# “QM19 summary talk”:
Outlook and future of heavy-ion collisions

Constantin Loizides  ORNL, Oak Ridge, USA

###### Abstract

A summary of the QM19 conference is given by highlighting a few selected
results. These are discussed as examples to illustrate the exciting future of
heavy-ion collisions and the need for further instrumentation. (The arXiv
version is significantly longer than the printed proceedings, with more
figures.)

##  1 Introduction

The goal of heavy-ion physics [1] is to understand the phase diagram of
Quantum Chromo Dynamics (QCD) as a function of the temperature ($T$) and
baryon chemical potential ($\mu_{\rm B}$), as shown in Fig. 1. At high
temperature and/or high baryon chemical a transition occurs, from ordinary
matter, where the hadronic degrees of freedom are dominant, to a Quark Gluon
Plasma (QGP), where the dominant degrees of freedom are quark and gluons,
which in ordinary matter are confined into hadrons. Lattice QCD calculations
[2] predict the presence of a Critical Endpoint (CE) somewhere in the region
$T<140,\mu_{\rm B}>300$ MeV with a first-order phase transition at higher
$\mu_{\rm B}$ and a cross-over transition at $T\approx 155$ MeV and lower
$\mu_{\rm B}$. Immense experimental and theoretical effort is underway to
characterize and understand the phase structure of QCD, and the emergence of
collectivity and matter properties, and the underlying equation of state (EoS)
from first principles. Following the presentation at the conference [3], these
proceedings are structured into the following topics:

  * 1.

the high-density frontier (Sec. 2) to study the onset of deconfinement, to
search for the CE and the first-order phase transition, and at very high
$\mu_{\rm B}$ to provide constraints for the neutron star structure and its
EoS;

  * 2.

the high-energy frontier (Sec. 3) to quantify the fluid properties of QGP and
to relate them to its microscopic structure;

  * 3.

the cold nuclear matter or small-$x$ frontier (Sec. 4) to characterize
properties of cold nuclear matter, and understand the structure of protons and
nuclei at small Bjorken-$x$;

  * 4.

the ultra-precision near and far future (Sec. 5) with the planned Electron Ion
Collider (EIC) and the proposed new future experimental equipment at the Large
Hadron Collider (LHC) and beyond.

Fig. 1:  The QCD phase diagram versus temperature and bayon chemical
potential. Indicated are the nuclear matter, hadron gas and QGP phases, as
well as the approximate region for the Critical Endpoint, and the first-order
and cross-over phase transitions. Current and future facilities probing
different region in $\mu_{\rm B}$ are: LHC [4], FCC [5], RHIC [6], and the
RHIC fixed-target program [7], SPS [8, 9], NICA [10], FAIR [11] and J-PARC
[12]. Figure from [13].

##  2 The high-density frontier

Predictions for the existence and exact location of the CE at finite $\mu_{\rm
B}$ have been continuously improved over the past years. New lattice QCD
results from the WB collaboration [14] using the imaginary $\mu_{\rm B}$
method are consistent with previous calculations by the HotQCD collaboration,
but have much smaller uncertainty, and hence add further evidence that the
location of CP is disfavoured for $\mu_{\rm B}<300$ MeV. Where possible,
lattice QCD predictions were already confronted with experimental data [15].
The experimental approach [16] is to probe different regions of the QCD matter
phase diagram by extracting the freeze-out parameters ($T$, $\mu_{\rm B}$)
from statistical model fits to hadron yields measured at LHC (few TeV) down to
SIS18 (few GeV) collision energies. In this way, the vicinity of the phase
boundary will be explored from $T\approx 155$ and $\mu_{\rm B}\approx 0$ MeV
down to about $T=80$ and $\mu_{\rm B}=900$ MeVas shown in Fig. 1.

