##  1 Introduction

In collisions of heavy nuclei (AA), such as they take place at LHC and RHIC,
enhanced production of strange and (multi-)strange hadrons with respect to the
baseline, is generally assumed to be a signal of Quark–Gluon Plasma (QGP)
production [Cabibbo:1975ig, Shuryak:1977ut, HotQCD:2014kol]. In proton-proton
(pp) collisions, the general assumption has historically been that temperature
and density is too low for a QGP to form. However, at the LHC it was revealed
that small system collisions exhibit several of the same characteristics as
collisions of ions [CMS:2010ifv, ALICE:2012eyl, ATLAS:2015hzw, ATLAS:2014qaj,
ALICE:2016fzo].

When the LHC started up, it was noted by several collaborations
[ALICE:2010vtz, LHCb:2010rlm, CMS:2011jlm] that the rates of strange mesons
and baryons to pions, in minimum bias (MB) collisions, exceeded those expected
from LEP [ALEPH:1996oqp] and SLD [SLD:2003ogn], see [Bierlich:2015rha] for a
side-by-side comparison. This culminated in the seminal measurements by the
ALICE collaboration [ALICE:2016fzo], revealing that not only strangeness is
enhanced with respect to the baseline, but also the increase is continuous as
function of event multiplicity, and connects smoothly between the smallest
multiplicities in , with only a single gluon exchange, up to the most central
heavy-ion collisions, producing a factor $10^{3}$ more particles at mid-
rapidity.

Two very different types of models have performed well in reproducing the rise
of strangeness. On the one hand core-corona models [Werner:2007bf] where the
basic assumption is that droplets of QGP can form down to the smallest
collision systems, given that the density is large enough, and on the other
hand models of microscopic interactions of strings [Sjostrand:1978dj,
Andersson:1979ij, Andersson:1983jt] or clusters [Webber:1983if] where
fragmentation dynamics is modified by the presence of other strings or
clusters. In this paper we will focus on the [Werner:2023zvo] implementation
of the core-corona model [Werner:2007bf] and the [Bierlich:2022pfr]
implementation of microscopic interactions, the rope-hadronization model
[Bierlich:2014xba].

The current state of affairs, where two types of models with qualitatively
different assumptions can explain the same signal, is unsatisfying. The key
question to be answered is if there are observables for which a) the two
models give very different predictions, that b) can be easily explained in
terms of model differences. Crucially, both of these conditions must be met in
order for an observable to be a good candidate for measurement. An observable
which meets a), but not b), can easily be an artifact from model
implementation. Conversely, an observable which meets b) but not a), would
place us in the exact same situation as after the original measurement of
multi-strange baryons [ALICE:2016fzo].

In this paper, we will develop a series of observables which satisfy both
conditions stated above. The observables are based on the observation that
while microscopically interacting strings preserve the strangeness quantum
number locally (i.e. in each breaking of a string), a macroscopic model will
necessarily only preserve it either globally, or locally in some volume of
created QGP. Concretely, if a signal is triggered by the presence of a
$\phi$-meson, a microscopic model will predict a local overabundance of
strangeness due to the presence of the $\phi$, whereas a QGP based model will
not.

The structure of this paper is as follows: in the beginning of Sec. 2 the Lund
strings and rope-hadronization model in , and the core-corona model in will be
described. Section 2.3 will explain the role of the $\phi$ meson trigger in
the Lund strings model, and its effects will be explained in Sec. 2.4.
Observables to distinguish among the models in strange hadrons production will
be shown in Sec. 3, and some final considerations will be drawn in Sec. 4.

##  2 Two mechanisms for strangeness enhancement

The models for strangeness enhancement in the two generators and are very
different in nature, with assuming a creation of small volumes of QGP, and
utilizing the rope formalism. However, they both perform well in reproducing
the overall strangeness enhancement results by ALICE [ALICE:2019avo], as shown
in fig:strangeness-enhancement111For details pertaining the settings of the
two generators see Appendix LABEL:app:parameters.. In this section we provide
a brief overview of the two models, and point out the main differences,
leading up to the development of distinguishing observables in the coming
sections.

