they are not completely independent. Not knowing the exactform of the correlations, a pragmatic solution is to minimise the total colour charge of the remnant(and hence the number of strings stretched to it), by allowing the different MPI systems to be colour-connected to each other along a “chain” in colour space, as illustrated in fig. 2b. Variations of this areused in the current forms of both the PYTHIA [46, 87] and HERWIG++ [47] MPI models, reducingthe number of strings/clusters especially in the remnant-fragmentation region at high rapidities. It is,however, still fundamentally ambiguous exactly which systems to connect and how. In the exampleof fig. 2, it is arbitrary that it happens to be the colour of MPI 1 and the anticolour of MPI 3 whichend up connected to the remnant. For a more detailed discussion of this aspect, see e.g. [87]. Aninteresting physics point is that, in this picture, the particle production at very forward rapidities iscontrolled essentially by how large one allows the colour charge of the remnant to become, which inturn depends on the number of MPI and their mutual colour correlations. This could presumably berevealed by studies correlating the particle production in the central region (sensitive to the number ofMPI) with that in the forward region (sensitive to the total charge of the remnant).In the absence of any further CR effects, the relationship between the number of MPI and theaverage particle multiplicity at central rapidities is still approximately linear. Consequently, the per-particle spectra in high-multiplicity events (with many MPI) are similar to those in (non-diffractive)low-multiplicity events (with few MPI) . This is what leads to the simple expectation of the flat (cid:104) p ⊥ (cid:105) ( n Ch ) spectrum exhibited by the “no CR” curves that were shown in fig. 1 in the previous sec-tion. However, as was also remarked on there, the experimental data convincingly rule out such aconstant behaviour. This observation is the main reason additional non-trivial final-state CR effectshave been included in both HERWIG++ and PYTHIA.In the original (non-interleaved) MPI model in PYTHIA 6 [28, 46], the parameters PARP(85) and
PARP(86) allowed to force a fraction of the MPI final states to be two gluons colour-connectedto their nearest neighbours in momentum space. The physical picture was that the hardest interactionbuilt up a “skeleton” of string pieces, onto which a fraction of the gluons from MPI were grafted (bybrute force) in the places where they caused the least amount of change of string length. This effec-tively minimised the increase in string length from those gluons. An important factor contributing tothe revival of the question of CR in hadron collisions was the tuning studies of this model, carriedout by Rick Field on underlying-event and minimum-bias data from the CDF experiment at the Teva-tron [50,88,89]. His resulting “Tune A” and related tunes [90,91] were the first to give good fits to theavailable data at the time, but the surprising conclusion was that in order to do so this “colour-spacegrafting” had to be done nearly all the time.An alternative set of CR models, which relied on physical analogies with overlapping strings insuperconductors, were developed only in the context of e + e − collisions [43, 66, 67], chiefly with theaim of studying potential CR uncertainties on the W mass, see [92] and references therein. As faras we are aware, this class of models has not yet been applied in the context of the more complexenvironment of hadron-hadron collisions.In the new (interleaved) MPI model in PYTHIA 6 [14, 28], showers and MPI were carried outin parallel, with physical colour flows. This was too complicated to handle with the old CR model.A new “colour annealing” CR scenario was developed [57, 69, 93] which, after the shower evolu-tion had finished, allowed for a fraction of partons to “forget” their LC colour connections, with newones determined based on the string area law (shorter strings are preferred), following a simplified For very low multiplicities, well-understood bias effects cause the average particle p ⊥ to increase (if the event is re-quired to contain only one particle, then that particle must be carrying all the scattered energy), while for high multiplicities,the contribution from hard-jet fragmentation also generates slightly harder spectra. PARP(78) . A further parameter,
PARP(77) , allowed to suppress the reconnection probability forfast-moving partons. Although still intended as a toy model, the new colour-annealing models ob-tained good agreement with the Tevatron minimum-bias and underlying-event data, e.g. in the form ofthe Perugia family of tunes [94, 95]. The most recent incarnations, the Perugia 2011 and 2012 tunes,also included LHC data and were among the main reference tunes used during Run 1 of the LHC [95].However, a study comparing independent MPI+CR tunings at different collider energies revealed dif-ferent preferred CR parameter values at different CM energies [96], implying that the modelling ofthis aspect, or at least its energy dependence, was still inadequate.In PYTHIA 8, the default MPI colour-space treatment is similar to that of the original PYTHIA6 model, although starting out from a more detailed modelling of the colour flow in each MPI. Witha certain probability, controlled by the parameter
ColourReconnection:range , all the gluonsof each lower- p ⊥ interaction can be inserted onto the colour-flow dipoles of a higher- p ⊥ one, in sucha way as to minimise the total string length [71]. The effects of this model was already illustrated infig. 1. A set of alternative CR scenarios was also presented in [71], but were still mostly intended astoy models in the context of estimating uncertainties on the top-quark mass.Finally, in the most recent developments of the HERWIG++ MPI model, an explicit scenario forcolour reconnections has likewise been introduced [47], based on a simulated-annealing algorithmthat minimises (sums of) cluster masses. In the context of the cluster hadronisation model [97], theminimisation of cluster masses fulfils a similar function as the minimisation of string lengths above.The two minimisations differ in that the string length measure is closely related to the product of theinvariant masses rather than the sum used in the cluster model. The main model parameter is theprobability to accept a favourable reconnection, p reco . The study in [47] emphasised in particularthat the largest pre-reconnection cluster masses are spanned between hard partons and the remnants(denoted n -type clusters), with inter-MPI ones (spanned directly between partons from different MPIsystems and denoted i -type) having the second-largest masses. The former again indicates that thereis a non-trivial interplay with the non-perturbative hadronisation of the beam remnant, while the latterreflects the lack of a priori knowledge about the colour correlations between different MPI systems.Similarly to the qualitative conclusions made with the PYTHIA CR models, the HERWIG++ studyfound that quite large values of p reco ∼ . were required to describe hadron-collider data. To illustrate the colour-space ambiguity between different MPI systems, and between them and thebeam remnant, let us take the simple case of double-parton scattering (DPS), with all the initiatorpartons being gluons. What happens in colour space when we extract two gluons from a proton? Evenif we imagine that the two gluons are completely uncorrelated, QCD gives several possibilities fortheir superpositions: ⊗ = ⊕ ⊕ ⊕ ⊕ ⊕ . (1)In this equation, the (a “viginti-septet”) effectively represents the LC term: incoherent addition ofthe two gluons, each carrying two units of (LC) colour charge (one colour and one anticolour), for atotal of 4 string pieces required to be attached to the remnant . However, note that the probability for Assuming each string can only carry one unit of flux, or equivalently a 4-unit “colour rope”, see [77]. P LC = 2764 < , (2)hence the naive expectation that subleading topologies should be suppressed by /N C is badly bro-ken already in this very simple case. The decuplets (octets) correspond to coherent-superpositiontopologies with a lower total colour charge and consequently only three (two) string pieces attachedto the remnant. The singlet represents the special case in which the two MPI-initiator gluons haveidentical and opposite colours, with total colour charge 0 (generating a diffractive-looking topologyfrom the point of view of the remnant). In QCD, for two random (uncorrelated) gluons, there is a 1/64probability for this to happen purely by chance.The other possible two-parton combinations are: ⊗ = ⊕ ⊕ , (3) ⊗ = ⊕ , (4) ⊗ = ⊕ , (5)where strict LC would correspond to populating only the (quindecuplet), (octet), and (sextet),respectively. The relative weights (probabilities) for each multiplet in each of these combinations areillustrated in fig. 3, along with diagrams exemplifying corresponding colour flows (with thick linesindicating partons, thin ones colour-flow lines). For each multiplet, three vertical bars indicate theprobability associated with that multiplet in strict Leading Colour (LC), in our model (defined below),and in SU (3) (QCD), respectively. The filled circles represent the ratio between our model and QCD,so for those unity indicates perfect agreement. Note that, since the subleading multiplets are absent inLC, only two non-zero bars appear for them. Below, we shall also consider the probability for threeuncorrelated triplets to form an overall singlet, which is / in QCD: ⊗ ⊗ = ⊕ ⊕ ⊕ , (6)while in our simplified model it will come out to be /
81 = (1 − ) / .We emphasise that we only use these composition rules for colour-unconnected partons, whichin the context of our model we approximate as being totally uncorrelated. LC-connected partons arealways in a singlet with respect to each other, and colour neighbours (e.g., the two colour lines of agluon or those of a q ¯ q pair produced by a g → q ¯ q splitting) are never in a singlet with respect to eachother.The approximation of colour-unconnected partons being totally uncorrelated, combined with a setof specific colour-space parton-parton composition rules, such as those of SU (3) or the simplifiedones defined below, allow us to build up an approximate picture of the possible colour-space corre-lations that a complicated parton system can have, including randomised coherence effects beyondLeading Colour. Due to the subleading correlations, there are many possible string topologies thatcould represent such a parton system, including but not limited to the LC one. The selection principlethat determines how the system collapses into a specific string configuration will be furnished by theminimisation of the string potential, as we shall return to below.Our model thus consists of two stages. First, we generate an approximate picture of the possiblecolour states of a parton system. Then, we select a specific realisation of that state in terms of explicitstring connections. This is done at the time when the system is prepared for hadronisation, i.e., afterparton showering but before string fragmentation.By maintaining the structure of the (LC) showers unchanged, we neglect any possibility of re-connections occurring already at the perturbative level. Though perturbative gluon exchanges and/or8 e i g h t LCOur ModelQCDOur Model/QCD
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Figure 3: Illustration of the possible colour states of two random (uncorrelated) partons. In strictLC, only the completely incoherent superposition is populated. Our model (described below) gives asystematically better approximation. Filled circles show the ratio between our model and full SU (3) .The diagrams below the histograms attempt to illustrate corresponding colour-flow configurations,with thick and thin lines denoting partons and colour-flow lines, respectively.full-colour shower effects might mediate such effects in nature, we expect their consequences to besuppressed relative to the non-perturbative ones considered here. This is partly due to the coherenceand collinear-enhancement properties already acting to minimise the mass of LC dipoles inside eachperturbative cascade, and partly due to the space-time separation between different systems (be theydifferent MPI systems, which are typically separated by transverse distances of order / Λ QCD insidethe proton, or different resonance-decay systems separated by / Γ res ). Thus, at high Q (cid:29) Λ QCD or Q (cid:29) Γ res we don’t expect any cross-talk between different MPI or different resonance systems,respectively. The case can be made that perturbative reconnection effects could still be active atlonger wavelengths, but we expect that such semi-soft effects can presumably be absorbed in thenon-perturbative modelling without huge mistakes.9 A) ¯ q (cid:48) q (cid:48) q g ¯ q vs. (B) ¯ q (cid:48) q (cid:48) q g ¯ q Figure 4: Illustration of a multi-parton state with a rather simple colour-space ambiguity. Subscriptsindicate colour-space indices. (A): the “original” (LC) string topology. (B): an alternative stringtopology, allowed by the accidentally matching “2” indices.