The CE is characterized as a divergence in the correlation length of the
underlying system, and hence manifests as a divergence of the associated
susceptibilities. Experimentally, ratios of susceptibilities are accessible
through event-by-event fluctuations in conserved quantities, such as electric
charge or baryon number. In particular, higher-order moments of net-proton
distributions, which are a proxy for baryon number, are expected to be
sensitive to the CE. First results of $C_{6}/C_{2}$ of net-proton
distributions were reported [17], which for central AuAu collisions at
$\sqrt{s_{\rm NN}}=200$ GeV are negative, as expected from lattice QCD, while
they were found to be positive for central AuAu collisions at $\sqrt{s_{\rm
NN}}=54.4$ GeV. Albeit being-only a $2\sigma$-effect at present, the result
may indicate the expected $O(4)$ criticality in the cross-over region [18].
Furthermore, a crucial study was performed [19] by checking that the freeze-
out parameters deduced from mean hadron yields agree with those from grand-
canonical fluctuations of conserved quantities for central collisions. This
demonstrates for the first time, using fluctuation observables, that a femto-
scale system attains thermalization. A low $\sqrt{s_{\rm NN}}$ ($<19.6$ GeV)
the presented study however indicates that the created system may be too
dilute to reach equilibrium or exhibits different relaxation times for
different moments. In view of the search for the CE this should be followed-up
on.

A first-order phase transition is characterized by an unstable co-existence
(spinodal) region corresponding to a softest point in the EoS. Direct flow
($v_{1}$) which is sensitive to the compressibility of the created matter, is
hence a key observable in the beam energy scan (BES) program. The slope of the
rapidity-odd component at mid-rapidity was reported as a function of beam
energy for various identified mesons and baryons [20]. Mesons and anti-baryons
exhibit negative slope over the whole range of beam energies, while baryons
(p, $\Lambda)$ exhibit a change of slope around $14.5$ GeV. In hydrodynamical
models a change of slope implies a minimum in $v_{1}$, and hence was proposed
as a signature of a first-order phase transition [21]. However, to
consistently treat effects from spectator matter and baryon stopping,
calculations in the frame of the BEST collaboration [22] would be needed using
more realistic 3D hydrodynamic calculations at finite $\mu_{\rm B}$ [23], with
an EoS with and without CE [24]. First data from the STAR fixed-target (FT)
program, the $v_{1}$ of the $\phi$-meson at $\sqrt{s_{\rm NN}}=4.5$ GeV were
also shown, but the statistical uncertainties are too large to allow for
additional conclusions. More results, from BES phase 2, and the fixed-target
program, are eagerly awaited.

##  3 The high-energy frontier

The high-energy experiments at RHIC and LHC continue to produce data with ever
increasing precision in measuring bulk properties. At the conference, for
example, new results of longitudinal flow decorrelations at LHC [25] and their
energy dependence at RHIC [26], linear and non-linear flow modes [27] and
higher order cumulant elliptic flow and fluctuations [28] of identified
particles in PbPb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV were presented.
These and many more data are now rigorously used in Bayesian extraction of
bulk properties, for example using the JETSCAPE framework [29]. It was argued
[30] that RHIC and LHC data add complementary constraints, and that there are
non-negligible uncertainties in the transition from fluid cells to particles
(“viscous corrections”). In developing community frameworks like JETSCAPE, it
is important to highlight and maintain the“plug-and-play” approach of the
framework, which allows one to interchange and study different descriptions of
the various stages of the collision. Furthermore, we need to insist that
contributing authors release their pieces of code as open source promptly.

The precision reached in the J$/\psi$ nuclear modification factor at mid-
rapidity from ALICE [31] is now good enough to clearly see predicted behavior
of the $R_{\rm AA}$ from statistical hadronization or regeneration [32, 33,
34] counteracting Debye screening: A clear minimum is observed in peripheral
collisions ($N_{\rm part}\sim 100$), beyond which the $R_{\rm AA}$ rises for
more central collisions almost reaching unity in central collisions.
Consequently, one also observes large flow as well as triangular flow due to
“approximately thermalized” charm at low $p_{\rm T}$ [35]. The total charm
cross section needs to be measured, since it is the natural normalization for
J/$\psi$ measurements. Knowing it precisely allows one to reduce model
uncertainties by constraining shadowing effects, and set precise limits on the
expected $R_{\rm AA}$. Unity is not apriori a limiting value; the J/$\psi$
$R_{\rm AA}$ can be larger than one [16] With a “brute-force” combinatorial
extraction of D meson production down to 0.5 GeV/$c$, first constraints of the
total charm cross section in PbPb were placed [36]. Better performance and
more statistics will be expected in Run-3 after the LS2 ALICE upgrades.