Figure 1: Yield ratios of $K_{S}^{0},\ \Lambda+\overline{\Lambda},\
\Xi^{-}+\overline{\Xi}^{+}$ and $\Omega^{-}+\overline{\Omega}^{+}$ to
$\pi^{+}+\pi^{-}$ in $|y|<0.5$ as a function of
$\bigl{\langle}dN_{ch}/d\eta\bigr{\rangle}_{|\eta|<0.5}$. Calculations from
and compared to ALICE results [ALICE:2019avo] to show that both models do well
in describing overall features of strangeness enhancement.

###  2.1 String and rope hadronization in PYTHIA

The Lund string model is the model underlying hadronization in the event
generator [Bierlich:2022pfr]. Several recent versions include the rope-
hadronization model [Bierlich:2014xba] relevant for this study. Both the
string model itself and the rope extensions have been thoroughly reviewed in
several recent papers (see e.g. [Bierlich:2016vgw, Bierlich:2017vhg,
Bierlich:2024odg]), and a full review will not be repeated here. However, the
most important facets for this study will be outlined in the following.

In , multiple 2 $\rightarrow$ 2 parton–parton scatterings radiate off
additional quarks and gluons in the parton shower, to finally be hadronized. A
$q\bar{q}$ pair will have a string spanned between them, with gluons acting as
“kinks” on the string. One string hadronizes to several hadrons, the amount
determined by the initial energy of the $q\bar{q}$ pair, and approximately one
hadron is produced per unit of rapidity per string.
The hadron flavour is determined by additional $q\bar{q}$ pairs tunneling out
of the vacuum when the string breaks (plus the flavour of the two additional
end-point quarks). In string breaks, charm quarks are too heavy to be produced
in any significant amount, and strange quarks are suppressed with respect to
up or down by a factor:

$\rho=\exp\left(-\frac{\pi(m^{2}_{s}-m^{2}_{u})}{\kappa}\right),$ (1)

where $m_{s}$ and $m_{u}$ are the strange and light quark masses respectively,
and $\kappa$ the string tension, which has a numerical value around 1 GeV/fm.
In practise, due to uncertainty about which value to use for quark masses,
$\rho$ is treated as a free parameter, estimated in $e^{+}e^{-}$ collisions at
the Z pole, i.e. a single string system. The current default value is
$\rho=0.217~{}$[Bierlich:2022pfr].

When several strings overlap with each other, two things happen,
simultaneously. Junction topologies which carry an intrinsic baryon number can
be dynamically formed [Sjostrand:2002ip, Christiansen:2015yqa], and the
effective string tension in the string breaks increases. Starting from the
value estimated in $e^{+}e^{-}$, the parameter $\rho$ in eq. 1 will increase
as the effective string tension increases [Bierlich:2014xba], giving rise to
an enhancement of strange hadrons being formed in high-density environments.

###  2.2 Core-corona production in EPOS

The core-corona model [Werner:2007bf] underlies hadronization in the event
generator [Werner:2023zvo]. In , multiple scatterings hadronize, leading to
the formation of prehadrons. The density of prehadrons determines the methods
that handle the event evolution. If the density of prehadrons is high, as it
will commonly be in high multiplicity pp or in AA collisions, they will be
treated as core. The core thermalizes and expands collectively. Instead
prehadrons close to the surface or escaping the bulk with a large transverse
momentum will be treated as corona, i.e. as in vacuum.

The corona part hadronizes using a string model, which is similar to the one,
but differs in some implementation aspects. One aspect, which is relevant for
this project, is that the string does not break as an iterative cascade, where
hadrons share break-up quarks. This implies that correlations along the string
are different, as it will be elaborated in the following sections.

