# The space–time structure of hadronization in the Lund model

## Abstract

The assumption of linear confinement leads to a proportionality of the energy–momentum and space–time pictures of fragmentation for a simple \(\mathrm{q}\bar{\mathrm{q}}\) system in the Lund string model. The hadronization of more complicated systems is more difficult to describe, and in the past only the energy–momentum picture has been implemented. In this article also the space–time picture is worked out, for open and closed multiparton topologies, for junction systems, and for massive quarks. Some first results are presented, for toy systems but in particular for LHC events. The density of hadron production is quantified under different conditions. The (not unexpected) conclusion is that this density can become quite high, and thereby motivate the observed collective behaviour in high-multiplicity \(\mathrm{p}\mathrm{p}\) collisions. The new framework, made available as part of the Pythia event generator, offers a starting point for future model building in a number of respects, such as hadronic rescattering.

## 1 Introduction

The Standard Model of particle physics is solidly established by now, and has been very successful in describing all perturbatively calculable observables for LHC \(\mathrm{p}\mathrm{p}\) collisions, i.e. those dominated by large momentum transfer scales [1]. But at lower scales the perturbative approach breaks down, and phenomenological models have to be developed.

One of the underlying assumptions for these models has been that the nonperturbative hadronization process, wherein the perturbatively produced partons turn into observable hadrons, is of a universal character. Then relevant nonperturbative parameters can be determined e.g. from LEP data, and afterwards be applied unmodified to LHC \(\mathrm{p}\mathrm{p}\) collisions. The hadronizing partonic state is quite different in the two processes, however. Firstly, the composite nature of the incoming protons leads to multiple semiperturbative parton–parton collisions, so-called MultiParton Interactions (MPIs) [2, 3], and also to beam remnants and initial-state QCD radiation. Secondly, the high number of interacting partons leads to the possibility of nontrivial and dynamically evolving colour topologies, collectively referred to as Colour Reconnection (CR) phenomena. Both MPIs and CR need to be modelled, and involve further new parameters. (CR has been observed in the cleaner \(\mathrm{e}^+ \mathrm{e}^- \rightarrow \mathrm{W}^+ \mathrm{W}^-\) process by the LEP collaborations [4], but that information is not easily transposed to the \(\mathrm{p}\mathrm{p}\) context.)

The most successful approach to providing a combined description of all relevant phenomena, at all scales, is that of event generators. Here Monte Carlo methods are used to emulate the quantum mechanical event-by-event fluctuations at the many stages of the evolution of an event [5]. For \(\mathrm{p}\mathrm{p}\) physics the three most commonly used generators are Pythia [6, 7], Herwig [8, 9] and Sherpa [10]. Fragmentation here proceeds either via strings [11], for the former, or via clusters [12], for the latter two. A note on terminology: “fragmentation” and “hadronization” can be used almost interchangeably, but the former is more specific to the breakup of a partonic system into a set of primary hadrons, whereas the latter is more generic and can also include e.g. decays of short-lived resonances.

In spite of an overall reasonable description, glaring discrepancies between data and models have been found in some cases. Most interesting is that high-multiplicity LHC \(\mathrm{p}\mathrm{p}\) events show a behaviour that resembles the one normally associated with heavy-ion collisions and the formation of a Quark-Gluon Plasma (QGP). In particular, ALICE has shown that the fraction of strange baryons increases with multiplicity, the more steeply the more strange quarks the baryon contains, while the proton rate is not enhanced [13]. Long-range azimuthal “ridge” correlations have also been observed by both CMS [14, 15] and ATLAS [16], as well as other signals of collective flow [17, 18, 19].

This is unlike conventional expectations, that QGP formation requires volumes and timescales larger than the one that can be obtained in \(\mathrm{p}\mathrm{p}\) collisions [20, 21, 22]. Nevertheless core–corona models have been developed, like the one implemented in EPOS [23], where a central high-density region can turn into a QGP, while the rest of the system remains as normal individual strings. Other mechanisms that have been proposed include rope formation [24] and shoving [25], or an environment-dependent string tension [26]. Common for all of them is that they introduce a space–time picture of the collision process.

In the traditional Lund string model [11] the linear confinement potential leads to a linear relationship between the energy–momentum and space–time pictures of a simple \(\mathrm{q}\bar{\mathrm{q}}\) fragmenting system. Many of the above models are based on the approximation of a number of such simple strings, parallel along the \(\mathrm{p}\mathrm{p}\) collision axis but displaced in the transverse plane by the collision/MPI geometry.

For a generic multiparton system, like \(\mathrm{q}\mathrm{g}_1\mathrm{g}_2\ldots \mathrm{g}_n\bar{\mathrm{q}}\), only an energy–momentum picture has been available until now [27]. The purpose of this article is to overcome that limitation, and provide a full space–time picture of the hadronization process, as part of the Pythia event generator.^{1} This will offer a natural starting point for more detailed future studies of a number of collective effects. The models mentioned above deal with the space–time structure before (like core–corona or shove) or during (like ropes or QGP) fragmentation. To this we would also add a possibility for studies of what happens after the first stages of the hadronization, when hadronic rescattering and decays can occur in parallel. In addition to the already mentioned observables, Bose–Einstein correlations could also be used to characterize final states.