The situation for the $Y$ is different as due to its much heavier mass one
does not expect a significant regeneration component. Broadening of its
spectral widths on the lattice for finite temperature, presented at the
conference [37], is compatible with the sequential dissociation picture.
Sequential dissociation is clearly established for the $Y$ family, and
observed to be significantly stronger in PbPb than pPb collisions [38]. In pPb
collisions, the suppression usually is attributed to comoving nuclear matter,
but could in principle also be partially from the onset of screening. Elliptic
flow was found to be consistent with zero [39, 40], however from extrapolating
Blast Wave fits of existing data, one expects significant $v_{2}$ only for
$p_{\rm T}>10$ GeV [41].

Understanding and characterizing parton energy loss continues to evolve from
rather qualitative to more-and-more sophisticated and precise level. A
milestone was reached with the first measurement of the jet fragmentation of
jets recoiling of an isolated photon [42]. The data clearly exhibit the
expected “text book” result that soft (low-$z$) particles are enhanced, while
hard (high-$z$) are suppressed, as predicted since more than 20 years (e.g.see
[43] and references therein). Detailed measurements using isolated photon–jet
correlations will play a key role to precisely quantify jet modification in
Run-3/4 at the LHC 111At LHC also Z–jet correlations are becoming available
[44] and in the sPHENIX era at RHIC. Measurements of inclusive jet $R_{\rm
AA}$ were performed from about 50 to 500 GeV/$c$ for $R=0.4$ in PbPb
collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV, and rather good agreement was
found between data and models [45, 46]. It was hence surprising that the
predictions [47] for larger radii and to higher $p_{\rm T}$ vary considerably
indicating that the limited available measurements in the past year attracted
the calculations. In order to avoid tuning of models to specific results, we
could define a set of reference measurements (accord) that every jet model has
to be able to describe before one trusts its prediction for a different
observable. The CMS collaboration released new results [47] of $R_{\rm AA}$
for high energy jets up to $R=1.0$, which yield $R_{\rm AA}\approx 0.8$ above
300 GeV/$c$ for central PbPb collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV.
Peripheral collisions are consistent with unity, in particular when taking
into account also the presence of the centrality bias [48]. The jet
suppression for large radii in central collisions is similar to that of
charged particles at $\sim 200$ GeV/$c$ [49], indicating the sought-after
consistency between inclusive charged particle and jet $R_{\rm AA}$ at high
$p_{\rm T}$ and large enough radius. Still, with new data from Run-3/4, it is
important to measure both to even higher $p_{\rm T}$, to reduce uncertainties,
and to investigate if $R_{\rm AA}\sim 1$ will be reached. Further progress
with jet measurements to lower $p_{\rm T}$ and large radii ($\geq 0.4$) can be
made by using Machine Learning methods trained on properties of the jet
constituents in pp collisions [50]. First results have already been
presented[51], and confirm the previously established picture of strong
quenching at low jet $p_{\rm T}$. However, in particular for large jet radii
of $R=0.6$ a potential bias induced by changed jet fragmentation needs to be
investigated.

Substructure and reclustering techniques are extremely promising window to
access the partonic structure of the shower [52]. New evidence for the
coherent/wide angle energy loss picture have been presented at the conference.
Symmetric splittings for wide-angles were reported to be suppressed relative
to vacuum [53] and single sub-jets less exhibit less suppression than jets
with multiple subjets [54].

One of the frontiers of the high-energy program is to study the evolution and
onset of QGP phenomena in smaller collision systems like pA and pp collisions
[55]. The yields of strange and multi-strange particles (relative to pions)
increase smoothly with multiplicity, rather independent of collision species
and energy [56] reaching a value predicted by statistical models in central
PbPb collisions. The increase with multiplicity in smaller systems can be
regarded as lifting of canonical suppression [57]. The rapid rise at low
multiplicity plus the presence of many QGP signatures in pp and pPb collisions
[58] lead to the development of “core/corona” models. In these models, the
core “hydrodynamizes”, while the corona does not and instead is treated as a
superposition of independent nucleon–nucleon collision. In this way, good
description of data across all systems is achieved with a rather universal
approach (similar to EPOS) [59].

