Jet modifications from colour rope formation in dense systems of non-parallel
strings

Christian Bierlich, Smita Chakraborty$\star$, Gösta Gustafson, and Leif
Lönnblad

Department of Astronomy and Theoretical Physics, Sölvegatan 14A, S-223 62
Lund, Sweden

⋆<EMAIL_ADDRESS>
$\dagger$ MCNET-22-02, LU-TP 22-09

## Abstract

We revisit our rope model for string fragmentation that has been shown to give
a reasonable description of strangeness and baryon enhancement in high-
multiplicity $\mathrm{\bf pp}$ events at the LHC. A key feature of the model
is that the enhancement is driven by the increased string tension due to
strings overlapping in dense systems. By introducing an improved space–time
picture for the overlap between fragmenting strings, where also non-parallel
strings are properly taken into account, we are now able to investigate the
enhancement both in jets and in the underlying event in a consistent way.

###### Contents

  1. 1 Introduction
  2. 2 String hadronization and colour fluxtubes
    1. 2.1 Lund string hadronization
    2. 2.2 Strings as colour fluxtubes
  3. 3 Rope hadronization with non-parallel strings
    1. 3.1 The parallel frame formalism
    2. 3.2 Overlap in the parallel frame
    3. 3.3 Monte Carlo implementation
    4. 3.4 Interplay with the Shoving model
  4. 4 Results in $\mathrm{pp}$ collisions
    1. 4.1 Model behaviour
    2. 4.2 Underlying event observables in reconstructed $\mathrm{Z}$ events
    3. 4.3 Strangeness yields in Z$+$jet events
  5. 5 Conclusion
  6. A Dependence of fragmentation parameters on $\kappa_{\mathrm{eff}}$

##  1 Introduction

One of the most characteristic features of Quark–Gluon Plasma (QGP) formation
in heavy ion (AA) collisions, is that of so–called “jet quenching” [1]. In
heavy ion collisions, jet quenching is mainly seen in energy loss or
dispersion effects, manifest as, for example, suppression of high $p_{\perp}$
particle yields, with respect to scaled proton–proton ($\mathrm{pp}$) case [2]
or the suppression of away-side jets in central collisions [3]. With the
higher energies available at LHC, the phenomenon has also been explored using
$\mathrm{Z}$ bosons plus jets, where the $\mathrm{Z}$ decaying to leptons is
used as an unaffected probe, to gauge the effect on the jet traversing the QGP
[4].

Several experimental signatures for QGP production have, however, also been
observed in high multiplicity pp collisions, including strangeness enhancement
[5] and long-range multi-particle correlations [6], more commonly known as
“collective flow”. Jet quenching effects have so far not been observed in
small systems (pp or proton–ion, $\mathrm{p}A$), which begs the question if
jet modification phenomena are completely absent in small systems, or if the
correct way to look for it has just not been established [7]. One obvious
reason for the difficulty, is that it is not possible, like in $AA$
collisions, to look at differences when comparing to similar measurements for
pp collisions. Comparisons with theoretical expectations are also difficult,
as the expected effects from quenching in small systems are very small, and
the signal is strongly affected by (uncertain) effects from initial state
radiation. It should also be mentioned that most theoretical descriptions of
jet quenching assumes that the jet is formed in a deconfined “bath” of free
partons (_i.e._ the QGP), which may not be appropriate in small systems, where
at most a few droplets would form. This includes approaches such as QGP-
modified splitting kernels [8] at high virtualities, coupled with shower
modifications by transport theory [9, 10] at lower ones, but also approaches
like the one offered by JEWEL [11], where partonic rescattering off medium
partons are combined with the Landau–Pomeranchuk–Migdal effect [12].

It should be noted that there are other mechanisms that may influence jet
production. In particular, some generators include “colour reconnections”,
especially in combination with multi-parton interactions (see, _e.g._ , [13,
14]), but it has been shown (_e.g._ , in [15] and [16]), that colour
reconnections may influence jet shapes even in $e^{+}e^{-}$ annihilations. In
addition it has been shown that the baryonic colour reconnection model in
HERWIG [17] can give rise to strangeness enhancement in dense environments in
$\mathrm{pp}$ collisions.

In a series of papers we have demonstrated, that collective flow and
enhancement of strangeness and baryons, can be reproduced in high multiplicity
pp events as a result of string-string interaction, when the infinitely thin
string is generalized to a confining colour fluxtube, similar to a vortex line
in a superconductor [18]. As discussed in ref. [19], models of string
interactions offers a novel and convenient framework for studying jet
modifications in small systems, as they are implemented in the general purpose
Monte Carlo event generator P  YTHIA , which allows the user to generate
realistic collision events, with the effects switched “on” or “off”. The study
of jet modification effects does therefore not need to rely on a (non-
existing) reference system.

The aim of this paper is therefore to look at possible effects of jet
modification via increased strangeness and baryon numbers in jets. A very
important tool is here the method developed in ref. [20], to account for the
interaction between strings which are not parallel to each other. This was not
possible in earlier versions of string-string interaction, but is naturally
very important for handling the interaction between string pieces connected to
a jet and strings in the underlying event.

The remainder of this paper is organised as follows. In section 2, we recap
the Lund string hadronization framework, taking into account the transverse
extension of strings, and discuss how the string tension increase when such
strings overlap, leading to strangeness and baryon enhancement. Then we
present the parallel frame and our updated rope model in section 3. In section
4 we investigate how the average string tension varies as a function of
multiplicity and transverse momentum, and then investigate the observable
modifications the updated rope model predicts for jets and the underlying
event in $\mathrm{pp}$ collisions at the LHC, before we present our
conclusions in section 5.

##  2 String hadronization and colour fluxtubes

In this section we will briefly introduce relevant parts of the Lund string
hadronization model, building up to the rope hadronization model used for the
model results. For more detailed reviews on Lund strings, we refer the reader
to the large body of existing literature. The original papers deal mainly with
hadronization of a single straight string [21, 22]. Gluons were introduced as
‘kinks‘ on a string in refs. [23, 24]. Somewhat dated reviews are presented in
refs. [25, 26], and a number of recent papers on Lund strings present the
model in a more modern context [27, 28, 20, 29], including our original paper
on rope hadronization [30].

The Lund string is a "massless relativistic string" (or a "Nambu-Goto
string"). Such a string has no transverse extension, and it also has no
longitudinal momentum, which implies that it is boost invariant. 111For the
kinematics of such a string see ref. [31]. This may be a good approximation
for a linear colour fluxtube, where the width is not important. In section 2.2
we will discuss going beyond this approximation.

###  2.1 Lund string hadronization

##### Hadronization of a straight string

We first look at a single, straight string stretched between a quark and an
anti-quark. The string can break via $q\bar{q}$ pair creation, in a process
which can be regarded as a tunneling process as discussed in ref. [32]. For a
single quark species the production probability is given by

$\frac{\mathrm{d}\mathcal{P}}{\mathrm{d}^{2}p_{\perp}}\propto\kappa\exp\left(-\frac{\pi\mu^{2}_{\perp}}{\kappa}\right).$
(1)

Here $\mu^{2}_{\perp}=\mu^{2}+p^{2}_{\perp}$ is the quark squared transverse
mass. The exponential conveniently factorizes, leaving separate expressions
for selection of mass and $p_{\perp}$ to be used in the Monte Carlo event
generator. With $\kappa\approx$ 1 GeV/fm, this result implies that strange
quarks are suppressed by roughly a factor 0.3 relative to a u- or a d-quark
(and that the probability to produce a c-quark with this mechanism is $\sim
10^{-11}$). It also means that the quarks are produced with an average
$p_{\perp}\sim 250$ MeV, independent of its flavour.