In the core part, the system is expanded using a hydrodynamic treatment. For
this study, the important part of the core evolution is the flavour
assignment, which for small systems is performed using a microcanonical
approach. In this approach, a volume $V$ with energy $E$ and net flavour
content $Q=(n_{u}-n_{\bar{u}},n_{d}-n_{\bar{d}},n_{s}-n_{\bar{s}})$, decays
into a configuration $\\{h_{1},...,h_{n}\\}$ of hadrons given by the
microcanonical partition function [Werner:1995mx, Liu:2003cs]:

$\Omega(\\{h_{1},...,h_{n}\\})=\frac{V^{n}}{(2\pi)^{3n}}\prod_{i=1}^{n}g_{i}\prod_{\alpha\in
S}\frac{1}{n_{\alpha}!}\int\prod^{n}_{i=1}d^{3}p_{i}\delta\left(E-\sum\varepsilon_{i}\right)\delta\left(\sum\vec{p}_{i}\right)\delta_{Q,\sum
q_{i}},$ (2)

where $\varepsilon_{i}=\sqrt{m^{2}_{i}-p^{2}_{i}}$ is the energy,
$\vec{p}_{i}$ the 3-momentum, $n_{\alpha}$ the number of particles of species
$\alpha$, $S$ the set of particle species considered, $g_{i}$ the degeneracy
of particle $i$. The $\delta$-function at the end of the partition function is
especially relevant for this study. Its role is to ensure flavour
conservation, with $q_{i}$ being the flavour composition of hadron $i$. The
two models are fundamentally different since here the flavour conservation is
over the full volume, while in the case of the Lund string, flavour is
conserved locally.

###  2.3 The special role of the $\phi$ meson

As mentioned in the introduction, the focus in this paper is on discriminating
observables making use of the special role of the $\phi$ meson in the string
model, sketched in Fig. 2.3.
In pp collisions, most strings are aligned with the rapidity axis of the
laboratory system. When a $\phi$ meson is produced, requiring two $s\bar{s}$
string breaks, the leftover (anti-)strange quarks must find their home in
hadrons produced in neighboring rank of the same string, i.e. close in
rapidity to the $\phi$.

tikzpicture [dashed] (0,0) – (2,0); (2,0) – (3,0); (4,0) – (6,0); (7,0) –
(8,0); [dashed] (8,0) – (10,0); /n 3/s, 4/¯s, 6/s, 7/¯s (,0) circle (2pt)
node[above] ; (5,0.4) circle (0.0pt) node[above=14pt] $\phi$; [¡-] (0,0.7) –
(2.7,0.7) node[left=38pt,label=[align=left, font=]associated
$K$, $\mathit{\Lambda}$, $\mathit{\Xi}$, $\mathit{\Omega}$, …] ; [-¿]
(7.5,0.7) – (10,0.7) node[left=30pt, label=[align=right, font=]associated
$K$, $\mathit{\Lambda}$, $\mathit{\Xi}$, $\mathit{\Omega}$, …] ; [-¿] (-2,-1)
– (12,-1) node[below right] rapidity;

Figure 2: In a pp collision, most strings will be aligned with the rapidity
axis of the laboratory frame. When a $\phi$ meson is produced, it requires two
neighboring $s\bar{s}$ string breaks. This, in turn, means that the
neighboring (associated) hadrons will include the leftover (anti-)strange
quarks from the breakup.

This is a unique prediction of the Lund string model. If the string degrees of
freedom would vanish, because they melt into a QGP, the structure carrying the
correlations in rapidity would vanish, and the correlations would therefore
not be observed.

The explanation given above, holds for a single string. In pp collisions, many
strings are produced in the same rapidity interval, and therefore one cannot
be certain that a given pair of neighboring particles are produced with string
breaks in common, even though they may share quark pairs.

In related methods, such as balance functions [Bass:2000az, Pratt:2011bc,
ALICE:2023asw], this is handled by subtraction procedures. Here, however, we
are aiming at a method that allows us to highlight directly the qualitative
differences arising between the models. A difference will be noticed comparing
particle ratios inside the same (high) multiplicity interval, and adding a
$\phi$ trigger to study the same particle ratios inside a narrow rapidity
window around the $\phi$.