Our simplified colour-space model is defined as follows. Rather than attempting to capture the full cor-relations (which we have emphasised are not a priori known anyway and would require a cumbersomematrix-based formalism), we note that the main subleading parton-parton combination possibilities ofreal QCD can be encoded in a single “colour index”, running from 1 to 9 (with corresponding indicesfor anti-colours).Quarks are assigned a single such colour index, antiquarks a single anticolour index, and gluonshave one of each, with the restriction that their colour and anticolour indices cannot be the same. Thus,formally our model has 9 different quark colour states and 72 kinds of gluon states. We emphasisethat these indices should not be confused with the ordinary 3-dimensional SU (3) quark colour indices(red, green, and blue); rather, our index labels the possible colour states of two-parton (and in somecases three-parton) combinations . Thus, for example, a quark and an antiquark are in an overallcolour-singlet state if the colour index of the former equals the anticolour index of the latter, otherwisethey are in an octet state, cf. eq. (4). We note that a similar index was used already in the models of“dipole swing” presented in [98, 99], though here we generalise to parton combinations involvingcolour-epsilon structures as well, cf. fig. 3.Confining potentials will be allowed to form between any two partons that have matching colourand anticolour indices. Since LC-connected partons are forced to have matching colour and anticolourindices, the “original” (LC) string topology always remains possible, but now further possibilities alsoexist involving partons that accidentally have matching indices, illustrated in fig. 4.Furthermore, two colour indices are allowed to sum coherently to a single anticolour index withinthree separate closed index groups: [1,4,7], [2,5,8], and [3,6,9]. E.g., two quarks carrying indices 2and 5 respectively, are allowed to appear to the rest of the event as carrying a single combined anti-8 index. These index combinations represent the antisymmetric ε ijk colour combinations that werepictorially represented as Y-shaped “colour junctions” in fig. 3. An explicit example of a parton systemwhose colour state includes such a possibility is shown in fig. 5. A model for string hadronisation ofsuch topologies was developed in [100] and has subsequently also been applied to the modellingof baryon beam remnants [87]. We reuse it here for hadronisation of junction-type colour-indexcombinations.The new model can be divided into two main parts: a new treatment of the colour flow in the beamremnant and a new CR scheme. The two models are independent and can therefore in principle becombined both with each other as well as with other models. (Note, however, that the old PYTHIA8 CR scheme is inextricably linked with the colour treatment of the beam remnant and thereforeonly works together with the old beam-remnant model.) Both of the models occasionally result in10 C) q (cid:48) ¯ q (cid:48) q g ¯ q vs. (D) q (cid:48) ¯ q (cid:48) q g ¯ q Figure 5: Illustration of a similar multi-parton state as in fig. 4, but now with index assignmentsresulting in a junction-type colour-space ambiguity. The orientation of the top q (cid:48) − ¯ q (cid:48) dipole hasalso been reversed relative to fig. 4. (C): the original (LC) string topology. (D): an alternative stringtopology with a junction and an antijunction, allowed by the cyclically matching “2” and “5” indices.complicated multi-junction configurations that the existing PYTHIA hadronisation cannot handle.Rather than attempting to address these somewhat pathological topologies in detail, this problem iscircumvented by a clean-up method that simplifies the structure of the resulting systems to a level thatPYTHIA can handle.The next two sections describe respectively the details of the new beam-remnant model and thenew CR scheme, including technical aspects and the algorithmic implementation. Afterwards thejunction clean-up method is described. It being an inherently non-perturbative object, we do not expect to be able to use perturbative QCDto understand the structure of the beam remnant. Instead, we rely on conservation laws; the partonsmaking up the beam remnant must, together with those that have been kicked out by MPI, sum upto the total energy and momentum of the beam particle, be in an overall colour-singlet state, withunit baryon number (for a proton beam), carrying the appropriate total valence content for each quarkflavour, with equal numbers of sea quarks and antiquarks. The machinery used to conserve all thesequantities should be consistent with whatever knowledge of QCD we possess, such as the standardsingle-parton-inclusive PDFs to which our framework reduces in the case of single-parton scattering.In this work the focus is on the formation of colour-singlet states, including the use of SU (3) epsilon tensors. This naturally leads to a modification of the treatment of baryon number conser-vation, due to the close link between baryons and the epsilon tensors in SU (3) . The modelling ofenergy/momentum and flavour conservation is not touched relative to the existing modelling of thoseaspects, and thus only a small review is presented here (for more details see [87]).The overall algorithm can be structured as follows:1. Determine the colour structure of the already scattered partons.2. Add the minimum amount of partons needed for flavour conservation.3. Add the minimum amount of gluons required to obtain a colour-singlet state.4. Connect all colours.5. With all the partons determined find their energy fractions.The conservation of baryon number is not listed as a separate point, but naturally follows from theformation of junctions. Let us now consider each of these points individually starting from the top.11 a) Q (cid:29) Λ (b) Q ∼ Λ Figure 6: A proton (black) with five distinct colour sources (e.g., four MPI-initiator partons plusone representing the beam remnant), chosen so that they add to a singlet (with magenta = antigreen).Shown are two different resolution scales representative of (a) the perturbative stage, during which theMPI systems are considered as being uncorrelated in colour space, and (b) the nonperturbative stage,at which the beam remnant is considered and we assign higher weights to states with lower total QCDcharge in order to mimic saturation effects.To calculate the colour structure of the beam remnant, let us return to the DPS example of earlier.With a probability of / , the two gluons form a completely incoherent state, leaving four colourcharges to be compensated for in the beam remnant (two colours and two anticolours). However thethree valence quarks alone are insufficient to build up a (eq. (6)), and therefore a minimum of oneadditional gluon is needed. Then two of the quarks can combine to form a , which can form an with the remaining valence quark, which then can enter in a with the added gluon. Conversely, ifthe two gluons had been in an octet state instead, the additional gluon would not have been needed.Thus, in order to determine the minimal number of gluons needed in the beam remnant, we need toknow the overall colour representation of all the MPI initiators combined.While it could be possible to choose this representation purely statistically, based on the (simpli-fied or full) SU (3) weights, we note two reasons that a lower total beam-remnant charge is likely to bepreferred in nature. Firstly, to determine the most preferred configuration the string length needs to beconsidered. Since the beam remnants reside in the very forward regime, strings spanned between theremnant and the scattered gluons tend to be long, and as such a good approximation is to minimise thenumber of strings spanned to the beam. This corresponds to preferring a low-charge colour-multipletstate for the remnant, and as a consequence also minimises the number of additional gluons required.Secondly, a purely stochastic selection corresponds to the assumption that the scattered partons are un-correlated in colour space. For hard MPIs (at Q (cid:29) Λ QCD ), this is presumably a good approximation,since the typical space-time separation of the collisions are such that two independent interactions donot have time to communicate. This is illustrated by fig. 6a. However, after the initial-state radiationis added, the lower evolution scale implies larger spatial wavefunctions, allowing for interference be-tween different interactions, illustrated by fig. 6b. An additional argument is that at a low evolutionscale the number of partons is low, thus to combine to an overall singlet the correlation between thefew individual partons needs to be large. To provide a complete description of this cross-talk, multi-parton densities for arbitrarily many partons would be needed, ideally including colour correlationsand saturation effects. Although correlations in double-parton densities has been the topic of severalrecent developments [101–109], the field is still not at a stage at which it would be straightforward12o combine explicit double-parton distributions with the standard single-parton-inclusive ones (whicha code like PYTHIA must be compatible with), nor to generalise them to arbitrarily many partons.The only formalism we are aware of that addresses all of these issues (in particular flavour and mo-mentum correlations for arbitrarily many partons) while reducing to the single-parton ones for thehardest interaction remains the one developed in the context of the current PYTHIA beam-remnantmodel [87]. In this study, we supplement the momentum- and flavour-correlation model of [87] witha simple model of colour-space saturation effects appropriate to the SU (3) -multiplet language usedin this work. Noting that saturation should lead to a suppression of higher-multiplet states, we use asimple ansatz of exponential suppression with multiplet size, M : p ( M ) = exp ( − M/k saturation ) (7)where p is the probability to accept a multiplet of size M and k saturation is a free parameter that controlsthe amount of suppression.Everything stated above for the two-gluon case can be generalised to include quarks and an arbi-trary number of MPI. The calculation just extends to slightly more complicated expressions: ⊗ ⊗ ¯3 ⊗ ⊗ . . . = . . . (8)where quarks enter as triplets, antiquarks as antitriplets and gluons as octets. The statistical probabilityto choose any specific multiplet can be calculated in a similar fashion, either in full or simplified QCD.There is however still an ambiguity in how the colours are connected. For instance consider two quarksand one antiquark forming an overall triplet state. The colour calculation will not tell us which of thequarks are in a singlet with the antiquark. In the case of ambiguities, the implementation is to chooserandomly. The CR algorithm applied later may anyhow change the initial colour topology, lesseningthe effect of the above choice.At the level of the technical implementation, the choice of colour state for the scattered partonsis transferred to the final state particles of the event using the LC structure of the MPI and PS. Forexample, if the two gluons are in the octet state, one of the LHE colour tags (see [5, 6]) is changedaccordingly, and is propagated through to the colour of the final state particles.As a special case the overall colour structure of the beam remnant is not allowed to be a singlet.This is to avoid double-counting between diffractive and non-diffractive events. In the DPS example,if the gluons form a colour singlet, they essentially make up (part of) a pomeron, and thus shouldfall under the single-diffractive description. It would be interesting to look into the interplay betweenMPI and diffraction in more detail using the colour-multiplet language developed here, but this wouldrequire its own dedicated study, beyond the scope of this work.The conservation of flavour is relative straightforward and follows [87]. The principle used isto add the minimum needed flavour. For example if only an s quark is scattered from a proton, theremnant will consist of an ¯ s plus the three valence quarks.With the flavour structure and colour multiplet of the beam remnant known, it is now possibleto calculate explicitly how many gluons need to be added to obtain a colour-singlet state. Againthe idea is to add the minimum number of gluons to the beam remnant. Colour-junction structures,which we have argued can arise naturally in the colour structure of the scattered partons, complicatethis calculation slightly. To achieve an overall colour-singlet state the number of junctions minus thenumber of antijunctions has to match that of the beam particle. Taking this into account the minimalnumber of gluons is given by N gluons = max (cid:18) , ( N colour − N quarks + (cid:107) N junctions − N antijunctions − b (cid:107) )2 (cid:19) ; (9)13here b is the beam baryon number (1 if the beam is a baryon, 0 if the beam is a meson, and -1 ifthe is an antibaryon). The division by two is due to the gluons carrying two colour lines. Withoutjunctions, the number of gluons is simply the number of colour lines to the remnant minus the numberof available quarks to connect those colour lines to. It is easiest to understand how junctions changethis, by noting that the creation of a junction basically takes two colour charges and turn into oneanticolour. Thus the number of required connections goes down by one for each additional junctionneeded.After the gluons are added, all the colour connections and junction structures are assigned ran-domly between the remaining colours, with one exception: if the beam particle is a baryon and ajunction needs to be constructed (similarly for an antibaryon and an antijunction), two of the valencequarks will be used to form the junction structure (possibly embedded