When the quarks and antiquarks from neighbouring breakups combine to mesons,
their momenta can be calculated as an iterative process. The hadrons are here
“peeled off” one at a time, each taking a fraction ($z$) of the remaining
light-cone momentum ($p^{\pm}=E\pm p_{z}$) along the positive or negative
light-cone respectively. The probability for a given $z$-value is here given
by

$f(z)\propto\frac{(1-z)^{a}}{z}\exp(-bm_{\perp}^{2}/z).$ (2)

Here $m_{\perp}$ is the transverse mass of the meson, and the two parameters
$a$ and $b$ are to be determined by tuning to data from
$\mathrm{e}^{+}\mathrm{e}^{-}$ collisions. In principle the $a$–parameter
could depend on the quark species, but in default P  YTHIA (the Monash tune)
it is the same for strange and non-strange quarks. Baryon–antibaryon pairs can
be produced via production of a diquark–antidiquark pair, and in this case the
$a$-parameter has to be modified. The parameter $b$ must, however, be
universal.

An important consequence of eq. (2) is the probability distribution in proper
time ($\tau$) for string breakup vertices. Expressed in terms of the quantity
$\Gamma=(\kappa\tau)^{2}$, the distribution is given by:

$\mathcal{P}(\Gamma)\mathrm{d}\Gamma\propto\Gamma^{a}\exp(-b\Gamma)\mathrm{d}\Gamma.$
(3)

The breakup-time is an important ingredient for string interactions, as the
hadronization time sets an upper limit on the available time for strings to
push each other and form ropes. As such, hadronization of a system of
interacting strings will not happen when the system has reached equilibrium,
but will be cut off when the string hadronizes. For strings hadronizing early,
one can then imagine a mixed phase of strings and hadrons, before the
transition to a pure hadron cascade. In this paper we consider only the effect
of string interactions, and leave the interplay with the hadronic cascade for
a future paper. We note, however, that a full hadronic cascade has recently
been implemented in P  YTHIA [33, 34], revealing only minor effects in proton
collisions. Typical values for $a$, $b$ and $\kappa$ give an average breakup
time of around 1.5 fm. This can not be identified as the hadronization time
(or freeze-out time). This could equally well be interpreted as the time when
the quark and the antiquark meet for the first time. In addition the breakup
times fluctuate, and each string will hadronize at different times.

##### Gluons and non-straight strings

An essential component in the Lund hadronization model is that a gluon is
treated as a point-like “kink” on the string, carrying energy and momentum. A
gluon carries both colour and anti-colour, and the string can be stretched
from a quark, via a set of colour-ordered gluons, to an anti-quark (or
alternatively in a closed loop of colour-ordered gluons).

When a gluon has lost its energy, the momentum-carrying kink is split in two
corners, moving with the speed of light but carrying no momentum, stretching a
new straight string piece between them. When two such corners meet, they can
“bounce off”; the string connecting them then disappears, but a new one is
“born”. In a pp collision a typical string will contain several gluons,
connected by string pieces which are stretched out, may disappear and then be
replaced by new string pieces. All these string pieces move transversely in
different directions, but at any time the string consists of a set of straight
pieces. For a description of how such a string hadronizes, we refer to refs.
[35, 20]. The interaction between strings with several non-parallel pieces is
discussed in section 3.

###  2.2 Strings as colour fluxtubes

The description of a confining colour field by an infinitely thin string is
necessarily an approximation, relevant only when the result is insensitive to
the width. In high multiplicity events this is no longer the case, and the
strings have to be treated as colour fluxtubes, with a non-zero width. We here
first discuss the properties of a single fluxtube, and then the interaction
between two or more parallel fluxtubes. The generalization to non-parallel
fluxtubes is presented in section 3.

####  2.2.1 A single fluxtube

The simplest model for a QCD fluxtube is the MIT bag model [36]. Here a
homogenous longitudinal colour-electric field is kept inside a tube by the
pressure from the vacuum condensate. An improved description is obtained in
lattice calculations. A common method is here to use the method of Abelian
projections, proposed by ’t Hooft [37], which is based on partial gauge
fixing. The result of these calculations show that the field is dominated by a
longitudinal colour-electric field, surrounded by a transverse colour-magnetic
current in the confining vacuum condensate [38, 39]. This picture is very
similar to the confinement of the magnetic field in a vortex line in a
superconductor (with electric and magnetic fields interchanged, see e.g. ref.
[40]).

As observed in ref. [20] the measured shape of the colour electric field
obtained in ref. [41] is well approximated by a Gaussian distribution:

$E(\rho)=E_{0}\exp\left(-\rho^{2}/2R^{2}\right),$ (4)

where $\rho$ is the transverse distance in cylinder coordinates. The width of
a fluxtube is difficult to estimate in lattice calculations, as it is
naturally given in lattice units, see e.g. ref. [42]. In the bag model, the
width is roughly given by $\sqrt{\langle\rho^{2}\rangle}\approx 1$ fm (where
$\rho$ is the radial coordinate), while lattice calculations give radii of
$\sim 0.5$ fm or even less [43, 44, 38].

The field density in eq. (4) is related to the string tension through

$\int d^{2}\rho E^{2}(\rho)/2=\pi E_{0}^{2}R^{2}=g\kappa,$ (5)

where $g$ is the fraction of the total energy of the string associated with
the colour electric field. We expect $g$ to be of the order 1/2, which is the
value obtained in the bag model, where the energy in the field and the
expelled condensate are of equal size. For a further discussion of the vacuum
condensate and colour fluxtubes we refer to ref. [45] and references therein.

####  2.2.2 Interacting parallel fluxtubes

High multiplicity collisions will give a high density of fluxtubes, with a
corresponding high energy density. In ref. [20] we discussed the collective
effects expected from the initial expansion, and in this paper we will
concentrate on the effects of rope hadronization, and in particular study the
production of strange hadrons. Here we first restate our treatment of
interaction between parallel fluxtubes, presented in ref. [30]. How this can
be generalized to a general situation with non-parallel fluxtubes will be
discussed in section 3 below.

##### Rope formation

For two overlapping parallel fluxtubes, separated by a transverse distance
$\delta$, we get from eq. (4) the interaction energy of the field

$\int d^{2}\rho(\mathbf{E}_{1}(\rho)+\mathbf{E}_{2}(\rho))^{2}/2-2\int
d^{2}\rho E^{2}(\rho)/2=\int
d^{2}\rho\mathbf{E}_{1}(\rho)\cdot\mathbf{E}_{2}(\rho)=2\pi
E_{0}^{2}R^{2}e^{-\delta^{2}/4R^{2}}.$ (6)

Such a system will expand transversely, and if it does not hadronize before,
it will reach equilibrium, where the energy density corresponds to the free
energy density in the vacuum condensate.

The expression in eq. (6) does not include the surface energy for the combined
flux tube. In the bag model this is zero, and in equilibrium the transverse
area will be doubled, and the interaction energy will be zero. For a vortex
line in a dual QCD superconductor, it depends on the properties of the
superconductor, but also here the interaction energy will be much reduced at
the time of hadronization. It will then be necessary to go beyond the Abelian
approximation. For two fluxtubes stretched by quarks, the two quarks can
either form a colour sextet or an anti-triplet, and with more fluxtubes also
higher multiplets are possible. Here lattice calculations show that a set of
overlapping strings form a "rope", with a tension proportional to the second
Casimir operator for the colour multiplet at the end of the rope [46].