At low multiplicities, on the other hand, where only two strings are present
in an event, the addition of a $\phi$-trigger has a somewhat more complicated
effect.

Take the simplest case of a $\phi$ produced together with an associated $K$
(either to the left or to the right). Since we have a condition on finding a
$\phi$ in the event, the $K/\pi$ ratio at low multiplicity will now increase
drastically compared to the case with no $\phi$ trigger. This is simply due to
the fact that the additional strange quark has to go somewhere, and the
easiest is to end up in a $K$. Yet again, this is a prediction of the string
model, but is not too relevant for distinguishing between QGP production or
not, since it is generally not expected at very small multiplicities. It can,
however, distinguish between different types of strings, and results about
this point will also be shown in Sec. 3 .

###  2.4 Effects of the $\phi$ trigger

The main effect of applying a $\phi$ trigger, will be the emergence of
different types of correlations, as outlined in the previous section. Before
going to results at final state particle level, we will study whether the
application of a $\phi$ trigger introduces further biases in the model. One
could, indeed, imagine that looking in a certain rapidity window around the
$\phi$ would bias the region to have a larger-than-normal string tension, in
case of the rope-hadronization model, and a larger fraction of particles
produced from the core, in case of core-corona.

For the rope-hadronization model, we study the effective string tension used
to produce particles of a given species, with and without applying a $\phi$
trigger, in the same multiplicity bins as in Fig. 1. In Fig. 3 results are
shown for $\Lambda$ and $\Xi$ ($K_{S}^{0}$ and $\Omega$ yield to similar
results). In Minimum-bias results (magenta) no $\phi$ trigger has been
applied, in the inclusive sample (black) a $\phi$ is required to be anywhere
in the event, within the selected acceptance ($|y|<2.5$). The three selections
of $|\Delta y|$ signifying that a $\phi$ meson is required to be within that
rapidity range from the particle of interest. These are the same criteria
which will be used later for developing final state observables (except for
the Minimum-bias curves which correspond to the ones in Fig. 1).

Figure 3: Effective string tension producing $(\Lambda+\overline{\Lambda})$
(left) and $(\Xi^{-}+\overline{\Xi}^{+})$ (right) in $|y|<2.5$ as a function
of $\bigl{\langle}dN_{ch}/d\eta\bigr{\rangle}$.

The first thing to notice is that the enhancement in $\kappa_{eff}/\kappa$ as
a function of multiplicity is different for the two particle species. This is
expected since $\Xi$ (with its additional strange valence quark) is easier
produced when the string tension is large. Secondly, we notice that the
presence of a $\phi$ is not indicative of a larger local string tension than
the one already indicated by the presence of a $\Lambda$ or a $\Xi$, except
for a very minor effect for $\Lambda$ at low multiplicity. Finally, the
effective string tension rises with multiplicity in both cases, and
irrespective of the presence of a $\phi$, without saturating at high
multiplicities.

Figure 4: Core fraction of $\Lambda+\overline{\Lambda}$ (left) and
$\Xi^{-}+\overline{\Xi}^{+}$ (right) in $|y|<2.5$ as a function of
$\bigl{\langle}dN_{ch}/d\eta\bigr{\rangle}$.

In Fig. 4 the same study is repeated for the core-corona model implemented in
. The quantity of interest in this case is the core fraction, defined as the
amount of particles of a given species coming from the core, divided by the
total amount of that species in the event. The core fraction is divided in
multiplicity bins, as previously for the rope-hadronization model in Fig. 3.
The conclusions are similar. The presence of the $\phi$ does not enhance the
core fraction for the particle of interest, although in general the strange
baryons are produced more copiously in the core. A noteworthy difference
between the two models is that the core fraction saturates at high
multiplicities, where the effective string tension does not.

The combined results of Figs. 3 and 4 confirm that the observables presented
in the next section will not bias towards a local enhanced string tension or
core fraction, but are well suited for studying particle correlations.