The observed enhancement in the strange particle yields versus multiplicity
result from the fact that there is a pedestal effect: The strange particle
yields depend approximately linearly on multiplicity but only above some
minimum multiplicity (threshold), while the pion yields do not exhibit any
apparent threshold. Hence, normalizing the strange particle yields with the
pion yields creates an enhancement $\propto(1-M_{0}/M)$, which leads to a rise
with $M$ that is most apparent at low multiplicities. In microscopic models
one can interpret the observed threshold in several ways: i) The yield
increases with the number of overlapping strings, i.e. as a multi-string
effect related to the string density driving the increase of strangeness
production that however quickly saturates at higher multiplicity. ii) There is
minimum associated multiplicity necessary for multi-strange particle
production and/or different scaling of soft and (semi-)hard processes, i.e. a
single string effect leading to suppression at low multiplicity that is
reduced at higher multiplicity. To further probe production mechanism of
strangeness can be achieved by measuring two-particle angular correlations
(associated production) between same and opposite sign strange and non-strange
particles [60]. These studies in particular when they involve also the
$\Omega$ baryon will greatly benefit from the large increase in statistics of
the 200/pb pp program planned for Run-3/4 [61].

Fig. 2:  The $\Lambda^{+}_{c}/D$ ratio versus multiplicity at mid-rapidity
measured in several $p_{\rm T}$-ranges for pp, pPb and PbPb collisions at
$\sqrt{s_{\rm NN}}=5.02$ TeV compared to e+e- collisions [36].

The $\Lambda^{+}_{c}/D$ ratio versus multiplicity at mid-rapidity measured in
several $p_{\rm T}$-ranges for pp, pPb and PbPb collisions at $\sqrt{s_{\rm
NN}}=5.02$ TeV [36] shown in Fig. 2exhibits surprising features: The ratios,
in particular for lower $p_{\rm T}$, smoothly increase with multiplicity from
low multiplicity pp to central PbPb collisions. At the lowest measured
multiplicity interval (below the pp average) the ratio at $2<p_{\rm T}<4$
GeV/$c$ is already significant (4 times) larger than that measured in e+e-
collisions, while for the highest multiplicity interval it approaches the
value seen in central PbPb collisions. The result is in qualitative agreement
with the “recombination” hypothesis saturating already few ($\sim 6$) times
mean the average pp multiplicity, which requires the presence of a local
space-time density. Alternative mechanisms like implemented in PYTHIA (“color
reconnection”) can describe the data (but are not universal). The new data
open up research on hadronization in pp collisions and future ep (eA)
colliders.

In pp collisions at 13 TeV, heavier particles up to D-mesons exhibit finite
$v_{2}>0$, while $v_{2}\approx 0$ for B-mesons [62, 63]. Unlike at RHIC, where
the geometry of the initial state can be engineered through the collisions of
proton, deuteron or helium on gold [64], at LHC the dependence on geometry can
only be more indirectly probed. Still, measurements of $v_{2}$ in Z-tagged pp
events (which lead to more “central” events), and in photo-nuclear reactions
(which can be regarded as a $\rho$–nucleus collision), reveal finite $v_{2}>0$
[65, 66], while $v_{2}\approx 0$ was found in ee or ep collisions [67, 68]. In
particular, for those systems where $v_{2}\approx 0$, it would be good to
follow up with other measurements like identified particle $p_{\rm T}$
spectra, mean $p_{\rm T}$, or yield ratios versus $p_{\rm T}$ to check for
absence of collectivity in observables related to $v_{2}$. New calculations
for small systems, where initial and final state effects were consistently
treated, suggest the dominance of the final state effects beyond ${\rm
d}N/{\rm d}\eta>~{}10$ [69]. Together with the large set of observables, which
in PbPb collisions undoubtedly would be interpreted as QGP signals, this makes
it plausible that we indeed observe the sQGP even down to multiplicities just
above that of minbias pp collisions at LHC energies. While in 2015, there was
a large debate whether it is possible [58], we should now focus on
understanding how it is realized in QCD and what it implies.