Biro, Nielsen, and Knoll pointed out [47] that if a rope is formed by a number
of strings with random charges, they add up as a random walk in colour space.
This implies that the net colour grows as the square root of the number of
strings. A rope stretched by $m$ colour charges and $n$ anti-charges can then
form a colour multiplet characterised by two numbers $p$ and $q$, such that an
arbitrary state, by a rotation in colour space, can be transformed into a
state with $p$ coherent colours (_e.g._ red) and $q$ coherent anti-colours
(_e.g._ anti-blue), such that the colour and the anti-colour do not form a
colour singlet. Such a multiplet is denoted $\\{p,q\\}$, and we always have
$p\leq m$ and $q\leq n$.

For any such multiplet we can write down the number of states, _i.e._ the
multiplicity222The multiplicity provides the standard nomenclature for
multiplets, where $N=1$ is called “singlet”, $N=3$ is called “triplet”, $N=6$
is called “sextet” etc. We will here, when necessary, use the slightly more
verbose notation $\\{p,q\\}$, which allows one to distinguish between _e.g._ a
triplet and an anti-triplet. of the multiplet:

$N=\frac{1}{2}(p+1)(q+1)(p+q+2).$ (7)

As mentioned above, the total tension of such a rope is proportional to the
second Casimir operator for the multiplet, which gives

$\kappa^{\\{p,q\\}}=\frac{C_{2}({p,q})}{C_{2}({1,0})}\kappa^{\\{1,0\\}}=\frac{1}{4}\left(p^{2}+pq+q^{2}+3p+3q\right)\kappa^{\\{1,0\\}},$
(8)

where $\kappa^{\\{1,0\\}}\equiv\kappa$ is the tension in a single string.

In the P  YTHIA treatment used here, there are, however, other effects also
addressing string coherence effects. Importantly, parts of this colour
summation is in an approximate way treated by “colour reconnection”. As a
simple example we can look at two anti-parallel strings with triplet–anti-
triplet pairs in each end. These can either form an octet or a singlet, with
probabilities 8/9 and 1/9 respectively. Here the octet (denoted $\\{1,1\\}$)
gives

$\kappa^{\\{1,1\\}}=\kappa\cdot
C_{2}^{\\{1,1\\}}/C_{2}^{\\{1,0\\}}=9\kappa/4.$ (9)

The singlet ($\\{0,0\\}$), with no string at all, gives
$\kappa^{\\{0,0\\}}=0$.

The colour reconnection process in a situation with several strings can be
related to an expansion in powers of $1/N_{c}$, as discussed in refs. [48,
49].

For the special case of $N_{c}=3$ there is also a different kind of
reconnection. For a rope formed by two parallel strings, the two triplets in
one end can give either a sextet or an anti-triplet (and a corresponding anti-
sextet or triplet in the other end) with probabilities 2/3 and 1/3
respectively. For the latter we simply have just a single string.

The two original colour triplets are connected in a “junction”, and such a
reconnection can be particularly important for baryon production. This
possibility is not implemented in the present version of our Monte Carlo, but
will be included in future work. We note that for an arbitrary number of
colours, the corresponding situation is only obtained when $N_{c}-1$ colour
charges combine to one anti-colour charge. The junction formation with three
strings does therefore, for $N_{c}\neq 3$, correspond to a configuration where
$N_{c}$ strings are connected, which cannot be directly interpreted as a
$1/N_{c}$ correction.

We will in the following adopt a picture where the process of string (rope)
fragmentation follows after a process of colour reconnections, and that this
will leave the system in a state with $p$ parallel and $q$ anti-parallel
strings forming a coherent multiplet $\\{p,q\\}$.

##### Rope hadronization

A rope specified by the multiplet $\\{p,q\\}$, can break via a succession of
single $q\bar{q}$ productions, through the tunnelling mechanism in eq. (1). In
each step a multiplet $\\{p,q\\}$ is changed to either $\\{p-1,q\\}$ or
$\\{p,q-1\\}$. It is here important to note that _the tunneling is not
determined by the total tension in the rope, but by the energy released,
determined by the reduction in the tension_ caused by the production of the
new $q\bar{q}$ pair. Hence, we get from eq. (8) an effective string tension,
when the field goes from $\\{p+1,q\\}$ to $\\{p,q\\}$, given by

$\kappa_{\mathrm{eff}}=\kappa^{\\{p+1,q\\}}-\kappa^{\\{p,q\\}}=\frac{2p+q+4}{4}\kappa.$
(10)

The consequence of this picture is that we can treat the rope fragmentation as
the sequential decay of the individual strings forming the rope, much in the
same way as an everyday rope would break thread by thread. Technically it
means that we can use the normal string fragmentation procedure in P  YTHIA 8,
with the modification that we in each break-up change the fragmentation
parameters according to the effective string tension calculated from the
overlap of neighbouring strings. The changes to these parameter explained in
detail in ref. [30], and are for reference also listed in appendix A. The
changes are somewhat convoluted, since most of the parameters only indirectly
depend on the string tension, but the main effect easily seen in eq. (1),
namely that an increased string tension will increase the probability of
strange quarks and diquarks relative to light quarks in the string breakup.

##  3 Rope hadronization with non-parallel strings

Our previous work on rope formation [30] relied on the assumption that strings
in high energy hadron collisions can be assumed to be approximately parallel
to each other and to the beam axes. This prevented a detailed investigation of
possible effects in hard jets, especially those traversing the dense
environment of an $AA$ collision. In our recent work on the shoving model [20]
we found a remedy where the interaction between any pair of strings can be
studied in a special Lorentz frame, even if they are not parallel to each
other or to the beam. We call it “the parallel frame”, and it can be shown
that any pair of straight string pieces can be transformed into such a frame,
where they will always lie in parallel planes.

Below we will use this parallel frame to calculate the increased string
tension in the rope formation of arbitrarily complex string configurations.

###  3.1 The parallel frame formalism

In the previous rope implementation [30], the way to determine if any two
string pieces are overlapping was to boost them to their common centre-of-mass
frame and here measure the distance between them at a given space-time point
of break-up. This was done in a fairly crude way, not really taking into
account that the two string pieces typically cannot be considered to be
parallel in this frame. In general there is no frame where two arbitrary
string pieces can be considered to be exactly parallel, but in the parallel
frame introduced in ref. [20] it can be shown that any two string pieces will
always be stretched out in parallel planes in a symmetric way. This works for
all pairs of string pieces, even if one piece is in a high transverse momentum
jet and the other is in the underlying event.

Figure 1: The parallel frame showing the parallel planes of two strings and
the opening angle $\theta$ and skewness angle $\phi$.

In figure 1 we show a space–time picture of two string pieces stretched
between two pairs of partons in this parallel frame. Since massless partons
are propagating at the speed of light irrespective of the magnitude of their
momenta, only the angles between them are important for the following. In the
parallel frame the two string pieces have the same opening angle $\theta$, and
the partons of one piece propagate with an angle $\theta/2$ w.r.t. the
$z$-axis. The partons of the other propagate in the opposite direction, with
an angle $\pi-\theta/2$. At any given time, both string pieces will lie in
planes parallel to the $xy$-plane and to each other. Looking at the
projections of the string pieces on the $xy$-plane, we denote the angle
between them by $\phi$, and the frame is chosen such that all partons form an
angle $\phi/2$ with the $x$-axis.