##  3 Results with final state particles

In this section, the results at hadron level will be discussed. The suggested
observables to distinguish between the two types of models are particle ratios
in rapidity intervals ($\Delta y$) around the triggered $\phi$ meson.
As in the previous section, the inclusive results (black) indicate that the
whole $|y|<2.5$ is considered, with the presence of a $\phi$-meson in the
event. All figures shown in this section are thus ratios of (multi-)strange
hadron yields to pions, as function of charged-particle multiplicity in the
rapidity range, $|y|<2.5$. The coloured lines represent the different rapidity
separations between the $\phi$ meson and the strange hadron of interest, in
red $|\Delta y|<0.1$, in green $|\Delta y|<0.5$, and in blue $|\Delta y|<1.0$.
is shown with fully drawn lines, with dashed lines.

First, in Fig. 5, we show results for $\Lambda$ (left) and $\Xi$ (right).
These hadrons clearly display the features of the models. As the core-corona
model is insensitive to local flavour conservation at the level of string
breaks, the curves collapse into one single curve. , on the other hand,
observes local flavour conservation at the level of string breaks, and a
marked enhancement is clearly seen moving from the inclusive sample, and
closer to the $\phi$. As one would expect, the effect quickly saturates – the
curves for the three $|\Delta y|$ selections are very close together – as
hadrons are produced with a separation of one unit of rapidity between them on
average222The narrow selection in $|\Delta y|$ is motivated by the desire to
target a realistic experimental scenario..

Figure 5: Yield ratios of $(\Lambda+\overline{\Lambda})/(\pi^{+}+\pi^{-})$
(left) and $(\Xi^{-}+\overline{\Xi}^{+})/(\pi^{+}+\pi^{-})$ (right) as a
function of $\bigl{\langle}dN_{ch}/d\eta\bigr{\rangle}$ in $|\eta|<0.5$.

All curves, for both models, show some degree of saturation at high
multiplicities, although this feature is much more noticeable for . As shown
in the previous section, the core fraction itself saturates (see Fig. 4),
giving a natural explanation for , where the effective string tension does not
(Fig. 3).

In Fig. 6 results for $K^{0}_{S}$ (left) and $\Omega$ (right) are shown. These
two results are quite different than the ones shown above. The $K^{0}_{S}$
yields with respect to pions are remarkably different.
The curves rise towards low multiplicities, completely opposite to what one
would naively expect from QGP strangeness enhancement and from what one
observes in Fig. 1. The explanation lies again in local conservation of
strangeness at the level of string breaks. At low multiplicity, there are no
baryon junctions. If a $\phi$ meson is requested the additional strange quark
will allow the production of a $K$ (see Fig. 2.3). In the extreme case of the
lowest multiplicity bin with just a few charged particles produced per unit of
rapidity, two of those charged particles are likely to be pions from the
$\phi$ decay. This means that it is much more likely to have a $K$ produced on
each side of the $\phi$, rather than producing a heavily suppressed strange or
multi-strange baryon.
The curves are again almost collapsed to a single curve, with a slight bend
upwards at low multiplicities. No ingredient in the hadronization model can
produce this effect, so it might be attributed to the extreme low multiplicity
probed that forces the presence of a handful of charged particles only, a
$\phi$ and a $K_{0}^{S}$ in the event.

Figure 6: Yield ratios of $2K_{S}^{0}/(\pi^{+}+\pi^{-})$ (left) and
$(\Omega^{-}+\overline{\Omega}^{+})/(\pi^{+}+\pi^{-})$ (right) as a function
of $\bigl{\langle}dN_{ch}/d\eta\bigr{\rangle}$ in $|\eta|<0.5$.