Although the presence of flow without observable jet quenching does not rule
out having a “mini” QGP, one of the key questions is the apparent absence of
parton energy loss in small systems. Above a few GeV/$c$, nuclear modification
factors measured in minimum bias pA collisions at mid-rapidity are consistent
with pQCD calculations including nuclear PDFs (see references in [58]).
Observing a suppression directly in light-particle flavor spectra is
complicated due to the selection biased introduced by the “event-activity
categorization” [70]. At high $p_{\rm T}>10$ GeV/$c$, $v_{2}$ is understood to
reflect the path-length dependence of parton energy loss in PbPb collisions
[71]. Therefore, the new data from ATLAS [72], which demonstrated significant
non-zero $v_{2}$ values up to high $p_{\rm T}\sim 50$ GeV/$c$ in pPb
collisions measured using the template fit method, are puzzling. They can not
be consistently interpreted as due to parton energy loss, since the latter
would imply $R_{\rm pPb}$ to be significantly below unity, not seen in data.
Furthermore, the similarity of the $p_{\rm T}$ shape of $v_{2}$ between the
pPb and PbPb systems provokes the question, whether there is a yet unknown
source of $v_{2}$ in both systems? In the intermediate region, the
anisotropies were found to be larger in minimum-bias than in jet-selected
events, and the observed $v_{2}$ may come from changing the admixture of
particles from hard scattering and the underlying event. At $p_{\rm T}>10$
GeV/$c$, the ATLAS data essentially report long-range correlation between hard
and soft particles on the level of 2%, which is remarkably similar to that in
PbPb collisions, maybe reflecting some source of residual correlation from the
likely presence of a di-jet. It is clear that this question will need to be
followed up, maybe by measurements in peripheral PbPb collisions, or by
studies using event generators trying to disentangle the various
contributions.

Peripheral PbPb collisions, most of the beam energy systems from STAR, as well
as small systems (pA and ppcollisions) exhibit $v_{2}$ but no measurable
effect from parton energy loss on the $p_{\rm T}$ spectra [73, 74, 75]. Hence,
one of the next key questions is to study the onset of parton energy loss at
RHIC and LHC with lighter ions, like for example in OO or ArAr collisions
[61]. Unlike in pp or pA collisions, multiplicity distributions in AA
collisions exhibit a pronounced plateau that simplifies the centrality
determination, and reduces effects from biases induced by the event selection.
Indeed, it will be very beneficial to perform the same measurements also at
$\sqrt{s_{\rm NN}}=0.2$ TeV, since the number of MPI per nucleon–nucleon
collision will be significantly lower than at LHC energies, while the expected
medium properties do not differ greatly. Furthermore, it may be useful to
describe the $R_{\rm AA}$ data from the RHIC beam energy scan [74], as well as
the SPS data [76] with models including state-of-the-art initial and final
state effects.

Fig. 3:  Approximate ($x$,$Q$) coverage of various experiments for regions
probed by DIS measurements including at the planned EIC, as well as possible
future direct photon and Drell-Yan measurements at RHIC and LHC, The estimated
saturation scales for proton and Pb are also indicated. The horizontal dashed
line and the dashed curve indicate the kinematic cuts above which data were
included in the nNNPDF fits. To calculate $x$ and $Q$ the approximate
relations $x=2p_{\rm T}/\sqrt{s_{\rm NN}}\exp(-y)$ and $Q=p_{\rm T}$ was used
with $\sqrt{s_{\rm NN}}=8.8$ TeV for LHC, and $\sqrt{s_{\rm NN}}=0.2$ TeV for
RHIC. Figure from [77].

##  4 The cold-nuclear matter or small-$x$ frontier

At small longitudinal momentum fraction $x$ and momentum transfer $Q$, parton
dynamics is expected to be affected by non-linear QCD evolution, where the
rate of gluon–gluon fusion is in competition with that of gluon splitting. In
this kinematic regime, the extremely high gluon density may even saturate,
possibly leading to the existence of another pre-collision state of matter ––
the so-called colour glass condensate (CGC) [78]. The saturation scale, where
for a given $x$ the competing processes are in balance, is enhanced in nuclei
by a factor $A^{1/3}$ compared to protons, and hence comparisons between
measurements in pp and pA collisions are of particular interest.