To simplify the calculations we write the momenta of the partons using their
transverse momentum, $p_{\perp}$, and pseudo-rapidity difference, $\eta$, with
respect to the $z$-axis, rather than the energy and opening polar angle (where
$p_{z}=e\cos\frac{\theta}{2}=p_{\perp}\sinh\frac{\eta}{2}$), and get, using
the notation $p=(e;p_{x},p_{y},p_{z})$,

$\displaystyle p_{1}$ $\displaystyle=$ $\displaystyle p_{\perp
1}\left(\cosh\frac{\eta}{2};\phantom{-}\cos\frac{\phi}{2},\phantom{-}\sin\frac{\phi}{2},\phantom{-}\sinh\frac{\eta}{2}\right),$
$\displaystyle p_{2}$ $\displaystyle=$ $\displaystyle p_{\perp
2}\left(\cosh\frac{\eta}{2};-\cos\frac{\phi}{2},-\sin\frac{\phi}{2},\phantom{-}\sinh\frac{\eta}{2}\right),$
$\displaystyle p_{3}$ $\displaystyle=$ $\displaystyle p_{\perp
3}\left(\cosh\frac{\eta}{2};\phantom{-}\cos\frac{\phi}{2},-\sin\frac{\phi}{2},-\sinh\frac{\eta}{2}\right),$
$\displaystyle p_{4}$ $\displaystyle=$ $\displaystyle p_{\perp
4}\left(\cosh\frac{\eta}{2};-\cos\frac{\phi}{2},\phantom{-}\sin\frac{\phi}{2},-\sinh\frac{\eta}{2}\right).$
(11)

Clearly we have six degrees of freedom, and we can construct six independent
squared invariant masses, $s_{ij}=(p_{i}+p_{j})^{2}$. This means that for any
set of four _massless_ partons we can (as long as no two momenta are
completely parallel) solve for $p_{\perp i}$, which will give us:

$p^{2}_{\perp
1}=\frac{s_{12}}{4}\sqrt{\frac{s_{13}s_{14}}{s_{23}s_{24}}},\,\,\,\,p^{2}_{\perp
2}=\frac{s_{12}}{4}\sqrt{\frac{s_{23}s_{24}}{s_{13}s_{14}}},\,\,\,\,p^{2}_{\perp
3}=\frac{s_{34}}{4}\sqrt{\frac{s_{13}s_{23}}{s_{14}s_{24}}},\,\,\,\,p^{2}_{\perp
4}=\frac{s_{34}}{4}\sqrt{\frac{s_{14}s_{24}}{s_{13}s_{23}}},$ (12)

and furthermore solve for the angles $\phi$ and $\eta$:

$\cosh\eta=\frac{s_{14}}{4p_{\perp 1}p_{\perp 4}}+\frac{s_{13}}{4p_{\perp
1}p_{\perp 3}}\quad\text{and}\quad\cos\phi=\frac{s_{14}}{4p_{\perp 1}p_{\perp
4}}-\frac{s_{13}}{4p_{\perp 1}p_{\perp 3}}.$ (13)

To further specify the frame we renumber the particles so that $\phi<\pi/2$ to
have the strings more parallel to the $x$-axis and not to the $y$-axis, and we
define the $x$-axis to be their combined _rope_ axis. The result is that for a
breakup at a given space–time point in one string piece, we can in the
parallel frame have a reasonable handle on the overlap with any other string
piece.

###  3.2 Overlap in the parallel frame

In eq. (6) we wrote down the interaction energy of two completely parallel
strings separated by a small distance. We now want to use this to estimate the
effective overlap of two strings that are not completely parallel, but lie in
parallel planes.

At a specific point along the $x$-axis in the parallel frame we denote the
separation between the strings in the $yz$-plane by $(\delta_{y},\delta_{z})$
and integrate the interaction of the field given the skewness angle $\phi$ to
obtain

$\displaystyle\mathcal{I}(\delta_{y},\delta_{z},\phi)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\int
d^{2}\rho\mathbf{E}_{1}(\rho)\cdot\mathbf{E}_{2}(\rho)$ (14) $\displaystyle=$
$\displaystyle\\!\\!E_{0}^{2}\cos\phi\\!\\!\int\\!\\!\\!dy\,dz\exp\left(-\frac{y^{2}\cos\frac{\phi}{2}+z^{2}}{2R^{2}}\right)\exp\left(-\frac{(y-\delta_{y})^{2}\cos\frac{\phi}{2}+(z-\delta_{z})^{2}}{2R^{2}}\right)$
$\displaystyle=$ $\displaystyle\\!\\!2\pi
E_{0}^{2}R^{2}\frac{\cos\phi}{\cos\frac{\phi}{2}}\exp\left(-\frac{\delta_{y}^{2}\cos\frac{\phi}{2}+\delta_{z}^{2}}{4R^{2}}\right).$

Here we note that the skewness angle enters both in the scalar product and in
the strength of the field along the $y$-axis, and that the overlap vanishes
for orthogonal strings.

We can now define the relative overlap as
$\mathcal{I}(\delta_{y},\delta_{z},\phi)/\mathcal{I}(0,0,0)$ and use it as a
probability (assuming that $\mathbf{E}_{1}\cdot\mathbf{E}_{2}>0$) that a
breakup in one string is affected by an increased string tension due to the
overlap with the other. This would then correspond to a
$\\{2,0\\}\to\\{1,0\\}$ transition giving an effective string tension
$\kappa_{\mathrm{eff}}=3\kappa/2$ in eq. (10). If the strings instead points
in the opposite directions along the $x$-axis
($\mathbf{E}_{1}\cdot\mathbf{E}_{2}<0$) this would correspond to a
$\\{1,1\\}\to\\{0,1\\}$ breakup with $\kappa_{\mathrm{eff}}=5\kappa/4$.

In this way we can for each breakup in one string piece, take all other string
pieces in an event, and for each go to the parallel frame to determine if it
will contribute to $p$ or $q$. In our implementation described below, we sum
the relative overlaps in $p$ and $q$ respectively and round them off to
integers, rather than treating them as individual probabilities for each pair
of string pieces, which on average gives the same result.

It should be pointed out that in the parallel frame we also have a handle on
which string breaks up first. If we assume that the string breaks at a common
average proper time along the string, $\tau_{H}$, we can in the parallel frame
calculate the proper time of the other string in space–time point where we
calculate the overlap. If the latter is at larger $\tau_{H}$, we conclude that
the other string has already broken up, and can no longer contribute to an
increased string tension in the break-up being considered.

###  3.3 Monte Carlo implementation

The main technical problem with implementing the rope model in P  YTHIA 8, is
the order in which the string fragmentation proceeds. First, the flavour and
transverse momentum of the break-up is chosen (eq. (1)) together with the type
of the chopped-off hadron. Only then is the momentum fraction, $z$, chosen
according to eq. (2), and only then do we know exactly where the string breaks
and can calculate the $\kappa_{\mathrm{eff}}$ in that point. But we need to
know $\kappa_{\mathrm{eff}}$ to be able to calculate a break-up, so we have a
kind of Catch-22 situation.

The way we solve this is to perform a trial break-up to pre-sample the overlap
of a given string, and use the overlap there to get an approximate
$\kappa_{\mathrm{eff}}$. Then we discard the sample break-up and produce a new
one using this $\kappa_{\mathrm{eff}}$. On the average we will then get a
reasonable estimate of the overlap around a break-up. For a general break-up
in the underlying event this should be good enough, but if we are interested
in details of the hadron production in, _e.g._ , the tip of a jet, this
procedure may be inappropriate (see further discussion below in section 4.3).

The procedure to calculate $\kappa_{\mathrm{eff}}$ looks as follows:

  1. 1.

Produce a trial break-up in the string being fragmented, and deduce from which
string piece it comes.

  2. 2.

Pair this piece with every other string piece in the event, make a Lorentz
transformation to the parallel frame of each pair.

  3. 3.

Using the pseudo-rapidity of the produced hadron in each such frame, and
assuming the break-up occurred at the proper time, $\tau_{H}$, find the space-
time point of the break-up of the first string piece.