in a diquark), if they have notalready been scattered in the MPIs.With finally the full parton structure known, including both flavours and explicit colours, the laststep of the construction of the beam remnant is the assignment of energy fractions ( x values) to eachremnant parton, according to modified PDFs. To obtain overall energy-momentum conservation, theindividual partons are scaled by an overall factor. The scaling becomes slightly more complicated bythe introduction of primordial k ⊥ . Details on the modified PDF versions and the scaling can be foundin [87]. As discussed above, the CR model is applied after the parton-shower evolution has finished (andafter inclusion of the beam-remnant partons as described in the preceding subsection), just beforethe hadronisation. The model builds on two main principles: a simplified SU (3) structure of QCD,based on indices from 1 to 9, to tell which configurations are possible; and the potential energy of theresulting string systems, as measured by the so-called λ measure [23], to choose between the allowedconfigurations.The starting point for the model is the LC configuration emerging from the showers + beamremnants. Thus between each LC-connected pair of partons a tentative dipole is constructed. Thisconfiguration is then changed by allowing two (or three) dipoles to reconnect, and this procedureis iterated until no more reconnections occur. In each step of the algorithm, four different types ofreconnections can occur, illustrated in fig. 7:1. simple dipole-type reconnections involving two dipoles that exchange endpoints (fig. 7a);2. two dipoles can form a junction-antijunction structure (fig. 7b);3. three dipoles can form a junction-antijunction structure (fig. 7c);4. two multi-parton string systems can form a junction and an antijunction at different points alongthe string and connect them via their gluons (fig. 7d).Note that, although mainly dipoles between quarks are shown in the illustrations, all dipoles ( q - ¯ q , q - g , g - ¯ q and g - g ) are treated in the same manner in the implementation. Within an LC dipole, the quark andantiquark are assumed to be completely colour coherent, so that the probabilities for two dipoles to bein a colour-coherent state can be found by the standard SU (3) products. In full QCD, the probabilitiesfor type I (dipole) and II (junction) reconnections for q - ¯ q dipoles are given by eq. (4) and (5) as P q ¯ q I = 1 / and P qq II = 1 / , respectively. For gg dipoles, the calculation is complicated slightly by thefact that eq. (1) takes into account both the colour and anticolour charges of both of the gluons. With a14 ¯ q ¯ qq ¯ qq ¯ qq (a) Type I: ordinary dipole-style reconnection q ¯ q ¯ qq ¯ qq ¯ qq J ¯ J (b) Type II: junction-style reconnection q ¯ q ¯ qq ¯ qq ¯ qq J ¯ J ¯ qq ¯ qq (c) Type III: baryon-style junction reconnection q ¯ qq q ¯ qq J ¯ J ¯ q ¯ qgg gg g g g g (d) Type IV: zipper-style junction reconnection Figure 7: The four different allowed reconnection types. Type I (a) is the ordinary string reconnection.Type II (b) is the formation of a connected junction antijunction pair. Type III (c) is the formation ofjunction and antijunction, which are not directly connected. Type IV (d) is similar to type II exceptthat it allows for gluons to be added between the two junctions.15robability of P gg I = 8 /
64 = 1 / each, either “side” (colour or anticolour) of the gluons are allowedto reconnect (for a / probability that CR is allowed on both sides). And with a total probabilityof P gg II = 20 /
64 = 5 / either one or the other side is allowed a junction-type reconnection (bothsides would be equivalent to a dipole-style reconnection already counted above). For simplicity, theindex rules described in the beginning of this section have been defined to treat q ¯ q , qg , and gg dipolesall on an equal footing. The result is a compromise of P I = 1 / for all dipole-type reconnectionsand P II = 2 / for all junction-type reconnections. Differences between q ¯ q and gg combinations stillarise due to gluons being prevented from having the same colour and anticolour indices, and sincethe combination of two type-II reconnections is equivalent to a type-I reconnection. A comparisonbetween the weights resulting from our simplified treatment and the multiplet weights in full QCD forthe simple case of two-parton combinations was illustrated in fig. 3. Note also that the probability fora type-III reconnection among three uncorrelated q ¯ q dipoles (essentially creating a baryon from threeuncorrelated quarks) is P qqq III = 1 / in full QCD (among the 27 different ways to combine 3 quarks,only one is a singlet) while it is only 2/3 as large in our model; P III = 2 / × / / . Althoughour model should be a significant step in the right direction, we therefore still expect a tendencyto underestimate baryon production. When tuning the model below, we shall see that we are ableto compensate for this by letting junction-type reconnections appear somewhat more energeticallyfavourable compared with dipole-type reconnections.To recapitulate the model implementation, each dipole is assigned a random index value between1 and 9. Two dipoles are allowed to do a type-I reconnection if the two numbers are equal, providingthe / probability. Type-II reconnections are allowed if the two numbers modulo three are equaland the indices are different (e.g. 1-4 and 1-7), thereby providing the / probability. Three dipolesare allowed to do the type-III reconnection if they all have the same index modulo three and are alldifferent (1-4-7). The type-IV reconnection follows the same principles: a dipole needs to have thesame index, and a junction needs to have different-but-equal-under-modulo-three index. The exactprobability for type-IV reconnections depend on the number of gluons in the string.The number of allowed colour indices can in principle be changed (the 9 above), e.g. to vary thestrength of CR. However, the type-II, -III, and -IV reconnections rely on the use of modulo, thus careshould be taken if junction formation is allowed. A different method to control the strength of the CRwill be discussed below.The above colour considerations only tell which new colour configurations are allowed and notwhether they are preferable . To determine this, we invoke a minimisation of the λ string-lengthmeasure. The λ measure can be interpreted as the potential energy of a string, more detailed it is thearea spanned by the string prior to hadronisation. It is closely connected to the total rapidity spanof the string, and thereby also its total particle production. The minimisation is carried out by onlyallowing reconnections that lower the λ measure, which ensures that a local minimum is reached.A further complication is that, while the λ -measure for a quark-antiquark system with any numberof gluon kinks in between is neatly defined by an iterative procedure [23], the measure defined theredid not include junction structures. The first extension to handle these were achieved by starting fromthe simple measure between a quark antiquark dipole [100]: λ q ¯ q = ln (cid:18) s q ¯ q m (cid:19) (10)where s q ¯ q is the dipole mass squared and m is a constant with dimensions of energy, of order Λ QCD .For high dipole masses, the “ ” in eq. (10) can be neglected, splitting the λ -measure neatly into two16arts: one from the quark and one from the antiquark end (in the dipole rest frame): λ q ¯ q s (cid:29) m → ln (cid:18) s m (cid:19) = ln √ E q m + ln √ E ¯ q m . (11)The extension to handle a junction system used the same method, going to the junction rest-frame andadding up the “ λ -measures” from all the three (anti-)quark ends. The end result became λ q q q = ln √ E m + ln √ E m + ln √ E m (12)where the energies are calculated in the junction rest frame . This procedure worked well in thescenarios considered in that study, since all the dipoles had a relative large mass. However, in thecontext of our CR model, we will often be considering dipoles that have quite small masses. In thatcase, continuing to ignore the “ ” in eq. (10) can lead to arbitrarily large negative λ measures. Amongother things, such a behaviour could allow soft particles with vanishing string lengths to have a dispro-portionately large impact on dipoles with a large invariant mass. Generalising this behaviour to softjunction structures results in similar effects, namely that soft particles can have a disproportionatelylarge effect.An alternative measure is here proposed to remove the problem with negative string lengths, λ (cid:48) = ln (cid:32) √ E m (cid:33) + ln (cid:32) √ E m (cid:33) (13)where the energies are calculated in the rest-frame of the dipoles. This measure is always positivedefinite. In the case of massless particles the λ (cid:48) -measure can be rewritten to λ (cid:48) = ln (cid:32) s m + √ sm (cid:33) (14)where again s is the invariant mass squared of the dipole and m is a constant. The two measuresagree in the limit of large invariant masses ( s (cid:29) m ). The implementation includes a few alternativemeasures as options, but the above is chosen as the default measure and therefore also the one that theparameters are tuned for.A final complication regarding the λ measure is that the form above cannot be used to describe thedistance between two directly connected junctions. Instead the same measure as described in [100] isalso used in this study ( λ = β β + (cid:112) ( β β ) − , where β and β are the 4-velocities of the twojunction systems).Since the λ -measure for junctions introduces additional approximations, a tuneable parameter isadded to control the junction production. Several options for this parameter are possible and we settledon a m (cid:54) = m in the λ -measure for junctions. A higher m means a lower λ measure, resulting inan enhancement of the junction production. We cast the free parameter as the ratio, junctionCorrection : C j = m /m , (15)thus a value C j above unity indicates an enhancement in junction production, and vice versa. Thepossibility of a junction enhancement can be seen as providing a crude mechanism to compensate forthe intrinsic suppression of junction topologies in the colour-space model. Indeed, in the section on17 ¯ qg g q ¯ q ( gg ) (a) Ordinary pseudoparticle q gg ( gg ) q q q qqJ (b) Junction pseudoparticle Figure 8: The figure shows how two gluons are turned into a pseudoparticle depending on whetherthey are connected via an ordinary string (a) or a junction (b). The (gg) represents the formed pseu-doparticle.tuning below we find that values above unity are preferred in order to fit the observed amounts ofbaryon production.In the context of CR, it is generally the dipoles with the largest invariant masses which are themost interesting; they are the ones for which reconnections can produce the largest reductions ofthe λ measure. However, as evident from the above discussion, dipoles with small invariant massescan actually be the most technically problematic to deal with. It was therefore decided to removedipoles with an invariant mass below m from the colour reconnection. Technically this is achieved bycombining the small-mass pair into a new pseudo-particle. For an ordinary dipole this is a trivial task(fig. 8a), but if the dipole is connected to a junction the technical aspects becomes more complicated(fig. 8b). The easiest way to think of this is as an ordinary diquark, but in addition to these we canhave digluons, which will have three ordinary (anti-)colour tags. Note that we do not intend theseto represent any sort of weakly bound state; we merely use them to represent a low-invariant-masscollection of partons whose internal structure we consider uninteresting for the purpose of CR. Thepseudo-particles are formed after the LC dipoles are formed, and also after any colour reconnectionsif the new dipoles have a mass below m . Increasing m will therefore lower the amount of CR.Only small effects occur for variations around the Λ QCD scale, however increasing m beyond 1 GeVintroduces a significant reduction of CR.The complete algorithm for the colour reconnection can be summarised as below.1. Form dipoles from the LC configuration.2. Make pseudoparticles of all dipoles with mass below m .3. Minimise λ -measure by normal string reconnections.4. Minimise λ -measure by junction reconnections.5. If any junction reconnections happened return to point 3.The choice to first do the normal string reconnections before trying to form any junctions is due to thealgorithm not allowing to remove any junction pairs.Since each reconnection is required to result in a lower λ -measure than the previous one, theminimisation procedure is only expected to reach a local minimum. A possible extension to reachthe global minimum would be to use simulated annealing [110]. This is, e.g., the approach adoptedin the HERWIG++ CR model [47]. However this would also require the implementation of inversereconnections (i.e. a junction and an antijunction collapsing to form strings, and the unfolding ofpseudo-particles.). Secondly the computational time needed to find the global minimum would slow Note: we use a slightly different definition of m here compared to the original paper [100] gq ¯ qq ¯ qJ ¯ J ¯ JJq qq ¯ q ¯ q ¯ q (a) Single gluon split + g q ¯ qq ¯ qJ ¯ J ¯ JJq q q ¯ q g g g ¯ q ¯ qg g (b) Multi gluon split Figure 9: The figure shows how connected junctions are separated if they are connected by respec-tively a single gluon (a) or multiple gluons (b). The indices indicate where the split happens and whichparticles each gluon splits into.down the event generation speed very significantly. For purposes of this implementation, we there-fore restrict ourselves to a local deterministic minimisation here, noting that an algorithm capable ofreproducing the full expected area-law exponential would be a desirable future refinement.