A warning is that we are applying semiclassical models to describe the quantum world. Formally the Heisenberg uncertainty relations impose limits on how much simultaneous energy–momentum and space–time information one can have on an individual hadron. Our approach should still make sense when averaged over many hadrons in many events, as will always be the case.

The plan of the article is as follows. Section 2 gives a brief summary of relevant earlier work, on the “complete” description of the simple \(\mathrm{q}\bar{\mathrm{q}}\) system [11], and on the energy–momentum picture of an arbitrary partonic system [27]. Section 3 then introduces the new framework that provides a space–time picture also in a general configuration. Several special cases need to be addressed, and technical complications have to be sorted out, with some details relegated to two appendices. Section 4 contains some first studies, partly for toy systems but mainly for LHC events. This is without any of the collective effects that may be added later, but still provides an interesting overview of the overall space–time evolution of hadronization at the LHC. Finally, Sect. 5 concludes with a summary and outlook.

## 2 The Lund String model

### 2.1 The linear force field in QCD

*r*is the distance between them and \(\alpha _{\mathrm{s}}\) is the strong coupling constant. The presence of a linear term was first inferred from hadron spectroscopy (Regge trajectories), from which a \(\kappa \approx 1\) GeV/fm can be extracted, and has later been confirmed by lattice QCD calculations.

The linear term dominates at large distances, and in the Lund string model only this term is used to describe the breakup of a high-mass \(\mathrm{q}\bar{\mathrm{q}}\) system into several smaller-mass ones. Then the full colour field can be approximated by a one-dimensional string stretched straight between the \(\mathrm{q}\) and \(\bar{\mathrm{q}}\), Fig. 1. This string can be viewed as parametrizing the center of a cylindrical region of uniform width along its full length, such that the longitudinal and transverse degrees of freedom almost completely decouple.

### 2.2 The two-parton system

#### 2.2.1 Simple string motion

The simplest yo-yo system can be generalized as illustrated in Fig. 2b, where the quark and the antiquark have different initial energies, \(E_{\mathrm{q}} \ne E_{\bar{\mathrm{q}}}\). Equivalently, this system can be viewed as a boosted copy of the rest-frame setup in Fig. 2a. The energy–momentum and space–time coordinates suffer simultaneous transformations under a longitudinal boost, and Eq. (3) holds also after the boost. The transformation is especially easily formulated in light-cone coordinates, where \(\tilde{p}'^{\pm } = k^{\pm 1} \tilde{p}^{\pm }\) with \(k = \sqrt{(1 + \beta )/(1 - \beta )}\) for a boost with velocity \(\beta \), and similarly for \(\tilde{z}^{\pm }\).

Note that a string piece with \(E = \kappa l\) but \(p_z = 0\) in the original rest frame will obtain a \(p_z \ne 0\) after a boost to the frame with \(E_{\mathrm{q}} \ne E_{\bar{\mathrm{q}}}\). This is in seeming contradiction with a description set up in a rest frame where \(E_{\mathrm{q}} \ne E_{\bar{\mathrm{q}}}\) from the onset, where one would again expect \(p_z = 0\). The solution is that a string piece is an extended object, so that the two ends of it, if originally simultaneous, will no longer be it after the boost. Only a string piece at constant time in the new frame will obey \(E = \kappa l\) and \(p_z = 0\) there.

#### 2.2.2 String breaking and hadron formation

#### 2.2.3 Selection of breakup vertices

*i*can be written as

*z*value of each new hadron, where \(z = z^+\) (\(z = z^-\)) for fragmentation from the \(\mathrm{q}_0\) (\(\bar{\mathrm{q}}_0\)) end. The

*a*and

*b*are parameters that should be tuned to reproduce the experimental data. Hence,

*f*(

*z*) determines how the individual vertices correlate in order to create a hadron of mass \(m_h\) by taking a fraction

*z*of the energy–momentum left in the system. Note that the form of

*f*(

*z*) does not depend on previous steps taken, which leads to a flat rapidity plateau of the inclusive hadron production.

*i*and \(i-1\) breakups, respectively. The \(\mathrm{q}_0\) and \(\bar{\mathrm{q}}_0\) turning points define \(\varGamma _0 = 0\). The inclusive \(\varGamma \) distribution, after some steps away from the endpoint(s), is

*a*and

*b*as in Eq. (12).

The breaks of the string can be determined from Eq. (11) by iteratively picking \(z_i\) values according to Eq. (12) for the hadrons with masses \(m_i\) . This works well for the simple \(\mathrm{q}_0 \bar{\mathrm{q}}_0\) system, but Eq. (11) will not hold in systems with more than two partons. To this end an alternative procedure can be introduced [27] via \(\varGamma \) recursion. Here a \(z = z_i\) is still selected by Eq. (12), and converted to a \(\varGamma _i\) by Eq. (13). As illustrated in Fig. 4, each fixed \(\varGamma \) value corresponds to a hyperbola with the origin as its center. Correspondingly each fixed \(m_i\) corresponds to a hyperbola with the \(i-1\) vertex as its center. Therefore a given \((m_i, \varGamma _i)\) pair corresponds to the unique crossing of two hyperbolae at the location of the next vertex.