  4. 4.

In the corresponding $yz$-plane determine the proper time of the other string
piece and if that is less than $\tau_{H}$, calculate the overlap according to
eq. (14), and determine if this overlap should contribute to $p$ or $q$ in the
breakup.

  5. 5.

With the summed $p$ and $q$ (rounded off to integer values), we now calculate
$\kappa_{\mathrm{eff}}$ according to eq. (10).

  6. 6.

Throw away the trial break-up with its produced hadron and change the P  YTHIA
8 fragmentation parameters according to the obtained $\kappa_{\mathrm{eff}}$
and generate the final break-up.

As mentioned in section 2.1, some care has to be taken when it comes to soft
gluons. Normally, all string pieces can be said to be dipoles between colour-
connected partons, and in any parallel frame this string piece is parallel to
the $xy$-plane. But a soft gluon may have lost all its momentum before the
string breaks, and the break-up can then occur in a piece of the string that
is not parallel to the string pieces of the connected dipoles. To include this
possibility we introduce secondary dipoles, so that if we have two dipoles
connected to a soft gluon, _e.g._ $q_{i}-g_{j}$ and $g_{j}-\bar{q}_{k}$, then
a secondary string will be included spanned between the momenta of $q_{i}$ and
$\bar{q}_{k}$, but using the space-time point where the gluon has lost all its
momentum to the connected string pieces, as a point of origin.

The problem with soft gluons is present also for our shoving model in [20],
and the solution with secondary dipoles is now also used there. This will be
described in more detail in a future publication, where we also describe the
procedure for including these higher order dipoles in cases where we have
several consecutive soft gluons along a string.

###  3.4 Interplay with the Shoving model

Clearly our rope model is very tightly connected with our shoving model. They
both rely on the parallel frame and technically they both use the same
infrastructure for looking at overlaps between string pieces. However, here
there is again a kind of Catch-22.

Physically the shoving precedes the hadronization, and pushes the strings
apart before they hadronize. As this affects the value of
$\kappa_{\mathrm{eff}}$, the shoving should be executed first. However, for
technical reasons the pushes are applied directly to the produced hadrons
rather than to the individual string pieces. Therefore we must calculate the
hadronization before we can execute the pushes.

We are currently working on a solution to this problem, and plan to present it
in a future publication. The main effect of the shoving is expected to be a
dilution of the strings resulting in a lowered $\kappa_{\mathrm{eff}}$. As
discussed in [20] the precise value of the string radius is not known, and in
that paper we simply used a canonical value of 1 fm. Also the string radius
will affect the values of $\kappa_{\mathrm{eff}}$, and preliminary studies
show that the effects of string dilution from shoving are of the same order as
moderate decrease of the string radius of around 10%.

##  4 Results in $\mathrm{pp}$ collisions

In this section, features of the rope hadronization model with the parallel
frame-formalism are investigated in $\mathrm{pp}$ collisions. Since the main
feature of this new formalism is the much improved handling of string pieces
which are not parallel to the beam axis (_i.e._ jets), we will mostly
concentrate on observables in events containing a process with high momentum
transfer, but in section 4.1 we first show the behaviour in minimum bias
collisions. Here the most fundamental check of the dependence of
$\kappa_{\mathrm{eff}}$ with final state multiplicity, but more relevant for
the parallel frame formalism, is the dependence of $\kappa_{\mathrm{eff}}$ on
particle $p_{\perp}$. In section 4.2 we compare to existing experimental
results in the underlying event (UE) for $\mathrm{Z}$-triggered events. This
is to ensure that the existing description of such observables is not altered
by our model. Finally in section 4.3, we present predictions for the jet
observables that are affected by rope formation in $\mathrm{pp}$.

###  4.1 Model behaviour

Figure 2: $\langle\kappa_{\mathrm{eff}}/\kappa\rangle$ vs. $N_{\text{ch.}}$
(left) and vs. $p_{\perp,\text{prim.}}$ for $N_{\text{ch.}}>10$ (right). Solid
lines have string radius $R=1$ fm and dot-dashed lines have $R=0.5$ fm. Blue
and red lines are for minimum-bias event at 7 and 13 TeV respectively.

In this section, we explore the variation of the effective string tension
$\kappa_{\mathrm{eff}}$ with rope hadronization, for minimum bias
$\mathrm{pp}$ events. The $\kappa_{\mathrm{eff}}$ is shown for primary
hadrons, _i.e._ the effective string tension used to form a given hadron,
produced directly in the hadronization process. Results are shown for two
collision energies, $\sqrt{s}=$ 7 and 13 TeV, and two values of string radius,
$R=0.5$ and 1 fm.

In figure 2, the dependence of $\langle\kappa_{\mathrm{eff}}/\kappa\rangle$
with respect to $N_{\text{ch.}}$ in $|\eta|<0.5$ is shown on the left, and
$p_{\perp,\text{prim.}}$ on the right. On the right, only events with
$\mathrm{d}N_{\text{ch.}}/\mathrm{d}\eta>10$ are shown, to focus on events
with several parton interactions. (At 13 TeV this corresponds to keeping
roughly the 30% of events with the highest multiplicity [50].)

On the left plot of figure 2, it is seen that
$\langle\kappa_{\mathrm{eff}}/\kappa\rangle$ rises with around 30% for $R=1$
fm and 10% for $R=0.5$ fm, almost irrespective of $\sqrt{s}$, with the rise at
13 TeV being only slightly higher. The two main points to take away from this
figure, is a) that $\mathrm{d}N_{\text{ch.}}/\mathrm{d}\eta$ is a good proxy
for string density irrespective of collision energy, and thus works as a good
scaling variable, and b) that any result will be very sensitive to the choice
of $R$.

On the right plot of figure 2, we observe that the increase in
$\kappa_{\mathrm{eff}}$ is larger for primary hadrons in the lower $p_{\perp}$
bins for both values of $R$. This means that the lower $p_{\perp}$ primary
hadrons are formed from regions with high density of strings with more
overlaps with adjacent strings. However, the higher $p_{\perp}$ partons
correspond to “mini-jet” situations and are more separated in space-time from
the bulk of strings. Such strings have less overlaps resulting in a lower
$\kappa_{\mathrm{eff}}$. Hence the high $p_{\perp}$ primary hadrons formed
from such string break-ups show this effect.

In the lowest $p_{\perp}$ bins of $\langle\kappa_{\mathrm{eff}}/\kappa\rangle$
vs $p_{\perp,\text{prim.}}$ plot, it is seen that $\kappa_{\mathrm{eff}}$
drops to lower values. This behaviour arises from the fact that low
$p_{\perp}$ particles are biased towards low $\kappa_{\mathrm{eff}}$ values
due to the $p_{\perp}$-dependence on $\kappa$ in the tunneling probability in
eq. (1).

Overall we observe that rope hadronization significantly increases the string
tension at high-multiplicities and for low $p_{\perp}$ final-state particles.
For higher $p_{\perp}$ the effect is smaller, but does not disappear
completely.

###  4.2 Underlying event observables in reconstructed $\mathrm{Z}$ events

Figure 3: Associated particle production in Z$\rightarrow\ell^{-}\ell^{+}$
events at $\sqrt{s}=7$ TeV compared to the default P  YTHIA tune and with rope
hadronization. Top row: Distribution of charged particle multiplicity,
$N_{\text{ch}}$, (top left) and summed scalar transverse momenta, $\Sigma
p_{\perp}$ (top right) measured for events with $p_{\perp}^{Z}$ range 0-6
GeV[51]. Bottom row: $\Sigma p_{\perp}$ distributions in different azimuthal
regions, in events with $p_{\perp}^{Z}$ range 10-20 GeV [52]. Left:
_transverse_ region, $\pi/3<|\Delta\phi_{Z}|<2\pi/3$, right: _towards_ region
$|\Delta\phi_{Z}|<\pi/3$.