The result for $\Omega$ shown in Fig. 6 (right) is also qualitatively
different from the results for $\Lambda$ and $\Xi$, shown in Fig. 5. Here all
curves are more or less collapsed into a single curve, even though additional
strange quarks are, in the case of , still present from the $\phi$ production.
The only way to produce the $\Omega$ baryon is by production of an $ss_{1}$
diquark (subscript indicates spin) followed by an $s$-quark
[Andersson:1981ce]. The enhancement observed indicates that production of the
additional strange quark, once an $ss_{1}$ diquark is produced, is not a
limiting factor for $\Omega$ production. There are no octet baryons with quark
content $sss$, therefore a SU(6) spin$\times$flavour Clebsch-Gordan weight of
zero is assigned to the $\Omega$ for the non-existing octet. This means that
for $\Omega$ in the decuplet the weight will be of unity.
If instead a $u$ or a $d$ quark was produced to combination with the $ss_{1}$
diquark, like for the $\Xi$, the corresponding weights with the same
normalization amount to 1/6 in the octet case and 1/3 in the decuplet case.
Since these two weights do not sum to unity, but rather to 1/2, it means that
one effectively rejects half the $u$ or $d$ string breaks, when they appear
next to an $ss_{1}$ diquark. Consequently, the regular strangeness suppression
factor no longer acts as a suppressor for $\Omega$-production.

0.49  0.49  0.49  0.49

Figure 7: Yield ratios of different strange hadron species measured in
$|y|<2.5$ as a function of $\bigl{\langle}dN_{ch}/d\eta\bigr{\rangle}$
measured in $|\eta|<0.5$ normalized to the value integrated over all the
multiplicity classes: $2K_{S}^{0}/(\pi^{+}+\pi^{-})$ (top left),
$(\Lambda+\overline{\Lambda})/(\pi^{+}+\pi^{-})$ (top right),
$(\Xi^{-}+\overline{\Xi}^{+})/(\pi^{+}+\pi^{-})$ (bottom left) and
$(\Omega^{-}+\overline{\Omega}^{+})/(\pi^{+}+\pi^{-})$ (bottom right).

Finally, in order to summarize the findings, and thus the suggestions for
discriminating observables, we show the four hadron to pion ratios as double
ratios in Fig. 7. The double ratios are constructed as the ratio of the
original curve to the value integrated over all multiplicity bins. It is clear
from this visualization, that curves for all rapidity selections overlap,
indicating that the enhancement is consistent from low to high multiplicity
for all the probed rapidity intervals. Beyond that, the double ratios do not
reveal many new insights, but serve as a better overview of the potential gain
of performing this measurement, with all predictions brought on the same
scale.

##  4 Considerations and outlook

The seminal strangeness enhancement results from ALICE started a whole new
direction of research linking flavour enhancement in small and large systems.
The apparent ability of microscopic string based models and models based on
QGP formation to both describe the data equally well has spurred a large
interest in finding observables which can discriminate between the two types
of models. In this paper we have developed observables for which the
underlying physics differences of the models are highlighted, and that produce
significantly different and measurable results.

The different mechanisms for strangeness enhancement implemented in and with
ropes have been described. It has been shown that studying the strange hadron
production in events with a $\phi$ meson effectively distinguishes among the
different strangeness production implementations.
Two main observations are made. Firstly, with ropes predicts a clear layering
of the strange hadron to pion ratios as a function of the charged particle
multiplicity which is not observed in . This is expected to originate from the
Lund strings and rope-hadronization model. Secondly, the $K_{0}^{S}/\pi$ ratio
is decreasing with multiplicity for with ropes, remaining rather flat for .
This is in disagreement with the results both from data and models in the case
of MB $K_{0}^{S}/\pi$ ratio, where no $\phi$ trigger is requested.

Both these observations will have to be tested in data. Run 3 and 4 at LHC
open the possibility for such measurements which are statistics eager. Indeed,
the amount of data that will be collected will be one to two orders of
magnitude larger with respect to the amount of data collected in Run 1 and 2.
Another improvement will come from the detector upgrades which will allow to
reconstruct particle yields at lower transverse momentum and with higher
momentum resolution. Innovative reconstruction algorithms will permit to
directly track strange particles, due to vicinity of inner trackers to the
beam pipe.
Thanks to all these improvements it will be now possible to detach the
different mechanisms that induce strangeness enhancement.