#### 2.2.4 The tunneling process

The combination of a \(\mathrm{q}_{i-1}\) and a \(\bar{\mathrm{q}}_i\) gives the flavour of a meson but does not fully specify it. The quark spins can combine e.g. to produce a pseudoscalar or a vector meson, and flavour-diagonal mesons mix, and so on. All of these aspects are relevant for the model as a whole, but for the considerations in this article we only need to know the masses of the produced mesons. Similarly for baryon production, where the production mechanisms are less well understood, whether “diquark” or “popcorn” [31, 32]. In the latter approach actually three different production vertices are involved, one for each of the quarks, but also here an effective description in terms of two, as for mesons, is meaningful. A diquark is taken to be a colour antitriplet, just like an antiquark, and we thus use the notation \(\bar{\mathrm{q}}\) as shorthand for either of them.

Since the string itself has no transverse motion, it is assumed that the transverse momentum is locally compensated inside each \(\mathrm{q}_i\bar{\mathrm{q}}_i\) pair. The transverse momentum of a hadron \(\mathrm{q}_{i-1}\bar{\mathrm{q}}_i\) is then given by the vector sum of its constituent transverse momenta. The hadron masses in Sect. 2.2.2 have to be replaced by the corresponding transverse masses.

#### 2.2.5 Massive quarks

*z*axis in opposite directions. The massive yo-yo system is depicted in Fig. 5a, along with the massless case for comparison. At time \(t = 0\),

Although the motion properties of the massless and massive cases hold in every longitudinal boosted frame, the effects of boosts are simpler to address for massless quarks. A useful trick is to replace the effect of the quark mass by an extra string piece of length \((E_{\mathrm{c}}(t) - p_{z,\mathrm{c}}(t))/\kappa \) at each endpoint. Its length is \(m_{\mathrm{q}}/\kappa \) at the turning point, see Fig. 5b, where the massive motion is illustrated by the hyperbolae, whose asymptotes are the straight lines of the massless case. Thereby time \(t = 0\) is also offset to account for the reduced oscillation time. The extra string piece is purely fictitious and does not break during the fragmentation process. The physical region, between the hyperbolae, is highlighted in grey in Fig. 5b. Given that the hadron created from the endpoint is always heavier than the endpoint quark, all the hadrons are automatically created inside the physical region.

### 2.3 Multiparton systems

In the Lund model the colour flow is based on the limit of infinitely many colours [33]. Then there is one string piece from the \(\mathrm{q}\) to the \(\mathrm{g}\) and another from the \(\mathrm{g}\) to the \(\bar{\mathrm{q}}\), and the two do not interfere. The gluon thus can be viewed as a kink on a single string stretched from the \(\mathrm{q}\) to the \(\bar{\mathrm{q}}\).

Although Fig. 6 is useful to visualize the time evolution of the system, the parameter plane picture is most convenient when addressing the kinematics [27]. This is a diagram that displays the different string regions in terms of the light-cone four-vectors defining each region, i.e. \(p^+_{\mathrm{q}} = p_{\mathrm{q}}\), \(p^-_{\bar{\mathrm{q}}} = p_{\bar{\mathrm{q}}}\) and \(p^+_{\mathrm{g}} = p^-_{\mathrm{g}} = p_{\mathrm{g}}/2\) in the \(\mathrm{q}\mathrm{g}\bar{\mathrm{q}}\) case, whose parameter plane is displayed in Fig. 7. The low regions represent the states in which none of the partons have lost their energy, corresponding to the two string regions in view 1 of Fig. 6, the \(\mathrm{q}\mathrm{g}\) and the \(\mathrm{g}\bar{\mathrm{q}}\) string pieces. The intermediate region corresponds to the new string piece created from the \(\mathrm{q}\) and \(\bar{\mathrm{q}}\) momenta once the gluon has lost all its energy. Finally, the upper regions are related to the two string pieces formed when \(\mathrm{g}\) re-appears. Although the complete parameter plane picture (for half a period) is the one shown in Fig. 7, the dashed upper regions are normally neglected, since the system is assumed to fragment before then. This reasonable assumption avoids a large number of complications for handling fragmentation in these regions. The three remaining regions are then formed by the combination of one \(+\) component and one − one, where ± no longer relates to motion along the \(\pm z\) axis, but more generically denotes the reference vector directed towards (\(+\)) or away from (−) the \(\mathrm{q}\) end of the system.

*n*partons, out of which \(n-2\) are gluons, as \(n-1\) initial regions and \((n-1) (n-2)/2\) intermediate regions. The expression for the hadron four-momentum can also be generalized to an

*n*-parton system by accounting for the momenta taken from each parton as

### 2.4 Fragmentation implementation summary

The fragmentation process in Pythia is based on the four-momenta of the partons created in the (semi)perturbative stages of the collision process, plus the partons in the beam remnants [34]. By the colour-connection between those partons, an LHC event is likely to contain several \(\mathrm{q}\mathrm{g}_1\mathrm{g}_2\ldots \mathrm{g}_{n-2}\bar{\mathrm{q}}\) systems, that can be handled separately.