Before moving on to study rope effects on jets, it is important to assess
whether rope formation drastically changes existing observables, currently
well described by the existing model. In events with a $\mathrm{Z}$-boson
present, the most likely place for such a change to occur, is in the UE. To
this end, we use a standard UE analyses implemented in the Rivet program [53].

In figure 3, $N_{\text{ch.}}$ and $\Sigma p_{\perp}$ for
Z$\rightarrow\ell^{-}\ell^{+}$ events in $\mathrm{pp}$ collisions at 7 TeV are
compared to ATLAS data[51, 52]. The $\mathrm{Z}$-boson is reconstructed from
the electron or muon channel with invariant mass $66<m_{\ell^{-}\ell^{+}}<166$
GeV in $|\eta|<2.5$.

The charged particle multiplicity and summed scalar $p_{\perp}$ distributions
for Z$\rightarrow\mu^{-}\mu^{+}$ channel with $0<p_{\perp}^{Z}<6$ GeV, are
shown in top row of figure 3. It is seen that adding rope hadronization,
overall preserves the distributions as produced by default P  YTHIA 8\. We
note that rope hadronization has a slight effect of pushing particles from
lower to higher $\Sigma p_{\perp}$ regions, which follows from the
$p_{\perp}$-dependence of the tunnelling probability in eq. (1).

The particle $p_{\perp}$ in the away region (opposite azimuthal region to that
of the Z boson), balances the $p_{\perp}^{Z}$. Hence the towards and
transverse regions with respect to the Z boson are much less affected by a
recoiling jet and therefore have cleaner UE activity.333It should here be
noted that the charged particle activity in events with a hard interaction
such as Z-production is generally higher than in minimum bias events. These
regions are sensitive to the hadronization mechanism, rope hadronization
effects will be apparent here. So we look at the UE-sensitive observables such
as scalar summed $p_{\perp}/\delta\eta\delta\phi$ distributions for charged
particles in events with $p_{\perp}^{Z}$ in the range 10-20 GeV in the bottom
row of figure 3. These plots show the $\Sigma p_{\perp}$ distributions in the
transverse $(\pi/3<|\Delta\phi_{Z}|<2\pi/3)$ and towards
$(|\Delta\phi_{Z}|<\pi/3)$ regions[52]. We see that the rope hadronization
curve follows the default P  YTHIA 8 curve, again preserving the overall
physics behaviour of P  YTHIA 8, except for a slight shift in $\Sigma
p_{\perp}$, as in the top right plot.

We conclude that UE measurements are equally well described with rope
hadronization as without, and it is therefore not necessary to re-tune
fragmentation parameters before proceeding to give predictions for jet
observables.

Figure 4: Pion yields in Z$+$jet events in 13 TeV $\mathrm{pp}$ collisions vs.
$p_{\perp,\text{particle}}$ in the UE (top left), vs. $p_{\perp,\text{jet}}$
in the jet cone (top right), as a function of
$z=p_{\perp,\text{particle}}/p_{\perp,\text{jet}}$ (bottom left), and vs.
$p_{\perp,\text{jet}}$ for $0.4<z<0.6$ (bottom right).

###  4.3 Strangeness yields in Z$+$jet events

To investigate experimentally observable consequences of our rope model in
terms of the yield of different hadron species inside jets, we have chosen to
study its effects in Z$+$jets events at LHC energies. It has been shown in,
_e.g._ , ref. [54], that such events are very useful for separating regions of
phase space dominated by the UE from the regions dominated by jets. By
selecting events where the Z boson is well balanced by a hard jet in the
opposite azimuthal region, we can study the UE in a cone around the Z, where
there should be very little activity related to the jet, and thus we can get a
good estimate of the UE activity on an event-by-event basis. In this way we
can get a reliable way of correcting jet observables for UE effects, not only
for the transverse momentum of the jet but also for the flavour content.

####  4.3.1 Overall jet features

To observe the modification in the flavour production in the jet, we want to
look at the yield ratios of different hadron species. Hence we have written a
Rivet analysis where we first locate a reconstructed Z boson for
$m_{\mu^{-}\mu^{+}}$ in the range 80-100 GeV and $|\eta|<$ 2.5 and search for
the hardest associated jet in the opposite azimuthal hemisphere. We further
restrict the Z boson by requiring it to be within $|\eta|<$ 1.9 and
$p_{\perp}^{Z}>8$ GeV using the standard Z-finding projection in Rivet. Once
we find such a Z boson in the event, we search for the associated hardest
(charged particle) jet using the anti-kT [55] algorithm with a radius
$R_{j}=0.4$ in $|\eta|<2.1$ with the azimuthal separation
$\Delta\phi_{\text{jet},Z}\geq 2\pi/3$.

Figure 5: Yield ratio of different strange hadron species and protons to pions
in the UE cone vs. $p_{\perp,\text{particle}}$, scaled by factors to show them
clearly. Solid lines are with rope hadronization and dot-dashed lines are for
default P  YTHIA 8.

To subtract UE contributions from the jet $p_{\perp}$, we calculate a
characteristic $\Sigma p_{\perp,\text{UE}}$, by summing up the $p_{\perp}$ of
the charged final state particles (not including muons from the Z decay) that
lie within a cone of radius $\sqrt{2}R_{j}$ around the Z boson. Therefore, for
a given event, the yields of the particles is calculated twice: once within
the jet cone, then within a cone of radius $\sqrt{2}R_{j}$ with respect to the
Z boson. The latter serves as our underlying event reference and we subtract
half of this yield from the yield inside the jet cone to get the final yield
of the hadrons in that event associated with the jet. Denoting the initial
jet-$p_{\perp}$ as $p_{\perp,\text{pseudojet}}$, the corrected
$p_{\perp,\text{jet}}$ becomes:

$p_{\perp,\text{jet}}=p_{\perp,\text{pseudojet}}-0.5\times\Sigma
p_{\perp,\text{UE}}$ (15)

and the corresponding yields:

$\text{yield}_{\text{jet}}=\text{yield}_{\text{pseudojet}}-0.5\times\text{yield}_{\text{UE}}$
(16)

This method of UE subtraction can easily be extended to $\mathrm{p}A$ and $AA$
collisions to give a comparable result among the three systems. Similar
methods have previously been used in heavy ion collisions [56]. We do this
analysis for $\mathrm{pp}$ collisions at $\sqrt{s}=13$ TeV with
$p_{\perp,\text{jet}}\geq$ 10 GeV for string radius $R=1$ fm.

Figure 6: Yield ratio of different strange hadrons and protons to pions in the
jet cone, for $R_{j}=0.4$ vs. $p_{\perp,\text{jet}}$, scaled by factors to
show them clearly. Solid lines are with rope hadronization and dot-dashed
lines are for default P  YTHIA 8.

To examine the model performance in reproducing general features of the jets,
such as particle multiplicity as a function of their transverse momentum and
of the transverse momentum of the jets, we look at the pions. Our rope model
is known to have very small effects on the overall multiplicity [30], and we
know that pions in general are dominating the particle production, even though
we expect a slight drop in pions, since high $\kappa_{\mathrm{eff}}$ will
favour strange hadrons and baryons over pions. In figure 4 we show the pion
yield as a function of particle $p_{\perp}$ in the UE cone, and the UE-
subtracted yield as a function of $p_{\perp,\text{jet}}$ in the jet cone in
figure 4. We also show the pion yield with respect to
$z=p_{\perp,\text{particle}}/p_{\perp,\text{jet}}$ and in the mid-$z$ region
as a function of $p_{\perp,\text{jet}}$. Indeed we find that the rope effects
are very small for pion production, both in the UE and in the jet, with the
possible exception of the lowest bin in the $z$ distribution. We will revisit
the bottom row plots in connection with strangeness yields in the jet cone in
section 4.3.2.