Figure 3 gives an overview of the approximate ($x$,$Q$) coverage of various
curent and future experiments for EM or DIS measurements.At RHIC and EIC a
region down to about $x\sim 10^{-4}$ will be probed with the future direct
photon and Drell-Yan measurements enabled by the RHIC cold nuclear program
[79], for which STAR and sPHENIX plan forward upgrades at $2.5<\eta<4$ [80,
81]. Shown are also regions probed by nuclear DIS measurements [82, 83, 84],
including at the planned EIC [85]. Due to the large beam energy available at
the LHC, instrumenting the forward region at the LHC enables measurements
probing parton densities at small $x$ of the proton or nucleus, down to $x\sim
10^{-6}$, over a large range in $Q^{2}$. The LHCb experiment [86] is a single-
arm spectrometer equipped with tracking and particle-identification detectors
as well as calorimeters with a forward angular coverage of about $2<\eta<5$.
The FoCal is high-granularity Si+W electromagnetic calorimeter and
metal+scintillator hadron calorimeter at $3.4<\eta<5.8$ proposed to be
installed in ALICE in LS3 [77]. The FoCal and LHCb measurements probe much
smaller $x$ than any of the other experiments. The FoCal will access the
smallest $x$ ever measurable until the possible advent of the LHeC [87] or FCC
[88]. Compared to RHIC, the LHC will give access to a significantly larger
region of phase space that is potentially affected by parton saturation. In
particular, the region of gluon saturation will extend to $p_{\rm T}$ values
high enough that perturbative QCD should be applicable.

Fig. 4:  Nuclear modification factor $R_{\rm pPb}$ as a function of $p_{\rm
T}$ for prompt $D^{0}$ integrated over $2.5<|y^{\ast}|<4.0$ for $p_{\rm T}<6$
GeV/$c$ and $2.5<|y^{\ast}|<3.5$ for $6<p_{\rm T}<10$ GeV/$c$ for pPb
collisions at $\sqrt{s_{\rm NN}}=5.02$ TeV as measured by LHCb compared to
theoretical predictions of different pQCD calculations using nuclear PDFs and
a recent CGC calculation. The figure is adapted from [89].

Precise information at forward rapidity probing small $x$ at the LHC is
provided by the measurement of prompt D-meson production at $2.5<y<4.0$ by
LHCb [89]. D-meson production is directly sensitive to the gluon density,
since the dominant production process for $c\bar{c}$ production is gluon
fusion $gg\rightarrow c\bar{c}$. The measured nuclear modification factor
$R_{\rm pPb}$ as a function of $p_{\rm T}$ at forward rapidities (Fig.
4)exhibits that the forward production of prompt D-mesons is suppressed
compared to pp collisions, with $R_{\rm pPb}\sim 0.6$ at low $p_{\rm T}$ and
increasing mildly with $p_{\rm T}$. The measured suppression is consistent
with expectations based on the various calculations using nuclear PDFs or the
CGC framework. The suppression of charm production in the calculations with
nuclear PDFs is a direct result of the reduced gluon density at $x\lesssim
10^{-2}$, which is commonly referred to as gluon shadowing. The calculated
values range from $R_{\rm pPb}$ about $0.3$ to 0.9, reflecting the current
uncertainties in the nuclear modification of the small-$x$ gluon density. This
directly confirms that shadowing at small $x$ is large, and that the data
place constraints on nuclear PDFs, as discussed at the conference [90].

Fig. 5:  Left panel: Expected uncertainty for the nuclear modification factor
of isolated photons at $\sqrt{s_{\rm NN}}=8.8$ TeV measured with FoCal. The
bands indicate the systematic uncertainty, mostly due to uncertainties on the
efficiency and energy scale, as well as the decay photon background
determination. The current EPPS16 and nNNPDF1.0 uncertainties are indicated by
the black line and the shaded band, respectively. Right panel: The nuclear
modification of the gluon distribution for Pb versus $x$ at $Q^{2}=10$
GeV${}^{2}/c^{2}$ for $x>10^{-6}$ compared between nNNPDF1.0 parameterization
and fits to the FoCal pseudo-data (red band) and “high energy” EIC pseudo-data
(green band). 90% confidence-level uncertainty bands are drawn, and the
nuclear PDFs are normalized by the proton NNPDF3.1. Figures from [77].