The existing junction hadronisation model [87] was developed mainly for the case of string systemscontaining a single junction (in the context of baryon-number violating SUSY decays like ˜ χ → qqq ).For such systems, the strategy of is to take the two legs with lowest energy in the junction rest-frameand hadronise them from their respective quark ends inwards towards the junction, until a (low) energythreshold is reached, at which point the two endpoints are combined into a diquark (which containsthe junction inside). This diquark then becomes the new endpoint of the last string piece, which canthen be fragmented as usual.The case of a junction-antijunction system was also addressed in [87] (arising e.g., in the caseof e + e − → ˜ t ˜ t ∗ → ¯ q ¯ q qq ), but the new treatment of the beam remnants presented here, as well asthe new CR model, can produce configurations with any number of colour-connected junctions andantijunctions. This goes beyond what the existing model can handle.The systems of equations describing such arbitrarily complicated string topologies are likely to bequite involved, with associated risks of instabilities and pathological cases. Rather than attempting toaddress these issues in full gory detail, we here adopt a simple “divide-and-conquer” strategy, slicingthe full system into individual pieces that contain only one junction each, via the following 3 steps: • If a junction and an antijunction are connected with a single gluon between them, that gluon isforced to split into two light quarks (u,d and s) that each equally share the 4-momentum of thegluon (corresponding to z = ). Since the gluon is massless, the two quarks will have to beparallel (fig. 9a). • If a junction and an antijunction are connected with at least 2 gluons in between, the gluonpair with the highest invariant mass is found, and is split according to the string-fragmentation19unction. The highest invariant mass is chosen due to it having the largest phase space andbeing the most likely to have a string breakup occur. The split is done in such a way that thetwo gluons are preserved but each of them give up part of their 4-momentum to the new quarkpair. The new quark that is colour-connected to one of the gluons will be parallel to the othergluon (fig. 9b, where the indices indicate who is parallel with whom.). • After the two rules above have been applied, only directly-connected junction-antijunctionsare left. If all three legs of both junctions are connected to each other, the system containsno partons and can be thrown away. If two of the legs are directly connected, the junction-antijunction system is equivalent to a single string piece and is replaced by such, see fig. 10a.Finally, the case of a single direct junction-antijunction connection is dealt with differently,depending on whether the system contains further junction-antijunction connections or not. Inthe former case, illustrated in fig. 10b, the maximum number of junctions are formed fromthe partons directly connected to the junction system. The remaining particles are formed intonormal strings. In the example of fig. 10b, three quarks are first removed to form a junctionsystem; the remaining q and ¯ q then have no option but to form a normal string. The currentmethod randomly selects which outgoing particles to connect with junctions. One extensionwould be to use the string measure to decide who combines with whom. (However the effectof this might be smaller than expected, since the majority of the multi-junction configurationcomes from the beam remnant treatment, which later undergoes CR.)For cases with a single direct junction-antijunction string piece and no further junctions in thesystem, illustrated in fig. 10c, the λ -measure is used to determine whether the two junctionsshould annihilate or be kept [87] (essentially by determining whether the strings pulling on thetwo junctions cause them to move towards each other, towards annihilation, or away from eachother). If the junctions survive, a new q ¯ q pair is formed by taking momentum from the otherlegs of the junction. Otherwise the junction topology is replaced by two ordinary strings. Anoption to always keep the junctions also exists. By default, we do not account for any space-time separation between different MPI systems. This ismotivated by the observation that, physically, the individual MPI vertices can at most be separated bytransverse distances of order the proton radius, which by definition is small compared with the lengthof any string long enough to fragment into multiple particles.We do note, however, that in order for reconnections to occur between two string pieces, theyshould be in causal contact ; if either string has already hadronised before the other forms, thereis no space-time region in which reconnections between them could physically occur. In the restframe of a hadronising string piece, we take its formation time to be given roughly by its invariantmass, τ form ∼ /m string . The time at which it begins to hadronise is related to the inverse of Λ QCD , τ had ∼ / Λ QCD . In order for reconnections to be possible between two string pieces, we requirethat they must be able to resolve each other during the time between formation and hadronisation,taking time-dilation effects caused by relative boosts into account. There are several ways in whichthis requirement can be formulated at the technical level, and accordingly we have implemented a fewdifferent options in the code. In principle, the two strings can be defined to be in causal contact if therelative boost parameter fulfils: γ τ form < C time τ had ⇒ γ cm string r had < C time (16)20 qq J ¯ J q ¯ q (a) Doubly-connected J ¯ J system ¯ qJ Jq q q ¯ Jq q ¯ qq q qJ + q (b) Multiple J ¯ JJ · · · Connections ¯ qJq q ¯ Jq ¯ q q J + q ¯ J ¯ q ¯ q ¯ q (c) A single J ¯ J Connection
Figure 10: The figure shows how directly connected junctions are separated. Fig. a shows the re-placement of a doubly-connected junction-antijunction system by an ordinary string piece. Fig. bshows the method used to reduce systems with more than two interconnected junctions. Fig. c showsthe split-up of a system containing exactly one junction-antijunction connection, into two separatejunction systems.where C time is a tuneable parameter and r had (= τ had c ≡ is a fixed constant given by the typicalhadronisation scale. There are however two major problems with this definition: first it is not Lorentzinvariant; the two dipoles will not always agree on whether they are in causal contact or not. Thiscan be circumvented, by either requiring both to be able to resolve each other (strict) or just either ofthem to be able to resolve the other (loose). Secondly, the emission of a soft gluon from an otherwisehigh-mass string changes m string significantly for each of the produced string pieces, which gives anundesirable infrared sensitivity to this measure, reminiscent of the problems associated with definingthe λ string-length measure itself. One way to avoid this problem is to consider the first formationtime of each colour line, i.e. the dipole mass at the time the corresponding colour line was first createdin the shower, which we have implemented as an alternative option. No matter the exact definitionof formation time and hadronisation time, all models agree that reconnection between boosted stringsshould be suppressed. A final extremely simple way to capture this in a Lorentz-invariant way is toapply a cut-off directly on the boost factor γ , which thus provides a simple alternative to the othermodels.These different methods have all been implemented and are available in PYTHIA, via the mode ColourReconnection:timeDilationMode . The C time parameter introduced above is speci-fied by ColourReconnection:timeDilationPar and controls the size of the allowed relative21oost factor for reconnections to occur. As such it can be used to tune the amount of CR. Its optimalvalue will vary depending on the method used, but after the methods are tuned they produce similarresults (see section 3 for details).A final aspect related to space-time structure that deserves special mention is resonance decays.By default, these are treated separately from the rest of the event. Physically, this is well motivatedfor longer-lived particles (e.g., Higgs bosons), which are expected to decay and hadronise separately.For shorter-lived resonances the separation of the MPI systems and resonance decays is physicallynot so well motivated. E.g., most
Z/W bosons and top quarks will decay before hadronisation takesplace, Γ (cid:29) Λ QCD , and as such should be allowed to interact with the particles from the MPI systems,ideally with a slightly suppressed probability due to the decay time.Currently, only two extreme cases are implemented, corresponding to letting CR occur before or after (all) resonance decays. The corresponding flag in PYTHIA is called PartonLevel:earlyResDec .When switched on, CR is performed after all resonance decays have occurred, and all final-state par-tons therefore participate fully in the CR. Since no suppression with resonance lifetime is applied, thisgauges the largest possible impact on resonance decays from CR. When switched off, CR is performed before resonance decays, hence involving only the beam remnant and MPI systems. It is equivalent toassuming an infinite lifetime for the resonances, and hence estimates the smallest possible impact onresonance decays from CR. An optional additional CR can be performed between the decay productsof the resonance decays, with the physics motivation being H → W W → qqqq studies.To summarise, we acknowledge that the treatment of space-time separation effects and causalityis still rather primitive in this model. The derivation of a more detailed formalism for these aspectswould therefore be a welcome and interesting future development. The tuning scheme follows the same procedure as for the Monash 2013 tune [34]. However at amore limited scope, since only CR parameters, and ones strongly correlated with them, are tuned.As a natural consequence of this, the Monash tune was chosen as the baseline. As discussed insection 2.3.4, several options are available for the choice of CR time-dilation method, which naturallyresults in slightly different preferred parameter sets. Here, we consider the following three modes: • Mode 0: no time-dilation constraints. m controls the amount of CR (mode 0); • Mode 2: time dilation using the boost factor obtained from the final-state mass of the dipoles,requiring all dipoles involved in a reconnection to be causally connected (strict); • Mode 3: time dilation as in Mode 2, but requiring only a single connection to be causallyconnected (loose).This allows to investigate the consequences of some of the ambiguities in the implementation of themodel. For the purpose of later studies that may want to focus on a single model, we suggest to usemode 2 as the “standard” one for the new CR. The parameters described in this section will thereforecorrespond to that particular model, with parameters for the others given in appendix A. Note that thissection only contains the main physical parameters; for a complete list we again refer to appendix A.