The production of each new hadron begins with the selection at random of whether to split it off from the \(\mathrm{q}\) end or from the \(\bar{\mathrm{q}}\) one of the system. The flavour of a new \(\mathrm{q}\bar{\mathrm{q}}\) break of the string (where \(\mathrm{q}\) may also represent an antidiquark), leads to the formation of a new hadron, as already described. Its mass is selected, according to a Breit–Wigner for short-lived particles with a non-negligible width. The transverse momentum is obtained as the vector sum of those of the hadron constituents, assuming that the old and new breakup vertices are in the same region. Then the longitudinal momentum fraction *z* is picked according to the probability distribution in Eq. (12), with the difference that the hadron mass has to be replaced by the transverse ditto, \(m_{h} \rightarrow m_{\perp h}\). In a simple \(\mathrm{q}\bar{\mathrm{q}}\) system, the new breakup vertex is easily obtained from the \((m_{\perp h},z)\) pair. Else the \(\varGamma _i\) value of the new breakup is calculated using Eq. (13), again with \(m_{h} \rightarrow m_{\perp h}\), and a solution is sought to the \((m_{\perp h}, \varGamma _i)\) pair of equations. Vertex *i* may end up in the same string region as \(i - 1\), or involve a search in other regions. Among technical complications of this search is that the transverse directions are local to each string region, which leads to discontinuities in the hyperbolae of constant \(m_{\perp h}\) at the borders between string regions, that would not be there for \(p_{\perp }= 0\).

The random steps from both string ends continue until the remaining invariant mass of the system is deemed so small that only two final hadrons should be produced. Details on this final step can be found in “Appendix A”, along with the challenges encountered when implementing the space–time picture and the methods applied to solve them. Had the fragmentation always proceeded from the \(\mathrm{q}\) end, say, the final step would always have been at the \(\bar{\mathrm{q}}\) end, with the minor blemishes of this step concentrated there. Now these are instead smeared out over the whole event.

## 3 The space–time description

So far, the fragmentation process in Pythia was developed in terms of the energy–momentum fractions \(x^{\pm }\) and \(z^{\pm }\) of breakup vertices and hadrons, presented in Sect. 2.2.2. Therefore, the location of the breakup vertices is only specified in the energy–momentum picture. In order to study the density of hadron production, this information should first be translated to the space–time one, which will be done in this section.

### 3.1 The two-parton system

*i*in a simple \(\mathrm{q}\bar{\mathrm{q}}\) system. Its location with respect to the origin of the energy–momentum picture, where \(\mathrm{q}\) and \(\bar{\mathrm{q}}\) have been created, is given by the \(\hat{x}^{\pm }\) fractions. Then, considering \(p^+\) to be the \(\mathrm{q}\) four-momentum and \(p^-\) the \(\bar{\mathrm{q}}\) four-momentum, the location of breakup

*i*in the energy–momentum picture is defined as \(\hat{x}_i^+ p^+ + \hat{x}_i^- p^-\). Recalling the linear relation between space–time and energy–momentum, Eq. (3), the space–time location of breakup point

*i*thereby is defined as

*i*and \(i+1\), with space–time coordinates \(v_i\) and \(v_{i+1}\), together define the \(\mathrm{q}_i\bar{\mathrm{q}}_{i+1}\) subsystem that forms hadron

*i*. One obvious choice is then to define the hadron production point as the average of the two,

### 3.2 More complex topologies

If the system is composed of more than one gluon, also more than one intermediate region has to be taken into account, as illustrated in Fig. 11. In such cases, more gluons have to be included when determining the space–time offset of some intermediate regions, such as the \(\mathrm{q}\mathrm{g}_3\) one. This region is created when both \(\mathrm{g}_1\) and \(\mathrm{g}_2\) have lost their energies, giving an offset \(v_{\mathrm {reg}} = (p_{g_1}+p_{g_2}) /2\kappa \).

*jk*are for ones containing four-momenta from partons

*j*and \(k, k\ge j\). The

*jk*region offset is found to be

*m*, and for a breakup vertex in this region it thus holds that,

### 3.3 Gluon loops

So far, gluons have only appeared in open strings between a \(\mathrm{q}\) and a \(\bar{\mathrm{q}}\) end, but it is also possible to have closed gluon loops, as exemplified in Fig. 12a for a \(\mathrm{g}\mathrm{g}\mathrm{g}\) system. In order to reduce the problem to a familiar one, an initial \(\mathrm{q}\bar{\mathrm{q}}\) is generated by string breaking in one of the string regions. This break should be representative of what ordinary fragmentation is expected to give. Thus the region is chosen at random, but with a bias towards ones with larger masses, where more ordinary string breaks are to be expected. Inside that region, the \(\varGamma \) value of the vertex is chosen according to Eq. (14), and a further random choice gives the longitudinal location of the breakup. Having taken this step, the *n*-gluon-loop topology is effectively mapped onto an \((n+1)\)-parton open string with \(\mathrm{q}\) and \(\bar{\mathrm{q}}\) as endpoints. The key difference is that, unlike open strings considered so far, \(\varGamma _{\mathrm{q}} = \varGamma _{\bar{\mathrm{q}}} \ne 0\).