However, a quantitative determination of the amount of gluon shadowing based
on hadron production measurements is complicated by the fact that hadronic
final state effects (rescattering) may also play a role in the observations.
Recent forward measurements from LHCb at $\sqrt{s_{\rm NN}}=8.16$ TeV include
the observations of a stronger suppression of the cross section ratio of
$Y(1S)$ to J/$\psi$ from b in pPb compared to pp collisions, as well as
stronger nuclear modification of $Y(2S)$ compared to $Y(1S)$ [91]. Both
require the presence of additional (final-state) effects to describe the data.
Also, the forward/backward ratio of D-meson production at high $p_{\rm T}$ in
pPb collisions at $\sqrt{s_{\rm NN}}=8.16$ TeV [92] differs from expectation
of models and data at 5 TeV.

A clean probe providing direct access to partons are direct photons, since
they couple to quarks, and are not affected by final-state nor hadronization
effects. At leading-order more than 70% are produced by Compton
($qg\rightarrow\gamma q$) process, hence directly sensitive to the gluon
density. An essential design goal of proposed FoCal [77] is ability to
reconstruct $\pi^{0}\rightarrow\gamma\gamma$ decays at forward rapidity up to
large transverse momenta $p_{\rm T}\sim 20$ GeV/$c$ with high efficiency. This
will enable precise discrimination between direct photons and decay photons,
hence enabling to measure direct photons from low transverse momentum up to
$\sim 20$ GeV/$c$ at large rapidity. The unique performance of a future
isolated photon measurement with FoCal is demonstrated in Fig. 5. An accuracy
of about 20% is expected at $4$GeV$/c$, improving to about 5% at 10 GeV/$c$
and above, which will strongly constrain nuclear PDFs below $x\sim 0.001$, in
a region complementary to the EIC.

##  5 The ultra-precision near and far future

An overview of the timeline of planned and proposed near- and far-future
instrumentation for high-energy nuclear physics, grouped into different
categories (high density, high energy, small-$x$ and ultra-precision future)
is given in Fig. 6. Besides the ongoing experimental program, in particular at
RHIC and SPS, new dedicated future experiments and facilities are being built
over the next 5–10 years to characterize the phase structure of strongly-
interacting matter at high $\mu_{\rm B}$ [93]. In addition, at the LHC, data
can be taken in fixed-target mode, for example by LHCb with the SMOG2 system
[94], allowing to probe the freeze-out curve up to $\mu_{\rm B}\approx 400$
MeV (see summary in [95]).

In 2021 the ALICE LS2 upgrades [96, 97] to improve the capabilities for rare
probes at low $p_{\rm T}$ will have be completed, which in particular include
a new inner tracking system based on MAPS [98], the GEM-based TPC readout
[99], and the forward muon tracker [100]. LHCb, will complete its LS2 upgrades
[101], with among more minor improvements, a new pixel vertex locator [102]
and a new high-granularity silicon micro-strip planes upstream and
scintillating-fibre downstream tracker [103] usable in up to 30–100% central
PbPb collisions. In 2023, sPHENIX [104] is expected to start operating
specifically designed for measurements of hard probes at RHIC. The major
upgrades for CMS [105] and ATLAS [106] are prepared for data taking in 2027
after LS3. For CMS, they mainly are a new high-granularity pixel and Si-strip
tracking system up to $\eta<3.8$ [107] with a MIP timing layer for time-of-
flight measurements [108] and a high-granularity calorimeter [109]. For ATLAS,
similarly, they mainly are a new high-granularity pixel and Si-strip tracking
system up to $\eta<4$ [110]. In LS3, the ALICE collaboration will replace the
inner-most pixel layers with truly cylindrical layers using thinned, wafer-
sized sensors [111], which will further improve the secondary vertex finding
as well as tracking efficiency and momentum resolution at low $p_{\rm T}$.