We begin with e + e − collisions. Only small effects are expected in this environment, due to the p ⊥ -ordering of the shower and the absence of MPIs. Only CR and string-fragmentation variables22
20 40 60 ) c h P r obab ili t y ( n -6 -5 -4 -3 -2 -1
10 110 Charged Multiplicity (udsc)
Vincia 1.200 + MadGraph 4.4.26 + Pythia 8.206Data from Phys.Rept. 399 (2004) 71L3 MonashNew CR model bins /N c – – V I N C I A R O O T hadrons fi ee ch n T heo r y / D a t a (a) ) | p / d | Ln ( x c h > dn c h / < n -3 -2 -1
10 110
Charged Momentum Fraction (udsc)
Vincia 1.200 + MadGraph 4.4.26 + Pythia 8.206Data from Phys.Rept. 399 (2004) 71L3 MonashNew CR model bins /N c – – V I N C I A R O O T hadrons fi ee )| p |Ln(x T heo r y / D a t a (b) Figure 11: Charged-particle multiplicity (a) and momentum fraction (b) spectra, in light-flavourtagged data from the L3 collaboration [111]. (Plots made with V
INCIA R OOT [112]. The ratio panesfollow the now-standard “Brazilian” colour conventions, with outer (yellow) bands corresponding to σ deviations and inner (green) bands corresponding to σ deviations.)were studied, since the shower was left untouched. The fragmentation model contains three mainparameters governing the kinematics of the produced hadrons: the non-perturbative p ⊥ produced instring breaks, controlled by the σ ⊥ parameter ( StringPT:sigma ), and the two parameters, a and b ,which control the shape of the longitudinal ( z ) fragmentation function. For pedagogical descriptions,see e.g. [2, 23, 30, 34]. Since the effects are expected to be small, we made the choice of keeping σ ⊥ = 0 .
335 GeV unchanged, adjusting only the longitudinal ( a and b ) parameters. Changing theminimal number of parameters also helps to disentangle the effects of CR from the retuning. As averification, a tune with a smaller σ ⊥ ( .
305 GeV ) was considered, however after retuning a and b thetwo tunes described the LEP data with a similar fidelity. (The choice of testing a lower σ ⊥ was madesince the CR model tends connect more collinear partons leading to shorter strings, but a harder p ⊥ spectrum of the produced hadrons [75].)The determination of the two parameters of the Lund fragmentation function, a and b , is com-plicated slightly by the fact that they are highly correlated; choosing both of them to be quite smalloften produces equally good descriptions of fragmentation spectra as choosing both of them large,corresponding to a relatively elongated and correlated χ “valley”. By simultaneously consideringboth variables and comparing them to both multiplicity and momentum spectra, cf. fig. 11 (with the “New CR model” curve showing our new model, and “Monash” the baseline Monash 2013 tune), wehere settled on a low-valued pair, as compared with the default Monash values:23 tringZ:aLund = 0.38 The new CR also alters the ratio between the identified-particle yields, especially so for baryonproduction due to the introduction of additional junctions. Thus the flavour-selection parameters ofthe string model also need to be retuned, by comparing with the total identified-particle yields, seee.g. [34]. As expected the effects are minimal in e + e − collisions, and only small changes are required.The modifications were therefore done with a view to providing a better description also for pp col-liders, but staying within the uncertainties allowed by the LEP data. This resulted in an adjustment ofthe parameters for the diquark over quark fragmentation probability and the strangeness suppression: StringFlav:probQQtoQ = 0.078
As expected the diquark over quark probability is reduced due to the introduction of junctions. Moresurprisingly is the increased suppression of strange quarks, since the model a priori should not in-fluence flavour selection. The technical implementation of the junction hadronisation does, however,introduce a slight enhancement of the strangeness production, due to an even probability for a gluonto split into an u, d or s quarks when separating junction systems. This is not visible at LEP, but at pp colliders the slightly lower strangeness fragmentation is favoured.The final set of fragmentation parameters we define is more technical. For junction systems andbeam remnants, a separate set of parameters controls the choice of total spin when two, already pro-duced, quarks are combined into a diquark. Unlike diquarks produced by ordinary string breaks(whose spin is controlled by the parameter StringFlav:probQQ1toQQ0 ), which can only con-tain the light quark flavours (u, d, s), and for which the significant mass splittings between the light-flavour spin-3/2 and spin-1/2 baryon multiplets necessitates a rather strong suppression of spin-1 di-quark production (relative to the naive factor 3 enhancement from spin counting), junction systemsin particular can allow the formation of baryons involving heavy flavours, which have smaller masssplittings and which therefore might require less suppression of spin-1 diquarks. We note also that di-quarks produced in string breakups are produced within the linear confinement of the string, whereasjunction diquarks come from the combination of two already uncorrelated quarks, so there is a priorilittle physics reason to assume the parameters must be identical.With the limited amount of junctions in the old model, none for ee and at most two for pp , theseparameters previously had almost no influence on measurable observables and were therefore largelyirrelevant for tuning. With the additional junctions produced by our model, these parameters can nowgive larger effects. Measurements of higher-spin and heavy-flavour baryon states at pp colliders arestill rather limited though, and so far we are not aware of published directly usable constraints fromexperiments. For the time being therefore, we choose to fix the parameters to be identical to those forthe production of ordinary diquarks in string breakups: StringFlav:probQQ1toQQ0join = 0.027,0.027,0.027,0.027
The four components give the suppression when the heaviest quark is u/d, s, c or b, respectively. Westress that this is merely a starting point, hopefully to be revised soon by comparisons with new datafrom the LHC experiments. 24 .2 Hadron Colliders
The retuning to hadron colliders consisted of tuning three main parameters: • C time ( ColourReconnection:timeDilationPar ): controls the overall strength of thecolour-reconnection effect via suppression of high-boost reconnections, see section 2.3.4. Canbe tuned to the (cid:104) p ⊥ (cid:105) vs n ch distribution. • C j ( ColourReconnection:junctionCorrection ): multiplicative factor, m /m , ap-plied to the string-length measure for junction systems, thereby enhancing or suppressing thelikelihood of junction reconnections. Controls the junction component of the baryon to mesonfraction and is tuned to the Λ /K s ratio. • p ref ⊥ ( MultiPartonInteractions:pT0Ref ): lower (infrared) regularisation scale of theMPI framework. Controls the amount of low p ⊥ MPIs and is therefore closely related to thetotal multiplicity and can be tuned to the d (cid:104) n ch (cid:105) /dη distribution.By iteratively fitting each parameter to its respective most sensitive curve an overall good agreementwith data was achieved (see fig. 12) with the following parameters: ColourReconnection:junctionCorrection = 1.2
Note in particular this is the first time that PYTHIA has been able to describe the Λ /K s ratio in pp collisions while remaining consistent with LEP bounds. We explore this in more detail in section 3.4.The C j = 1 . parameter shows that a slight enhancement of junction reconnections (i.e., baryonproduction) is needed, relative to mesonic ones. However, given the approximations used in theimplementation of especially the junction structures, such a difference is not unreasonable. Smalldifferences between the modes can be seen in the (cid:104) p ⊥ (cid:105) vs n ch and more significant differences formultiplicity distributions , cf. fig. 13. With respect to the latter, however, we note that the differencesin the tails of the multiplicity distributions can be tuned away by modifying the assumed transversematter density profile of the proton, which was kept fixed here to highlight the differences with theminimal number of retuned parameters.The new colour treatment of the beam remnant (BR) introduces a single new parameter con-trolling the amount of saturation, cf. the discussion in section 2.3.1. Due to the low p ⊥ of the BRparticles, the effects are largest in the forward direction. We therefore use the forward charged multi-plicity as measured by the TOTEM experiment [116] to compare different modelling choices of thisaspect, see fig. 14a. The difference between no saturation ( k saturation → ∞ ) and maximal saturation ( k saturation = 0 . is about and exhibits no shape difference over the TOTEM pseudo rapidityrange. For illustration and completeness, we may also consider what happens over the full rapidityrange, at least at the theory level. This is illustrated in fig. 14b. In the central region, the effect ofapplying saturation is a slight decrease of the particle yield, and thus would already have been tunedaway by p ref ⊥ . It was therefore chosen to use a relative high saturation to mimic the effect of the earlierPYTHIA beam remnant model: Technically:
BeamRemnants:saturation = 1E9. b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b Data b MonashMode Mode Mode . . . . . . . . Mean p ⊥ vs charged hadron multiplicity, | η | < √ s = h p ⊥ i [ G e V ]
20 40 60 80 100 120 140 160 1800 . . . . . . . . n M C / D a t a (a) b b b b b b b b b b Data b MonashMode Mode Mode
Charged hadron η integrated over p ⊥ at √ s = d N c h / d η - - . . . . η M C / D a t a (b) b b b b b b b b b b Data b MonashMode Mode Mode . . . . . . . Λ /K versus rapidity at √ s = TeV N ( Λ ) / N ( K S ) . . . . . . | y | M C / D a t a (c) Figure 12: The average p ⊥ as a function of multiplicity [52] (a), the average charged multiplicity asa function of pseudorapidity [113] (b), and the Λ /K s ratio [114] (c). All observables from the CMScollaboration and plotted with the Rivet framework [115]. All PYTHIA simulations were non singlediffractive (NSD) with a lifetime cut-off τ max = 10 mm/c and no p ⊥ cuts applied to the final stateparticles. 26 b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b Data b MonashMode Mode Mode − − − − − − Charged hadron multiplicity, | η | < √ s = P n . . . . n M C / D a t a (a) | η | < . b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b Data b MonashMode Mode Mode − − − − − Charged hadron multiplicity, | η | < √ s = P n . . . . n M C / D a t a (b) | η | < . Figure 13: The two plots show the multiplicity distributions for respectively very central tracks (a)and the full CMS tracker coverage (b), compared with CMS data [52]. All PYTHIA simulations wereNSD with a lifetime cut-off ( τ max = 10 mm/c) and no p ⊥ cuts were applied to the final state particles. b b b b b b b b b b b b b b b b b b Data b MonashNo saturationMax saturation . . . . . Charged particle | η | at TeV, track p ⊥ >
40 MeV, for N ch ≥ d N / d η . . . . . . . . . | η | M C / D a t a (a) TOTEM Comparison All (max sat.)All (no sat.)Baryons (max sat.)Baryons (no sat.)10 − − − / N e v t d N / d | y | AllBaryons0 2 4 6 8 100.60.811.21.4 | y | n o s a t / m a x s a t (b) Generator-Level Figure 14: (a): different extreme saturation choices compared with the TOTEM forward multiplicitydata [116]. (The PYTHIA simulation includes all soft-QCD processes and a particle lifetime cut-off τ max = 10 mm/c.) (b): MC rapidity distributions for respectively all particles and baryons only.(For simplicity only non diffractive (ND) events were used, hadron decays were turned off to reflectprimary hadron production, and no p ⊥ cuts were imposed.)27 tring string q ¯ q g (a) /N C → string string q ¯ q g (b) Figure 15: (a): illustration of the correspondence between colour connections and string pieces inan ordinary (LC) 3-jet topology. b): if the gluon jet is composed of (at least) two gluons, there isprobability for the q ¯ q system to be in an overall singlet. (The LC string topology remains a possibilityas well, with the string-length measure λ used to decide between them.) Notation: g ac denotes a gluonwith colour (anticolour) index c ( a ). For (anti)quarks, we use q c ≡ q c and ¯ q a ≡ ¯ q a . BeamRemnants:saturation = 5