As an example, the parameter plane for a gluon-loop consisting of three gluons is displayed in Fig. 12b. In this case, the string between \(\mathrm{g}_1\) and \(\mathrm{g}_3\) has broken into two string pieces, generating two new string regions, \(\mathrm{g}_1\mathrm{q}\) and \(\mathrm{g}_3\bar{\mathrm{q}}\). Although the full \(\mathrm{g}_1\mathrm{g}_3\) region is duplicated in the parameter plane, in the right endpoint region only the “active area” between \(\mathrm{q}\) and \(\mathrm{g}_1\) is open to fragmentation, while the left endpoint region only uses the complementary area between \(\mathrm{g}_3\) and \(\bar{\mathrm{q}}\). Apart from that, the fragmentation process can now play out in the same way as for an open string, with the same rules for the space–time locations of the breakups. Note that the \(\mathrm{q}\) and \(\bar{\mathrm{q}}\) “endpoints” correctly will be assigned the same creation vertex in this procedure.

### 3.4 Smearing in transverse space

*x*and

*y*are transverse spatial coordinates and \(\sigma \) is the width of the distribution.

### 3.5 Massive quark implementation

*z*-component of space, represented as \(\varDelta t\) and \(\varDelta z\). The former can be determined from the difference between the time coordinates at which the massless and massive quarks lose their three–momenta, \(t_{\mathrm {massless}}\) and \(t_{\mathrm {massive}}\) in Fig. 13a, i.e.

### 3.6 Other implementation details

Up until now, only open \(\mathrm{q}\mathrm{g}_1\mathrm{g}_2\ldots \mathrm{g}_{n-2}\bar{\mathrm{q}}\) and closed \(\mathrm{g}_1\mathrm{g}_2\ldots \mathrm{g}_n\) strings have been considered. A third possibility is junction topologies, wherein three string pieces meet in a common vertex [36], and whereby the junction effectively carries the baryon number of the system. Such topologies can arise e.g. when the three valence quarks are all kicked out of an incoming proton, but there are also scenarios in which further junctions and antijunctions may be formed [37].

A junction system consists of three different “legs”, each stretched from an endpoint quark via a number of gluons in to the junction. In Pythia the fragmentation process is most conveniently defined in the rest frame of the junction. Here the total energy of each leg is determined, and the two legs with the lowest energies are fragmented from the respective \(\mathrm{q}\) end inwards. The process stops when the next step would require more energy than left in the leg. Once the two initial legs have fragmented, the two leftover \(\mathrm{q}\) from the respective last breaks are combined to create a diquark. Together with the third leg and its original endpoint \(\mathrm{q}\), this diquark defines a final string system, which now fragments as a normal open string.

The assignment of space–time locations in junction topologies introduces no new principles, but requires some extra bookkeeping. The three junction legs are considered as three different systems, to be dealt with in the same order as they fragmented, starting from the leg with the lowest energy.

Low-invariant-mass systems hadronize about as high-mass ones, even if kinematics is more constrained. The exception is when the invariant mass of the system is so low that only one hadron can be formed. In such cases, the “early” hadron production point is at the origin of the \(\mathrm{q}\bar{\mathrm{q}}\) system, i.e. \(v_e^h= (0; 0, 0, 0)\). Note that smearing in transverse space will give rise to negative squared invariant times in such cases. This is not a problem if the reason is that the collision of two Lorentz-contracted proton “pancakes” naturally would lead to a spread of *x*, *y* coordinates of collisions at \(t=0\). The “middle” and “late” definitions are calculated from the four-momentum \(p_h\) of the hadron as \(v^h = p_h/2\kappa \) and \(v_l^h = p_h/\kappa \), respectively.

Truly stable particles are only \(\mathrm{e}^{\pm }\), \(\mathrm{p}\), \(\bar{\mathrm{p}}\), \(\gamma \) and the neutrinos. Also some weakly decaying particles with long lifetimes are effectively treated as stable by default: \(\mu ^{\pm }\), \(\pi ^{\pm }\), \(\mathrm{K}^{\pm }\), \(\mathrm{K}_L^0\) and \(\mathrm{n}/\bar{\mathrm{n}}\).

### 3.7 A comparison of time scales

*E*and transverse momentum \(p_{\perp }\) [38]. This expression is conveniently split into a “Heisenberg uncertainty” factor (\(p_{\perp }\) is a measure of the off-shellness of intermediate propagators) and a “time dilation” factor, as indicated. Similar relations hold for emissions off the two incoming partons.

Typically, parton shower descriptions in event generators such as Pythia stop at scales of the order \(\mathrm{p}_{\perp \mathrm {min}}\,=\,\)0.5–1 GeV, mainly because \(\alpha _{\mathrm{s}}\) becomes so big that perturbation theory cannot be trusted below that. (The current default value for Pythia final-state radiation is 0.5 GeV, but that is the \(p_{\perp }\) for each daughter of a branching with respect to the mother direction, meaning a separation of 1 GeV between the two daughters. Eq. (41) should not be trusted up to factors of 2 anyway.) This corresponds to a \(\tau _{\mathrm {regen}} \approx 0.25\) fm, say, to be compared with the average hadronization time \(\langle \tau _{\mathrm {had}} \rangle \approx 1.3\) fm (see Sect. 4.2 below), i.e. about a factor five difference. To a good first approximation, the simulated perturbative activity can therefore be viewed as happening in a single point as far as the hadronization process is concerned. This is even more so for the hard perturbative activity that gives rise to separate jets, for which \(p_{\perp }\gg 1\) GeV. The emissions that possibly are simulated below 1 GeV can only give small wrinkles on the strings stretched between the main partons.