In the UE region, the density of strings is high resulting in a higher number
of overlaps among them. As a result, we would expect large effects due to rope
hadronization in the UE. In order to observe this effect, we look at the yield
ratio of the strange hadrons to pions in the UE cone. In figure 5, we show the
yield ratio to pions for strange mesons ($K^{0}_{S}$) and baryons ($\Lambda$,
$\Xi$ and $\Omega$) and protons with respect to $p_{\perp,\text{particle}}$.
Yields of $\Xi$ and $\Omega$ baryons have been scaled by a multiplicative
factor to show them in comparison to the other species. As expected, the
different yields are higher with rope hadronization turned on as compared to
default P  YTHIA 8\. The highest enhancement for each species is observed for
the lowest $p_{\perp,\text{particle}}$ ranges which subsequently decreases for
higher particle $p_{\perp}$ (which follows figure 2 in section 4.1).
Therefore, this plot show us that with rope hadronization, we get increased
yields of baryons and strangeness. This plot also shows us the UE contribution
to strangeness yields to that of within the jet.

Turning to flavour production inside the jet cone in figure 6, we show the UE-
subtracted yield ratio to pions for the same set of hadron species as before,
now with respect to $p_{\perp,\text{jet}}$. As rope hadronization will
increase both strangeness and baryon production, the largest enhancement is
expected for multistrange baryons. For $K^{0}_{S}$, only a slight increase is
observed, while the increase for protons is higher. The $\Lambda$ yield due to
rope hadronization is even higher due to combined baryon and strangeness
enhancement. The yield of $\Xi$ is $\sim$ 20% higher due to rope
hadronization, than default P  YTHIA 8 and the $\Omega$ yield with rope
hadronization is more than 50% higher. This shows that both baryon and
strangeness yields are enhanced by rope hadronization. We note that the
increase in the yield ratio due to rope hadronization is rather constant over
all $p_{\perp,\text{jet}}$. Hence if we look at the enhancement as a function
of the transverse momentum ratio of the particle species to that of the jet,
that would help us identify the $p_{\perp}$ ranges where rope effects are
higher.

Figure 7: Yield ratios as a function of
$z=p_{\perp,\text{particle}}/p_{\perp,\text{jet}}$ for $\mathrm{pp}$
collisions at $\sqrt{s}$=13 TeV: $2K^{0}_{S}/(\pi^{+}\pi^{-})$ (top left)
$(\Lambda+\bar{\Lambda})/(\pi^{+}\pi^{-})$ (top right),
$(\Omega^{-}+\Omega^{+})/(\pi^{+}\pi^{-})$ and
$(\Xi^{-}+\Xi^{+})/(\pi^{+}\pi^{-})$ (bottom left).

####  4.3.2 Jet substructure observables

Now we take a closer look at the particle to pion yield ratios as a function
of $z$ and $p_{\perp,\text{jet}}$. Studies have been performed where the ratio
of $p_{\perp}$ of the individual sub-jets to that of the leading jet serves as
a distinguishing observable for jet modification[57]. Since we want to look at
the strange flavour yields in the jet cone, we take a simpler approach. We
only plot the yield ratios in bins of $z$, which is the ratio of the particle
$p_{\perp}$ to the jet $p_{\perp}$.

In figure 7, we show the yield ratio of strange hadrons to pions vs. $z$. We
observe that the particle yields are increased at low (close to the UE) to
intermediate $z$ values. Furthermore, this enhancement is smaller for
$K^{0}_{S}$ and larger for the strange baryon $\Lambda$, and for multistrange
baryons $\Xi$ and $\Omega$ as expected. However, strangeness and baryon
enhancement drops at higher $z$. This highlights the behaviour that rope
hadronization effects decrease with higher $p_{\perp}$, as we noted in figure
2 in section 4.1.

Figure 8: Yield ratios of particles with $0.4<z<0.6$, as a function of
$p_{\perp,\text{jet}}$ for $\mathrm{pp}$ collisions at $\sqrt{s}$=13 TeV:
$2K^{0}_{s}/(\pi^{+}\pi^{-})$ (top left),
$(\Lambda+\bar{\Lambda})/(\pi^{+}\pi^{-})$ (top right),
$(\Xi^{-}+\Xi^{+})/(\pi^{+}\pi^{-})$ (bottom left) and
$(\Omega^{-}+\Omega^{+})/(\pi^{+}\pi^{-})$ (bottom right).

We note that, even though the parallel frame formalism allows the calculation
of $\kappa_{\mathrm{eff}}$ in events with jets, the current implementation is
lacking in the region $z\approx 1$, as already mentioned in section 3.3. The
previously mentioned Catch-22 situation, is purely related to the
implementation, and can be further understood by considering the shape of the
Lund symmetric fragmentation function in eq. (2), which is vanishing near
$z=1$. For a particle with $z$ close to one, the pre-sampled overlap is
therefore likely to have been calculated with a too-small $z$, which in turn
means that it is calculated for the wrong part of the string. In $\mathrm{pp}$
collisions this effect is small but non-negligible, which we have confirmed by
an a posteriori check (as the correct overlaps can be calculated after the
fact, but too late to be used in event generation). Another issue, which would
be present even in a perfect implementation, and therefore potentially more
severe, is the absence of interactions between hadrons formed early in time,
and their surrounding environment. For most of the produced particles, and in
particular in $\mathrm{pp}$, this effect should also be small. But in the case
of high $z$, the particle is always produced early, and the effect could be
larger. We plan to develop the model further in this direction, but in the
meantime we will in the following show results for particles at intermediate
$z$ values ($0.4<z<0.6$) where the effects arising from both these issues,
should be negligible.

To test the modification in flavour yields at mid-$z$ values, we look at
particle yields as a function of $p_{\perp,\text{jet}}$. Since these particles
are neither close to the tip of the jet, nor to the UE, it is more reasonable
to the trial-hadron sampling of $\kappa_{\mathrm{eff}}$ in these regions.
Moreover, as the jet $p_{\perp}$ increases, the particles get further and
further away from the UE. In figure 8, we show the yield ratio of strange
hadrons to pions in the 0.4 < $z$ < 0.6 region vs. $p_{\perp,\text{jet}}$. We
observe that the yields from the rope hadronization case are distinct compared
to default P  YTHIA 8\. The individual strange hadron yield to pion yield
ratio increases as we go from the $K^{0}_{s}$ meson to the $\Lambda$ baryon
(top row plots). For multistrange baryons, $\Xi$ and $\Omega^{-}$ (bottom row
plots), rope effects are amplified due to higher number of strange quarks,
resulting in a 20% - 50% increase in their yields in low
$p_{\perp,\text{jet}}$ ranges. However, as mentioned before, we would expect
the enhancement in the yields to drop at higher $p_{\perp,\text{jet}}$ bins.
This effect is rather small for $\Lambda$ but prominent for $\Xi$ and
$\Omega$. $\Omega$ (bottom right plot) is only shown up to 45 GeV due to
statistics.