We emphasise however that this is merely a starting point, and that a different balance between p ref ⊥ and k saturation may be preferred in future tuning efforts, especially ones taking a more dedicated lookat the forward region. In such a study the sharing of momentum between the partons in the remnantshould also be considered, since it is known to alter both particle production in the forward region andthe multiplicity distributions.An interesting signal that may help to break the relative degeneracy between p ref ⊥ and k saturation , isto look for baryons at high rapidities, which, due to the introduction of junction structures in the BRcan act as further tracers of the degree to which the BR has been disturbed. This is illustrated by thelower set of curves in fig. 14b. The effects are indeed seen to be slightly larger for baryons, howeverthe total cross section is also significantly lower. From this simple MC study we are not able to sayclearly whether such a measurement, which requires the additional non-trivial ingredient of baryonidentification, would be experimentally feasible. In the preceding sections, we constrained (“retuned”) the fragmentation parameters using observableslike the charged-particle multiplicity and fragmentation spectrum, which are indirectly sensitive toCR effects via the modifications caused to these spectra by the minimisation of string lengths. Butwhat about observables with more direct sensitivity to CR effects? There are two main categoriesof dedicated CR studies at LEP: Fully hadronically decaying
W W events (looking for reconnectionsbetween the two W systems), and colour-flow sensitive observables in three-jet events. In this studywe restrict our attention to the latter of these. A follow-up dedicated study of CR effects at e + e − colliders, both earlier as well as possible future colliders, is planned.Without CR, the three produced jets will in general be represented by a colour string stretched fromthe quark via the gluon to the antiquark, illustrated in fig. 15a. The string pieces spanned between thequark and gluon jets lead to a relatively large particle production between these jets, while there is asuppression of particle production in the phase-space region directly between the quark and antiquarkjets. However, if CR is allowed, there is a /N C chance that two (or more) sequentially emitted gluons28 o CR (Monash)New CR modelMax CR − − − Jet fraction passing rapidity cut-off N ( ≤ y m i n ) / N a ll . . . . . . . . . ≤ y min C R / n o C R (a) Rapidity along jet axis No CR (Monash)New CR modelMax CR Fraction of zero-charge jets N ( c h = , ≤ y m i n ) / N ( ≤ y m i n ) . . . . . . . . . ≤ y min C R / n o C R (b) Neutral fraction Figure 16: Observables constraining CR in 3-jet events at LEP. (a): the minimum rapidity of theconstituents of the third jet with respect to its jet axis. (b): the fraction of the third jet with total chargeequal to zero as a function of minimum rapidity of particles in the third jet.end up cancelling each others’ colour charges. Thus, if at least one additional gluon was produced inthe FSR, the “gluon jet” may effectively become overall colour neutral, allowing it to decouple fromthe quark-antiquark system in colour space. This is illustrated in fig. 15b. There is a caveat to theabove, namely, if the two gluons originate from a single gluon, ie., g → gg , the two gluons must forma colour octet. In the gluon-collinear limit, this colour structure dominates and the probability for thejet to end up colour neutral should therefore be strongly suppressed, below /N C . This aspect is notincluded in our model, since the history of the final state particles is not considered. The model maytherefore overestimate the CR effect on three-jet events.An additional consequence is that the jet will also have a total electric charge of zero, if all particlesfrom the fragmentation fall within the jet. The best observable uses both of these ideas. Firstly arapidity gap is required in order to select events with minimum radiation between the jets. This alsoenhances the probability that all the particles from the fragmentation falls within the jet. Secondlythe jet is required to have a total charge of zero. This observable was first proposed during the LEPruns [117] and successively followed up by several of the experiments [35, 37, 40]. The studies werelimited to excluding only the most extreme CR models, with no conclusions drawn between moremoderate CR models and no CR; the data was located in between the two predictions.Rather than comparing our model directly on the LEP data, we took a slightly simpler approach byonly considering the difference between no CR and the new CR model in the relevant observables. Thedifference is found to be negligible on those observables, cf. the “No CR” and “New CR” histogramsin fig. 16. Thus we do not expect that the new CR model could be ruled out by these LEP constrains.We note that the small difference between with and without CR can be understood, by rememberingthe large focus on junction structures in the new CR model. Junction structures do not produce colour-singlet jets in the same manner as ordinary strings, and thus are not sensitive to this observable in thesame way as ordinary strings reconnections. It is possible to consider a more extreme version ofthe new CR model where all dipoles are allowed to connect with each other, i.e. effectively replacing /N C by unity! This is illustrated by the “max CR” histogram in fig. 16. For this unphysically extreme29 ll (old CR model)All (new CR model)Junctions (new CR model)Diquarks (new CR model) Baryon production as a function of multiplicity Multiplicity h N b a r y o n s i Figure 17: average baryon multiplicity as a function of hadron multiplicity (generator-level, includingall particles, hadronic decays switched off).case the difference between the two models becomes so large it most likely would have been ruledout by the experiments. However such an extreme case would also be eliminated by just consideringLHC measurements (e.g. (cid:104) p ⊥ (cid:105) vs n ch ). As discussed in section 3.2, the new CR model is able to reach agreement with some key observablesthat have otherwise proved difficult for the string model (as implemented in PYTHIA) in the pp environment. In particular, subleading dipole connections that minimise the string-length measurecan account for the rise in the (cid:104) p ⊥ (cid:105) ( n ch ) distribution (a feature also present in earlier CR modelsin PYTHIA, though without the connection with subleading colour), and subleading string-junctionconnections can account for the observed increase in e.g. the Λ /K ratio between ee and pp collisions.The price is (a few) new free parameters governing the CR modelling, so the question naturallyarises to what extent this type of model can be distinguished clearly from other other phenomenolog-ical modelling attempts to describe the same data. All models we are aware of that simultaneouslyaim to describe both the LEP data and the LHC data (or ee and pp data more generally) rely on thehigher colour/energy densities present in pp collisions to provide the extra baryons. The multiplicityscaling of the baryon production is therefore expected to be higher than the linear scaling of the di-quark model. This is also what we observe, cf. fig. 17. For low multiplicity, both of the CR modelsagree with each other, however the increase happens faster for the new model. This shape differencein the scaling with particle multiplicity could provide an additional probe to test the new model.One also notices that there is a significant difference between the baryon production of the oldmodel and baryon production from diquarks in the new model. This is somewhat surprising since thehadronisation model is essentially left untouched. The explanation for this is two-fold: 1) the new CRmodel produces a different mass spectrum of strings (with generally lower invariant masses), and 2)low-mass strings and junction structures produce fewer additional diquarks.The first point is illustrated in fig. 18a, which shows the invariant-mass distribution of strings inthe new and old model. In the old model, the distribution is essentially flat, and includes a signifi-cant plateau towards very large invariant masses, whereas the distribution is strongly peaked at smallinvariant masses in the new model. The differences arise both from the junction cleanup procedure30 ll (old CR model)all (new CR model)Junctions (new CR model) q ¯ q strings (new CR model)- . . . . . String invariant mass distribution Log ( m [ GeV ]) / N e v t d N / d L o g ( m ) (a) ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut utututututututututututututututututut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut ut utututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututututrs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rs rsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsrsbc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bc bcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbcbc n junbar − n stringbar ut n junpar / n stringpar rs ( n junbar − ) / n stringbar bc ( m [ GeV ]) (b) Figure 18: (a): invariant mass distribution of string systems (note logarithmic x axis). (b): the produc-tion of baryons from respectively a q ¯ q system (“string”) and a qqq system (“junction”) with the sametotal invariant mass. The red triangles show the difference of total primary-baryon multiplicities, theblue squares show the ratio of total primary-hadron multiplicities (mesons+baryons), and the greencircles show the ratio for the primary-baryons multiplicities, subtracting off the extra baryon that thejunction topology always produces.(by which longer strings can be split by insertion of an additional quark-antiquark pair), and fromthe minimisation of the λ -measure. The old model also minimised the λ -measure, however this isachieved by combining strings, giving fewer but higher-mass string systems than before the CR.Due to energy-momentum conservation (and a greater relative importance of the quark endpoints),low-invariant-mass strings produce fewer baryons. Despite the fact that each junction system producesat least one baryon, we therefore note that this does not automatically lead to an increase in thetotal number of produced baryons. Since the invariant mass of a qqq junction system is distributedon three string pieces, whereas that of a q ¯ q system is carried by a single string, diquarks, whichare relatively heavy, can actually be quite strongly phase-space suppressed in junction topologies,especially at low invariant masses of the string system (where the majority of the junction topologieslie cf. fig. 18a). In addition the diquarks need to be pair produced to conserve baryon number, andthe current implementation requires the pair to be on the same junction leg, leading to an even largerphase-space suppression. This effect is illustrated in fig. 18b, where for instance a junction systemwith E CM = 10 GeV (in the peak of the mass distribution) has a five times lower probability toproduce an (extra) diquark pair compared to a 10-GeV dipole string. At fixed multiplicity this effectis hidden in the tuned parameters, but can be observed by the different scaling.Considering baryon production in more detail, the relative yield of different types of baryons ishighly revealing. A collection of such yields for the different models are listed in tab. 1. For most ofthe baryons, the new CR model predicts about − above the old model. This is in agreementwith result for Λ production shown earlier (fig. 12). There are however also some clear order-of-magnitude differences, for charm and bottom baryons. One example is Σ c production , for which thenew CR model predicts a rate more than a factor of 20 above that of the old model!To understand how such large differences can occur, we need to recall how baryons are producedin ordinary string fragmentation. Since no charm is produced in string break-ups, the only way toproduce a Σ c is to produce a dd -diquark and combine it with c quark from the shower. However, since The Σ c is a cdd state with spin S = 1 / , mass ∼ . GeV and PDG code 4112 [26]. N par /N events ) Old CR modelstring junction all N par /N events (all) π + . · . · . · p . . . . n . . . . ++ . · − . · − . . · − ∆ + . · − . · − . . · − ∆ . · − . · − . · − . · − ∆ − . · − . · − . · − . · − K + . . .