The comparison of invariant time generalizes to hold everywhere in an event, since time dilation works the same way for showers and hadronization. That is, a perturbative splitting at high energy and low \(p_{\perp }\) may occur at large time scales as measured in the rest frame of the event, when hadronization already started in the central region, but still well before it will begin in the part of the event that could be affected by the splitting.

At the end of the Pythia showers, the total number of partons in a typical LHC event is roughly half of the number of primary hadrons later produced. Given that the size, in each of three spatial dimensions, is only a fifth for the partonic system compared with the hadronic one, it might seem that the partonic density is much higher than the the hadronic one, and that partonic close-packing would be a more severe issue than hadronic ditto. Partons don’t have a well-defined size, however. A newly created parton could be assigned a vanishingly small size, and then the colour field surrounding it would expand with the speed of light. Thus the partonic size could be equated with the time since creation, multiplied by a standard time dilation factor.

At early times the partonic system of a collision therefore expands in size at about the same rate as the size of partons, and any net effect comes from the rise of the total number of partons as the cascade evolves from early times. Here the colour coherence phenomenon enters, however [38]. It is the obervation that the two daughters of a \( \mathrm{q}\rightarrow \mathrm{q}\mathrm{g}\) or \(\mathrm{g}\rightarrow \mathrm{g}\mathrm{g}\) branching share a newly-created colour-anticolour pair, that cannot contribute to the radiation until the partons are more separated than the wavelength of the further radiated partons. This gives a mechanism for close-packing avoidance, in event generators implemented in terms of angular or \(p_{\perp }\) ordering of radiation.

Had the parton shower been allowed to evolve further than the current cutoff, the partonic multiplicity and the partonic overlap would have increased as the \(\Lambda _{\mathrm {QCD}}\) scale of \(\approx 0.3\) GeV is approached. By then the naive size of partons would be of the order of 0.7 fm, which is about the expected transverse size of strings, and soft partons emitted at this stage form part of the emergent strings. We do not know how to model these late stages of the cascade, but any effects coming from them are included in the tuned parameters of the string fragmentation framework.

The picture painted here is based on studying one partonic cascade. Since protons are composite object, however, several partonic subcollisions can occur when two protons collide – MPIs. One therefore also should worry about the overlap of cascades from different MPIs – partonic rescattering. In part this issue has been studied [39], and shown to give small effects. That study only included the effects of parton multiplication by initial-state radiation, as encoded in parton distribution functions, and thus did not address the effects of collisions between partons from two separate MPI subcollisions. In general, however, MPIs occur at different transverse locations when the two Lorentz-contracted protons collide, and the products move out in different rapidities and azimuthal directions. Also here it is therefore plausible with only minor overlap at early times and large perturbative scales. (In a relative sense; most MPIs do not have all that large \(p_{\perp }\) values.) The overlap becomes relevant at later scales, where colour reconnection is the currently favoured mechanism for interactions between the emerging colour fields.

This reasoning generalizes: an event with few, fast particles can only be obtained when the \(\varGamma \) values and the breakup times are small. Conversely, events with high multiplicities of lower-momentum hadrons require high \(\varGamma \) values and late hadronization times. Whether early or late invariant times, however, the hadronization will still start in the middle and spread outwards.

## 4 Hadron density studies

### 4.1 Longitudinal and transverse distributions

*y*:

*y*and \(y_{\tau }\), as illustrated in Fig. 14 for the default “middle” definition of production points. The spread from the diagonal comes from a number of effects, such as the probabilistic fragmentation process, given by Eq. (12), and hadronic decays.

Figures 15 and 16 display the longitudinal and transverse spectra for \(\mathrm{p}\mathrm{p}\) collisions at \(\sqrt{s} = 13\) TeV given by the “early”, “middle” and “late” definitions of hadron production points, represented in green, red and blue, respectively. In the same figures, the spectra for a single string, at the same CM energy, using the “middle” definition are also illustrated in black. Both primary and secondary hadrons are taken into account.

The longitudinal spectra for the different definitions are very similar, as can be seen in Fig. 15. The largest disagreement is visible around \(y_{\tau } \approx 0\), where the spectra of the “early” definition peaks more, but “early” also has more particles at the very largest \(y_{\tau }\) values. In short, the “early” alternative maximizes the extreme behaviour of hadron production, whereas the “late” one minimizes it. The differences are not bigger than that we can consider the “middle” definition a fairly reliable one.

*r*values, as can be seen in the difference between the two “middle”

*r*distributions, while the distribution at larger

*r*values is rather insensitive. The difference between the “early”, “middle” and “late” production points is larger than for the longitudinal spectra, but still sufficiently close as to give confidence that meaningful results can be obtained. In the following, all plots will be for the “middle” definition.