##  5 Conclusion

We have here presented a study on how an effect from a dense system of colour
fluxtubes might be observed as strangeness enhancement in jets in high
multiplicity pp events. In such events it is essential to properly estimate
the interaction between non-parallel strings, including strings connected to a
hard scattered parton and strings in the underlying event. This problem was
solved in ref. [20], where the interaction of all string pairs can be
calculated in a Lorentz frame, where two string pieces lie symmetrically in
two parallel planes. We here show results for jet-triggered high-multiplicity
$\mathrm{pp}$ collisions. The generalization to $\mathrm{p}A$ and $AA$
collisions (using the Angantyr model [58]) will be presented in a future
publication.

The interacting strings can form “colour ropes”, which hadronize in a stepwise
manner by $q\bar{q}$ pair creation. The increased energy in the rope gives a
higher "effective string tension", $\kappa_{\mathrm{eff}}$, which increases
the number of strange quarks and diquarks in the breakups. In section 4.1 we
found that this results in an increase of $\kappa_{\mathrm{eff}}$ with
multiplicity in $\mathrm{pp}$ events at LHC energies. It is interesting to
note that the increase for a given multiplicity is almost independent of the
collision energy.

As expected we also found that the increase is quite dependent on the
transverse momentum, since high-$p_{\perp}$ particles are typically produced
in jets where the strings are not parallel with the bulk of the strings in the
underlying event, thus reducing the effective overlap with these. The
important question is then if the rope model, despite being reduced in jets,
anyway will result in a modification of the hadron composition of jets.

To study the effects on jets we focused our investigation on Z+jet events,
with the Z decaying to lepton pairs. As pointed out in _e.g._ ref. [54], it is
possible, in such events, to get a relatively clean separation between the
jets and the particle production in the underlying event. In particular the
hadrons produced in a cone around the direction of the Z particle should have
very little to do with the recoiling jet, and can therefore be used to correct
any observable in the jet cone for underlying-event contributions on an event-
by-event basis.

The modified $\kappa_{\mathrm{eff}}$ also affects the fragmentation
parameters. In section 4.2 results for multiplicity and the transverse
momentum distribution in the underlying event in pp Z+jet events, were
compared with results from default P  YTHIA 8 and with data from ATLAS. After
confirming that the rope hadronization gives negligible effects on these
general features of the underlying event, we feel comfortable that we can
study strangeness and baryon enhancement in the jets in a way, which is not
biased by the underlying-event corrections.

In section 4.3 our main results for strangeness and baryon number enhancement
in jets were presented, with the underlying event subtracted. We note that the
effect is most important for strange baryons, and growing with the number of
strange quarks. Thus it is largest for $\Omega$ baryons, and from the plots
showing the $\Omega/\pi$ ratio as a function of the jet transverse momentum,
we note that rope effects are very small for large jet $p_{\perp}$ as
expected, but quite noticeable for low jet $p_{\perp}$.

From this we conclude that it may indeed be possible to find jet modifications
due to collective effects, in our rope model, in small collision systems. The
size of the effect is, however, a bit uncertain. In part this is due the
uncertainty in the transverse size of the string, and our canonical choice of
$R=1$ fm may be a bit large. Although it should be possible to tune this
parameter to fit the overall strangeness and baryon enhancement, it is then
also important to also take into account the effects of repulsion between the
strings. Both of these effects will be addressed in future publications.

Looking ahead, it is also interesting to investigate the effects of colour
reconnection, in particular models that include junction formations, which
will also influence the baryon production. In the end we hope to develop a
picture where most collective effects can be interpreted as interactions among
strings, not only in $\mathrm{pp}$ collisions but also in $\mathrm{p}A$ and
$AA$.

## Acknowledgements

This work was funded in part by the Knut and Alice Wallenberg foundation,
contract number 2017.0036, Swedish Research Council, contracts number
2016-03291, 2016-05996 and 2017-0034, in part by the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme, grant agreement No 668679, and in part by the MCnetITN3 H2020 Marie
Curie Initial Training Network, contract 722104.

##  Appendix A Dependence of fragmentation parameters on
$\kappa_{\mathrm{eff}}$

There are several hadronization parameters in P  YTHIA 8, and even if they are
in principle independent, several of them has an implicit dependence on the
string tension. In our implementation of the rope hadronization, we take the
parameters as tuned to $\mathrm{e}^{+}\mathrm{e}^{-}$ data, where we expect no
rope effects, and for each breakup in the string fragmentation we rescale the
parameters according to the estimated change in string tension at that point,
due to the presence of overlapping string fields. The parameters under
consideration is the same as in our previous implementation [30], and the
dependence of the string tension is also the same. For completeness we list
them here, but for further details we refer to [30].

In the following we will denote the change in string tension by $h$, according
to $\kappa\mapsto\kappa_{\mathrm{eff}}=h\kappa$. The following parameters is
affected:

  * •

$\rho$ (StringFlav:probStoUD444This is the parameter name in P YTHIA 8.): the
suppression of $s$ quark production relative to $u$ or $d$ type production.
This parameter has a simple scaling

$\rho\mapsto\tilde{\rho}=\rho^{1/h}.$ (17)

  * •

$x$ (StringFlav:probSQtoQQ): the suppression of diquarks with strange quark
content relative to diquarks without strange quarks (in addition to the factor
$\rho$ for each extra $s$-quark) also scales like

$x\mapsto\tilde{x}=x^{1/h}.$ (18)

  * •

$y$ (StringFlav:probQQ1toQQ0): the suppression of spin 1 diquarks relative to
spin 0 diquarks (not counting a factor three due to the number of spin states
of spin 1 diquarks) again scales like

$y\mapsto\tilde{y}=y^{1/h}.$ (19)

  * •

$\sigma$ (StringPT:sigma): the width of the transverse momentum distribution
in string break-ups. This is directly proportional to $\sqrt{\kappa}$, giving

$\sigma\mapsto\tilde{\sigma}=\sigma\sqrt{h}.$ (20)

  * •

$\xi$ (StringFlav:probQQtoQ): the global probability of having a diquark
break-up relative to a simple quark break-up. This has a somewhat more
complicated $\kappa$ dependence and also has uncertainties related to the so-
called popcorn model as described in [30]. We decompose it as three different
parameters, $\xi=\alpha\beta\gamma$ with different $\kappa$-dependence, where
$\beta$ is related to the probability to have a $q\bar{q}$ fluctuation in
general in the popcorn model which is independent of $\kappa$ and is treated
as an independent parameter, while $\gamma$ is related to the masses and
scales as

$\gamma\mapsto\tilde{\gamma}=\gamma^{1/h},$ (21)

and $\alpha$ is related to the different di-quark states with an indirect
dependence on $\rho$, $x$, and $y$

$\alpha\mapsto\tilde{\alpha}=\frac{1+2\tilde{x}\tilde{\rho}+9\tilde{y}+6\tilde{x}\tilde{\rho}y+3\tilde{y}\tilde{x}^{2}\tilde{\rho}^{2}}{2+\tilde{\rho}}.$
(22)

Taken together we get the following dependence:

$\xi=\alpha\beta\gamma\mapsto\tilde{\xi}=\tilde{\alpha}\beta\left(\frac{\xi}{\alpha\beta}\right)^{1/h}.$
(23)

  * •

$b$ (StringZ:bLund): the parameter in the symmetric fragmentation function eq.
(2) scales with the $\rho$-parameter as follows

$b\mapsto\tilde{b}=\frac{2+\tilde{\rho}}{2+\rho}\,b.$ (24)

  * •

$a$ (StringZ:aLund): the other parameter in eq. (2) has an indirect dependence
on $b$ through the normalisation of the splitting function, $f(z)$. Keeping
the normalisation unchanged does not give a simple analytic form for the
scaling of $a\mapsto\tilde{a}$, and instead we use a numeric integration
procedure.

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