1Λ 4 . · − . · − . · − . · − Σ + . · − . · − . · − . · − Σ . · − . · − . · − . · − Σ − . · − . · − . · − . · − Σ ∗ + . · − . · − . · − . · − Σ ∗ . · − . · − . · − . · − Σ ∗− . · − . · − . · − . · − Ξ − . · − . · − . · − . · − Ω − . · − . · − . · − . · − D + . · − . · − . · − Λ + c . · − . · − . · − . · − Σ ++ c . · − . · − . · − . · − Σ + c . · − . · − . · − . · − Σ c . · − . · − . · − . · − Σ ∗ ++ c . · − . · − . · − . · − Σ ∗ + c . · − . · − . · − . · − Σ ∗ c . · − . · − . · − . · − ccq . · − . · − . · − B + . · − . · − . · − Λ b . · − . · − . · − . · − Σ + b . · − . · − . · − . · − Σ b . · − . · − . · − . · − Σ − b . · − . · − . · − . · − Σ ∗ + b . · − . · − . · − . · − Σ ∗ b . · − . · − . · − . · − Σ ∗− b . · − . · − . · − . · − bcq . · − . · − bbq . · − . · − Table 1: Primary particle and antiparticle production of identified hadrons. Ten million ND eventswere simulated and all particles and antiparticles were counted. Hadron decays were switched off toonly look at the primary production. Double heavy baryons where the last q can be any quark.32 d diquarks must have spin 1 (due to symmetry), their production is heavily suppressed relative to ud ones which can also exist in the far lighter spin-0 state. Enter junctions, for which there is a priori nospecific penalty associated with having two legs end on same-flavour quarks as compared to differentflavours. Up to some combinatorics and symmetry factors the production of ud and dd is thereforeexpected to be the same for junction systems, in good agreement with the observed predictions of thenew CR model. As such the production of baryons like Σ c theoretically provides an excellent probeto study the relative importance between diquark- and junction-driven baryon production.Similar results are also observed for Σ ++ c as well as for the production of the analogous b -baryons.And with the large b -physics programme at the LHC, we believe this could be an interesting study,both for its physics value as well as a possible source of background for other measurements. Theeffect can also be seen for Ω − , however not as clearly as for charm and bottom baryons. Moreover thepresence of additional suppression of strange diquarks in the string fragmentation model makes Ω − connection more complicated. We note that it can be an important validation channel however.Due to the majority of baryons being produced by the junction mechanism in the new CR model,the baryon yields also provides a clear probe to test the spin structure of the diquarks formed from thejunctions. The large difference in yield between Σ ∗ + c and Σ c is due to the choice of spin suppressionmentioned earlier in the tuning section. An actual measurement could be directly applied as a con-straint for this variable. At least with the development of the present model, we now have a vehiclethat allows to explore this type of phenomenology and interpret the findings.We should note that the baryons considered above are excited states that rapidly decay throughthe emission of a pion, e.g. Σ c → Λ + c π − . As such they may be quite challenging to observe experi-mentally. It is therefore not a given that it will be easy to utilise measurements of these yields. But itdoes provide a theoretical motivation for studying the production and measurement of heavy baryons.Another special class of baryons is the double (or triple) heavy baryons containing at least two cor b quarks. These baryons can not be formed in ordinary string fragmentation and is therefore almostnon-existent in the old CR scheme. The only production mechanism is via the junctions from thebeams (which also means that for p + p + collisions no double-heavy antibaryons are predicted). This isalso observed in tab. 1, where only a single double-heavy baryon is produced in the 10 million events.With the large amount of junctions, the new CR model provides a natural production mechanismfor double-heavy baryons, and as such the expected amount is also significantly higher than for theold CR model. The effect of massive quarks in the λ -measure is not well understood, however, andpossible other production mechanism might also contribute, thus the estimate is most likely rathercrude. Irrespectively, a measurement of double-heavy baryons probes a region of hadronisation thatthe current models do not describe. And it could possible also shed some light on whether the junctionmechanism might be a reasonable production mechanism.So far, we only considered total particle yields; more knowledge is available by studying moredifferential distributions. A natural next extension is the transverse momentum distributions. Asthe junctions are formed by minimising the λ -measure, the particles defining the junction may beexpected to preferentially be moving in the same direction and thereby create a boosted baryon. Thisin turn leads to an expected increase in transverse momentum for such junction baryons. This is alsoobserved in the low- p ⊥ region (below roughly p ⊥ ∼ GeV), where the particle production peak ishigher for junction baryons (fig. 19). In the region of very high p ⊥ (above roughly p ⊥ ∼ GeV) theparticle production is dominated by jets, for which the hard high- p ⊥ partons are more important thanthe overall boost. In addition, the perturbative gluons associated with the jet already provides a low λ -measure and as such limited CR is expected inside the jet regions. This leads to the high- p ⊥ regionbeing occupied predominantly by baryons produced in ordinary (diquark) string-breaks.Transverse momentum spectra have already been measured for some of the more common baryon33 ll (old CR model)All (new CR model)Junctions (new CR model)Diquarks (new CR model) − − − − − Primary Λ p ⊥ spectrum p ⊥ [GeV] / N e v t d N / dp ⊥ [ G e V − ] Figure 19: The Λ p ⊥ -distribution separated by production mechanism. Only ND events were includedand hadron decays were switched off. b b b b b b b b b b b b b b b b b b b b b b b b Data b Old CR modelNew CR model − − − − Λ transverse momentum distribution at √ s = TeV ( / N N S D ) d N / dp T ( G e V / c ) − . . . . Λ p T [GeV/ c ] M C / D a t a (a) b b b b b b b b b b b b b b b b b b b b b b b b Data b Old CR modelNew CR model . . . . . Λ /K versus transverse momentum at √ s = TeV N ( Λ ) / N ( K S ) . . . . p T [GeV/ c ] M C / D a t a (b) Figure 20: The (a) Λ p ⊥ -distribution and (b) the Λ /K s p ⊥ -distribution as measured by the CMSexperiment [114]. All PYTHIA simulations were NSD with a lifetime cut-off ( τ max = 10 mm/c) anda rapidity cut on 2 ( | y | < ). 34 b b b b b b b b b b b b brs rs rs rs rs rs rs rs rs rs rs rs rs bc bc bc bc bc bc bc bc bc bc bc bc bc π − K − K ρ K ∗ ¯ p φ Λ ¯ Λ Ξ − ¯ Ξ + Σ Λ ( Ω + ¯ Ω ) Data b Old CR model rs New CR model bc . . . . . . Mean p ⊥ vs particle mass h p ⊥ i [ G e V ] rs rs rs rs rs rs rs rs rs rs rs rs rs bc bc bc bc bc bc bc bc bc bc bc bc bc . . . . . . . . . . . . mass [GeV] M C / D a t a Figure 21: Average p ⊥ versus hadron mass at E CM = 200 GeV , compared with STAR data [118].species and a comparison with the Λ p ⊥ spectrum measured by CMS is given in in fig. 20. Sadly, theimprovement is far less satisfactory here. The new CR model (as well as the old model) overshootsthe production in the very low p ⊥ and the high p ⊥ region, whereas too few Λ baryons are predictedin between. Thus the Λ baryons from junctions tend to fall in the right region, however the effect isnot large enough. An interesting observation is that the ratio Λ /K s is now well described in the low p ⊥ region. This shows that the problem with the p ⊥ distribution is not specific to baryons but is moregeneric. The discrepancy between data and the model for large p ⊥ still exists, however the baryonproduction in this region is primarily from diquark string breaks in jets and as such is not really uniquefor the new CR model. It may point to a revision needed of the spectrum of hard (leading?) baryonproduction in jets, which may not be unique to the pp environment, see [34].The problem in the low p ⊥ domain is a common theme for all heavier hadrons (i.e. anything butpions) and would be interesting to explore further. (E.g., a measurement of ρ spectra could revealwhether it depends on the presence of strange quarks.) The PYTHIA models predict a p ⊥ -distributionthat peaks at lower values than what is actually observed. To study this in more detail, one cancalculate the average p ⊥ for the identified hadrons and plot it a function of their mass, as done e.g.by the STAR collaboration for pp collisions at E CM = 200 GeV [118]. In purely longitudinal stringfragmentation the expected result is a roughly flat curve, since no correlation between the mass ofthe particle and p ⊥ is present. The flat prediction is altered when hadron decays and jet physics areincluded, leading to the curve seen in fig. 21. The prediction is also altered if the string is boosted(e.g., by partonic string endpoints), the boost is transferred to the final particles and for the same boostvelocity a heavy particle will gain more p ⊥ than a light one. This effect can be enhanced by CR, sinceminimisation of the λ -measure prefers reconnections among partons moving in the same direction,thus creating boosted strings [75]. CR is therefore expected to give a sharper rise of the (cid:104) p ⊥ (cid:105) vs massdistribution. Unfortunately, we do not observe this expected effect at any significant level (fig. 21). Tobe candid, it is disappointing that the new model does not appear to address this problem at all. At thevery least, it leaves room open for criticism and possibly additional new physics. Of special interest inthis context are possible collective phenomena, such as (gas-like) hadron reinteractions or (hydro-like)flow, either of which could provide a (weak or strong, respectively) velocity-equalising component,and at least the latter has been applied successfully in the context of the EPOS model [72]. These35 b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b Data b Old CR modelNew CR modelNon diffractiveDouble diffractiveSingle diffractive − Rapidity gap size in η starting from η = ± p T >
200 MeV d σ / d ∆ η F [ m b ] . . . . . . . . ∆ η F M C / D a t a (a) b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b Data b Old CR modelNew CR modelNon diffractiveDouble diffractiveSingle diffractive − Rapidity gap size in η starting from η = ± p T >
800 MeV d σ / d ∆ η F [ m b ] . . . . . . . . ∆ η F M C / D a t a (b) Figure 22: The rapidity gap survival for a low p ⊥ cut (a) and a high p ⊥ cut (b). The differentcomponents are also shown for the new CR model (DD/SD/ND). The gathered data is from the ATLASexperiment [119].effects are generally not expected to be present at the relatively low energy density in pp collisionsat a centre-of-mass energy of 200 GeV, however. It would therefore be of great interest to redothe measurement in detail at LHC energies, for high- and low-multiplicity samples and/or in theunderlying event, to study whether the slope is steeper at the higher energy densities.Finally, we emphasise that the rapidity gap survival is heavily affected by the choice of CRmodel [68, 84]. The explanation is similar to that of the three-jet LEP measurement, where the re-connected colour-singlet jet produces a rapidity gap to the other jets. The old CR model combinesdifferent strings into a single larger string, and thereby covers the same rapidity span, essentiallyadding kinks to an already existing string topology. Instead the new model can produce the colour-singlet union of particles in the same rapidity region. The new model is therefore expected to producemore rapidity gaps compared with the old model, which is also what we observe in fig. 22. The newmodel predictions are significantly above the data in the mid-range rapidity region for low p ⊥ cut-offs. (For higher p ⊥ cuts the effect vanishes, due to the partonic description being more influentialon the rapidity gap survival.) We therefore emphasise that the new CR scenario should in principlebe accompanied by a retuning of diffractive parameters. This is not straightforward however, andinvolves not only the shown rapidity-gap survival distributions, but several other measurements at dif-ferent energies. It was therefore deemed beyond the scope of this study to perform a retuning of thediffractive components and we limit ourselves to pointing out the interplay. A future dedicated studyof this aspect could also well incorporate a study of the interference between the new BR model anddiffractive events. Colour reconnections contribute one of the dominant uncertainties on current experimental top-massextractions in hadronic channels (see e.g. the mini-review in [65]), and their size was recently reex-36odel ˆ m top [GeV] ∆ ˆ m top [GeV] ∆ ˆ m rescaledtop [GeV]no CR . ± . . ± . − . ± . − . ± . default . ± . − . ± .
09 +0 . ± . new model ERD . ± . − . ± .
09 +0 . ± . new model . ± . − . ± .
09 +0 . ± . max CR ERD . ± . − . ± . − . ± . no CR (tuned) . ± . − . ± . − . ± . max CR (tuned) . ± .
07 +0 . ± .