### 4.2 Temporal and radial evolution of hadron production

The number of hadrons is shown as a function of time for a single string with \(\sqrt{s} = 20\) GeV in Fig. 17. The red curve corresponds to the number of primary hadrons, formed by the string fragmentation, that have not decayed at the time, while the green curve represents the number of secondary hadrons, from particle decays. The total number of hadrons, illustrated in blue, is the sum of primary and secondary hadrons. The brown curve represents the number of final (i.e. stable) hadrons, see Sect. 3.6. Finally, the black curve depicts the number of hadrons with \(|z| < 0.5\) fm, to be discussed in Sect. 4.3.

*a*and

*b*in Eqs. (12) and (14) are set to \(a=0.68\) and \(b=0.98\) GeV\(^{-2}\) [41], giving rise to a suppression of small \(\varGamma \) values of breakup vertices, and thereby also of small hadron production times. In detail, the relation between \(\varGamma \) and \(\tau \), Eq. (9), implies \(P(\varGamma ) \propto \varGamma ^a \mathrm{d}\varGamma \propto \tau ^{2a} \, \tau \, \mathrm{d}\tau = \tau ^{2a+1} \, \mathrm{d}\tau \) for \(\tau \rightarrow 0\). Furthermore, the expectation value of \(\langle \varGamma \rangle = (1 + a)/b \approx 1.7\) GeV\(^2\) gives \(\langle \tau \rangle \approx \sqrt{ \langle \varGamma \rangle } /\kappa \approx 1.3\) fm, in agreement with Fig. 18. Because those aspects are typical of the fragmentation process, a similar behaviour is also observed in \(\mathrm{p}\mathrm{p}\) collisions.

*z*axis, i.e. the |

*z*| distribution of hadron production would look similar to the

*t*one in Fig. 20, except for the lack of a suppression at \(z = 0\). It is therefore interesting to show the radial evolution separately, Fig. 21, for the same

*t*range. Overall the two figures resemble each other, but all the relevant features have been compressed owing to the lower radial velocities. The \(\pi ^0 \rightarrow \gamma \gamma \) decay is shifted from \(t\approx 10^8\) fm to \(r \approx 10^6\) fm, for instance. The impact of weak \(\mathrm{s}\), \(\mathrm{c}\) and \(\mathrm{b}\) hadron decays are better visible in the range between 1 and 100 mm; beyond that scale essentially all relevant decays have already occurred. At the other end of the scale, note that around half of the hadron production occurs in \(r < 1\) fm; there is no equivalent dynamical suppression of small

*r*as there is of small

*t*.

### 4.3 Close-packing of hadron production in the central region

*r*and

*t*,

*r*-integrated number as a function of

*t*is shown in Figs. 17 and 19. This number only increases up to \(t \approx 2\) fm, a time after which the longitudinal expansion leads to a steady decrease. Therefore, in Fig. 22a, the

*r*distribution is only shown for a few different \(t \le 2\) fm. The hadron density at times \(t = 0.5\) fm is extremely low both for 20 GeV \(\mathrm{q}\bar{\mathrm{q}}\) systems and for 13 TeV \(\mathrm{p}\mathrm{p}\) events, since they hardly have had time to start hadronizing yet. From this point on, hadrons are generated from fragmentation and particle decays, giving an increasing hadron density in the central region. The maximal value is at \(t \approx 1.5\) fm, a value that relates well with typical hadronization time scales, and where the density at \(r = 0\) approaches 2 hadrons per fm\(^3\). A proton has a volume \(V_h = 4\pi r_{\mathrm{p}}^3/3 \approx 2.76\) fm\(^3\) if we use \(r_{\mathrm{p}}=0.87\) fm [35] so, assuming the same volume for all hadrons and disregarding potential Lorentz contraction effects, this implies that five hadrons overlap in the center of the collisions. That number increases rather slowly with the collision energy; it is around four hadrons at 2 TeV and seven at 100 TeV. Also other measures of close-packing are expected to display only a mild energy dependence, so our results at 13 TeV should offer guidance for a wide range of collider energies.

### 4.4 Hadron production at different multiplicities

*r*spectra are presented in Figs. 23 and 24, respectively. For the sake of clarity, some intermediate multiplicity bins are left out of the figures. By energy–momentum conservation the \(y_{\tau }\) (and

*y*) spectra are more peaked around the middle for increasing multiplicities. Not so for the

*r*spectra, where the distribution shifts towards larger values for the higher multiplicities. It is here useful to remind that the basic MPI framework implies that high multiplicities primarily come from having more MPIs, rather than e.g. from a single hard interaction at a larger \(p_{\perp }\) scale, and that therefore \(\langle p_{\perp }\rangle (n_{\mathrm {charged}})\) is expected to be reasonably flat. The experimental observation of a rising \(\langle p_{\perp }\rangle (n_{\mathrm {charged}})\) actually was the reason to introduce colour reconnection (CR) as a key part of a realistic MPI modelling [2].

Figure 26 shows the hadron density, defined as in Eq. (46), for the three same scenarios as above. The \(n_{\mathrm {had}}\), \(r_m\) and \(\varDelta y_\tau \) are calculated in each multiplicity range. The space–time hadron density increases with hadronic multiplicity, but significantly faster in the two scenarios without CR, as a direct consequence of the inverse quadratic dependence on \(r_m\). The lower values with CR on may be partly misleading, however; only because strings are spread across a bigger transverse area when CR is on, it does not mean that there are strings everywhere in that area. The typical average density of 5 hadrons per Lorentz invariant space–time element should therefore be viewed as a lower estimate.

### 4.5 Close-packing analysis in the hadron rest frame

The number of overlapping hadrons is shown in Fig. 27 for different hadronic multiplicity ranges, as presented in Sect. 4.4. Although close-packing also takes place in low-multiplicity \(\mathrm{p}\mathrm{p}\) events, the number of hadrons overlapping with a newly created one is not so high. For high-multiplicity events, on the other hand, close-packing often arises with a significant number of nearby hadrons, likely leading to collective effects that are not taken into account in Pythia.

## 5 Summary and outlook

The motivation for this article is the mounting evidence for several collective effects in high-multiplicity \(\mathrm{p}\mathrm{p}\) collisions, similar to those usually associated with the formation of a Quark–Gluon Plasma in heavy-ion collisions. Whether we are witnessing QGP also in \(\mathrm{p}\mathrm{p}\) or not remains an open question, but the need to allow for some kind of collective mechanisms can hardly be in doubt. It should not even come as a surprise, given that already order-of-magnitude estimates of the size of the fragmentation region told us that strings would be formed close-packed and fragment into close-packed hadrons within any realistic MPI-based scenario. Colour reconnection was introduced as a partonic-state mechanism to describe some signals of collectivity, notably the rise of \(\langle p_{\perp }\rangle (n_{\mathrm {charged}})\). But the rising fraction of multistrange baryon production with event multiplicity implies that collective effects are needed also in or after the fragmentation stage, or both.

To be able fully to explore various such scenarios it becomes important to understand the space–time structure of hadronization in more detail than hitherto. The aim of this article has been to develop the necessary framework, and implement it as part of the public Pythia event generator. Specifically, we have determined the space–time location of the string breakup vertices and compared three alternative definitions for primary hadron production points. Although the implementation of the space–time picture in a simple \(\mathrm{q}\bar{\mathrm{q}}\) string topology is straightforward, the picture gets much more intricate when more complicated topologies are addressed.

To illustrate the usefulness of the new framework, some simple first studies have been presented, notably exploring space–time hadron densities. Initially, inclusive longitudinal and transverse space-time distributions were shown, and the production and decay patterns from fm to m scales were traced. Next the density in a central slice \(|z| < 0.5\) was studied as a function of *t* and *r*. While not explicitly Lorentz invariant, it gave some first hints of close-packing problems. Moving from a volume element \(\mathrm{d}^3 x\) to \(\mathrm{d}^3 x/t\) gave access to Lorentz-invariant density measures. It was shown that the median radius of the fragmentation region is increasing with multiplicity, but almost only because of the colour reconnection effects. The flip side is that density is increasing significantly with multiplicity without CR, whereas it remains at an average of about five hadrons overlapping with CR included.

The close-packing of hadrons was finally analysed by counting the number of hadrons overlapping with a newly generated one in its rest frame, again for different event multiplicities. In this case, the number of nearby hadrons does increase with multiplicity, with CR included, implying that close-packing becomes increasingly important with multiplicity also here. The overlap is largest for low-\(p_{\perp }\) hadrons, in the MPI-dominated region, whereas it drops for larger \(p_{\perp }\) scales, dominated by hard QCD jets.

A few corners have been cut in the current \(\mathrm{p}\mathrm{p}\) implementation. Notably no space–time vertices have been assigned to the individual MPI collisions, although such assignments are implicit in the MPI impact-parameter and matter-profile framework [3]. A sensible space–time picture of parton-shower evolution would introduce offsets, although presumably not major ones. Similarly, the CR between different MPIs implies that the two ends of a string may start out from different space–time points. For now, all such effects have implicitly been made part of the generic smearing step in Sect. 3.4.

To these minor corrections should be added the potentially much larger dynamical ones that could generate collective effects, be it before, during or after the string fragmentation stage. The shove and rope mechanisms are two examples for the first two stages, but the immediate continuation of the current article would be to study the consequences of hadronic rescattering in a dense hadronic gas. Models for hadronic rescattering already exist [42], such as UrQMD [43] and SMASH [44], and could possibly be interfaced. For better control, however, it would be useful to implement relevant aspects of such a framework as an integrated part of the Pythia program.

The longer-term expectation is that continued experimental studies will provide further information on all kinds of collective phenomena in LHC \(\mathrm{p}\mathrm{p}\) events, and that model building will try to rise to the challenge. Especially interesting is to figure out which phenomena can be explained without invoking QGP, and which cannot. This would then reflect back on the LHC heavy-ion program.

## Footnotes

## Notes

### Acknowledgements

Work supported in part by the Swedish Research Council, contracts number 621-2013-4287 and 2016-05996, and in part by the MCnetITN3 H2020 Marie Curie Initial Training Network, grant agreement 722104. This project has also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme, grant agreement No 668679.

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