09 +0 . ± . Table 2: Values of m top as predicted by the different CR models. The rescaled top mass is obtainedby ˆ m rescaledtop = . m W ˆ m top .amined in the context of several simplified CR schemes [71]. It is thus interesting to consider our newCR model in the same framework. We follow ref. [71], to which we refer for details on selection ofthe events and top mass reconstruction procedure.Briefly stated, the idea is to select semi-leptonic top events, using the charged lepton and escapingneutrino mainly for tagging and then reconstruct the top mass from the hadronic decay. A masswindow around the W mass is required and the raw top mass is extracted by fitting the invariant massdistribution of the three jets with a skewed Gaussian distribution, which fits the distributions betterthan a standard Gaussian [71]. For the models/tunes considered in this work (details below), theresulting shift on the calibrated top mass is below 200 MeV, which is comparable with the currentlevel of CR uncertainty on the measurements [58, 59, 63, 64] and far below some of the “devil’s-advocate” toy models considered in [71]. Characteristic for those models is that they allow somefraction of reconnections where the λ measure is increased, whereas all the models considered hereinvolve a minimisation of λ , one way or another.We compare six models: no CR, the existing (default) PYTHIA 8 CR model with and withoutearly resonance decays (ERD), the new baseline CR model with and without ERD, and the new CRwith maximal CR and ERD. Since the largest effect is expected for ERD, the maximal CR scenario isonly considered with ERD switched on. Two versions of the “no CR” scenario were considered, onein which CR was simply switched off without any further changes (resulting in significant increasein central hadron multiplicity) and the other a semi-tuned version, where the activity ( (cid:80) E ⊥ ) in thecentral region ( | η | < ) for ND events was retuned. A similar approach is used for the maximal CRmodels, where again a non-tuned version and retuned version are considered. It should be noted thatneither of the retuned models provide a good description of all MB data (for instance (cid:104) p ⊥ (cid:105) ( n charged ) is described by neither tune). Since none of the models considered exhibit any top-specific behaviour,no additional retuning to top events was needed. The results are collected in tab. 2.The first observation is the large difference between the no CR model and the non ERD models for ∆ ˆ m top ( ∼ MeV). Since no CR is performed for the top decay products, not much of differencewas expected, and any difference observed has to reside entirely in the underlying event (UE). TheCR models considered lower the total string length, and thereby the activity in the UE, which directlyinfluences the non-scaled top mass. Since the “no CR” model uses the same tune parameters as the CRmodels, it has a too high activity in the UE, leading to the negative mass shifts seen. After retuning toa similar activity in the central region, the “no CR (tuned)” model agrees with the default CR scenariowithin the statistical uncertainty. This emphasises the importance of using consistent UE tunes for thistype of exercise, though we also note that after recalibration by the hadronic W mass, the rescaled top37ass ˆ m rescaledtop is remarkably stable.For the ERD models the above shift in UE is still present, but the UE is now also allowed toreconnect with the top decay products. Therefore, an UE parton in close proximity to a jet from thetop decay will have a large probability to be reconnected with the jet. This will result in narrowerjets, leading to less of the energy falling outside the jet cone, and thereby a larger top mass. This is inagreement with the simulations, where the ERD results are above the non-ERD results. But the effectis smaller than that of the UE activity, thus the overall shift is still negative compared to the non-tunedno CR model.Both of the above effects are magnified in the maximal CR scenario. We remind the reader thatin this scenario, the /N C suppression of subleading connections is switched off, hence this shouldbe considered an unphysical extreme variation. Coincidentally, however, the two effects end up ap-proximately canceling. A similar retuning of the central activity as above, “max CR (tuned)”, showsa significant increase in the top mass shift of more than one GeV with respect to the tuned no CRmodel, though again, the W mass calibration removes most of it.The fact that the rescaled top mass ( ˆ m rescaledtop ) is less sensitive to CR is due to a cancellationbetween CR in the W mass and the top mass. This is in perfect agreement with the simulations, wherethe deviation for all rescaled masses are below their respective non-rescaled deviations. On the otherhand, this means that any interpretation of where the variations between the models arise becomesextremely difficult. We will therefore refrain from attempting this, and instead purely discuss thenumerical values in term of the uncertainty on the top mass.For the rescaled top mass, the differences between the models stay below 200 MeV. This is slightlyless than what was observed earlier even for identical CR models (default) [71]. The variations in theresults can be attributed to a new tune combined with a change in the PS for t ¯ t events. We regardthe smaller differences as somewhat coincidental however, and further work is needed to genuinelyimprove our understanding of CR effects in the top mass measurement. What we can say at least isthat the results from the new model lie within those from the default CR model, and therefore do notgenerate a need for larger uncertainties. Even the maximal CR, which is our attempt to mimic thevery large shifts seen for models that ignore MB/UE constraints, does not change this picture. Insteada pattern emerges, namely that whenever the minimisation of the λ measure is used as a guidelinefor the CR, the shifts stay below 300 MeV (taking the models studied in [71] into account as well).The reason for this is two-fold: firstly the coherence of the PS ensures that the jet structure is not toosignificantly altered; secondly, the alterations are realised in a systematic fashion leading to a similarshift in both the top and the W mass, implying that the hadronic W mass recalibration is highly robust.An increased understanding of this interplay could potentially lower the uncertainty even further.Since these models are mainly constrained by measurements, further gains can also be achievedby improving and extending the programme of measurements sensitive to CR effects. A few newobservables targeting top events specifically were already suggested in [71]. In the context of topmass uncertainties, such observables are of course especially relevant, as this is the closest to in-situconstraints as can be obtained, mimising the “extrapolation” that the model has to cover between theconstraint and measurement environments.In order to establish whether the small effects on m t predicted by λ -minimising models are indeedconservative or not, it would be of crucial importance to test these models as directly as possible ina variety of environments, top included. The fact that our new CR models do not yet give gooddescriptions of identified-particle p ⊥ spectra should, in this context, be seen as a warning that therecan be additional non-perturbative uncertainties left unaccounted for, possibly of a dynamical origin.38 Summary and Outlook
The question “between which partons do confining potentials arise?” is a fundamental one in non-perturbative QCD, which any attempt at modelling the process of hadronisation must address. Inthe leading-colour approximation and neglecting beam-remnant correlations, this is relatively simple:there is a one-to-one mapping between perturbative QCD dipoles and string pieces / clusters. In thispaper, we have attempted to take a first step beyond leading colour, by including a randomisation overthe set of possible subleading-colour topologies, with probabilities chosen according to a simplifiedversion of the SU (3) colour algebra.We present the argument that while the LC approximation may be quite good in the environmentof e + e − collisions (more specifically in the absence of multi-parton interactions), we expect verysignificant deviations from it in pp collisions, where the survival of the strict LC topology should beheavily suppressed. Although the probability for a subleading colour connection to be possible be-tween any given pair of (uncorrelated) partons is only roughly /N C , it becomes increasingly unlikely not to have any such connections as the number of uncorrelated partons increases, as e.g. in the casewhen considering MPI.This implies that a complex multi-parton system will in general have several different string/clusterconfigurations open to it, at the time of hadronisation; the LC one is only one among many possibil-ities. We invoke the string-length λ measure to choose which one is preferred, so far via a simplewinner-takes-all algorithm that does not purport to always find the global minimum. Nonetheless, webelieve that this model represents a significant step in the right direction, allowing us to probe for thefirst time the effects of subleading colour on hadronisation in a way that may be said to be systematicand consistent with (a simplified version of) QCD.One noteworthy new aspect of our work is the use of string junctions to represent antisymmetriccolour combinations, such as two colours combining to form an overall anticolour. This provides anew source of baryon production, with properties qualitatively different from the standard diquarkscenario. We have shown that this aspect allows to reconcile measured baryon/meson ratios with thestring model in both pp and ee collisions simultaneously. However, we caution that the shapes ofthe p ⊥ spectra are still not well described. We had anticipated that the preference of reconnectionsto produce boosted string pieces should lead to an enhancement of the (cid:104) p ⊥ (cid:105) especially for heavierhadrons, but the magnitude of this effect observed in our model is still far too low to explain the data.We emphasised that there is a conceptual difference between colour connections and colour re-connections . The former is related to colour-space ambiguities , such as the unknown colour corre-lations between different MPI initiators or the subleading-colour connections explored in this work.Colour reconnections are related to dynamical reconfigurations of the colour/string space, via pertur-bative gluon exchanges or non-perturbative string interactions; i.e., they involve momentum exchangeas well. We did not explore effects of the latter type directly in this work, though we note thatthe fairly realistic string-interaction scenarios constructed by Khoze and Sj¨ostrand in the context of e + e − → W + W − studies [43, 66, 67] also feature string-length minimisation; hence it is possible thatour tuned parameters effectively attempt to cover both types. If so, the fact that the momentum spectraremain discrepant may point to the need for dynamical CR.Finally, we presented a few suggestions for additional observables, the measurement of whichwould give further insight and possibly help to distinguish both physical and unphysical CR models,as well as other ideas such as models based on colour ropes [76, 77], hydrodynamics [72], or (non-hydro) hadron rescattering. We ended by considering the simplified top-mass analysis of [71] andconclude that the models presented here lead to shifts in the top mass of order 200 MeV, which iswithin the current level of non-perturbative uncertainties on the measurements.39 cknowledgements Many thanks to T. Sj¨ostrand for his valuable comments on both our physics and our code. JRC thanksthe CERN theory unit for hospitality during the main part of this study. Work supported in partby the Swedish Research Council, contract number 621-2013-4287, in part by the MCnetITN FP7Marie Curie Initial Training Network, contract PITN- GA-2012-315877, and in part by the AustralianResearch Council, contract FT130100744.
A Model parameters
A complete list of all the parameters that differ from the Monash tune for the three different modelsare listed in the table below.Parameter Monash Mode 0 Mode 2 Mode 3StringPT:sigma = 0.335 = 0.335 = 0.335 = 0.335StringZ:aLund = 0.68 = 0.36 = 0.36 = 0.36StringZ:bLund = 0.98 = 0.56 = 0.56 = 0.56StringFlav:probQQtoQ = 0.081 = 0.078 = 0.078 = 0.078StringFlav:ProbStoUD = 0.217 = 0.2 = 0.2 = 0.2StringFlav:probQQ1toQQ0join = 0.5, = 0.0275, = 0.0275, = 0.0275,0.7, 0.0275, 0.0275, 0.0275,0.9, 0.0275, 0.0275, 0.0275,1.0 0.0275 0.0275 0.0275MultiPartonInteractions:pT0Ref = 2.28 = 2.12 = 2.15 = 2.05BeamRemnants:remnantMode = 0 = 1 = 1 = 1BeamRemnants:saturation - = 5 = 5 = 5ColourReconnection:mode = 0 = 1 = 1 = 1ColourReconnection:allowDoubleJunRem = on = off = off = offColourReconnection:m0 - = 2.9 = 0.3 = 0.3ColourReconnection:allowJunctions - = on = on = onColourReconnection:junctionCorrection - = 1.43 = 1.20 = 1.15ColourReconnection:timeDilationMode - = 0 = 2 = 3ColourReconnection:timeDilationPar - - = 0.18 = 0.073
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