Fig. 6:  Timeline of planned and proposed near- and far-future instrumentation
for high-energy nuclear physics, grouped into: the high-density frontier
(green) with BESII [112], BM@N [113], MPD [114], LHCb fixed-target [94], NA62
[8], CEE [115], CBM [11], NA60+ [9], J-PARK-HI [12]; the high-energy frontier
(red) with ALICE LS2 upgrades [96, 97], sPHENIX [104], CMS LS3 upgrades [105]
and ATLAS LS3 upgrades [106], ALICE ITS3 [111]; the small-$x$ frontier (blue)
with LHCb phase-1 upgrade [101], forward sPHENIX [80], forward STAR [81],
FoCal [77], EIC [85]; the ultra-precision future (black): LHCb phase-2 upgrade
[116], NGHI [117], FCC [5], SppS [118].

Besides the LHCb after its phase-1 upgrade [101], the proposed forward
upgrades [80, 81] at RHIC and FoCal [77] at the LHC, there is the planned EIC
[85] expected to begin data-taking around 2030. The EIC will be ep and eA
polarized (up to small nuclei) collider dedicated to the study of proton and
nuclei structures. The collider will enable DIS off a proton or a nucleus with
a range of $20<\sqrt{s}<140$ GeV and a luminosity of $10^{34}$ cm-2s-1. It
will allow us to systematically explore correlations inside protons and
nuclei, as well as to study saturation and hadronization with a controllable
initial state. For an overview of the EIC physics program and the connection
to heavy-ion physics, given at the conference, see [119].

For data-taking at similar timescale as the EIC, the successor of ALICE, the
Next Generation Heavy Ion (NGHI) has been proposed [117]. It is designed as a
fast, ultra-thin detector with precise tracking and timing, and will provide
the ultimate performance for measurements related to (multi-)heavy-flavor,
soft hadrons and thermal radiation over large range in rapidity $\eta\leq 4$.
Eventually one can add far-forward tracking and particle identification
capabilities to access the high net-baryons and/or a far forward calorimeter
for ultra-soft-$x$ photons. In 2030, LHCb will also have concluded its phase-2
upgrade [116], after which its tracking detectors will be able to cope with
even higher track densities of the HL-LHC. This will also allow the
reconstruction of central PbPb data, and open up the full suite of particle
identification provided by the LHCb spectrometer at forward rapidity inPbPb
collisions. Together with ATLAS and CMS, these experiments will provide an
incredibly rich, precision high-density QCD and heavy-ion program at the LHC,
in parallel to the precision cold nuclear matter program the EIC. As indicated
also in Fig. 6, the far-future perspective is equally bright, given the
activities related to the FCC [5] including preparations for a heavy-ion
program [120] and the CEPC-SppC [118].

##  6 Summary

Instead of a “summary of the summary”, let me just point out that —without any
doubt— we are experiencing the golden age of high-density QCD and heavy-ion
physics, with a numerous interesting problems to solve and an extremely bright
future in instrumentation ahead.

## Acknowledgements

I would like to thank the conference organizers for their endless effort to
create such a stimulating and successful conference, and for being so extra-
ordinary hosts. Support by the U.S. Department of Energy, Office of Science,
Office of Nuclear Physics, under contract number DE-AC05-00OR22725, is greatly
appreciated.

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# Intersecting hexagons in $3$-space

József Solymosi  Department of Mathematics, University of British Columbia,
Vancouver, BC, Canada V6T 1Z2<EMAIL_ADDRESS>and  Ching Wong
Department of Mathematics, University of British Columbia, Vancouver, BC,
Canada V6T 1Z2<EMAIL_ADDRESS>
###### Abstract.

Two hexagons in the space are said to intersect badly if the intersection of
their convex hulls consists of at least one common vertex as well as an
interior point. We show that the number of hexagons on $n$ points in $3$-space
without bad intersections is $o(n^{2})$, under the assumption that the
hexagons are ‘fat’.

The work of the first author has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 741420, 617747, 648017). His research is also
supported by NSERC and OTKA (K 119528) grants.

##  1\. Introduction

The general problem of finding the maximum number of hyperedges in a geometric
hypergraph in $d$-dimensional space with certain forbidden configurations
(intersections) was considered by Dey and Pach [1]. We are interested in
finding the maximum number of (planar and convex) polygons on some vertex set
of $n$ points in $3$-space, where no two of them are allowed to intersect in
certain ways. In this paper, we confine ourselves in the $3$-space, and by
polygons we mean planar polygons, i.e. the vertices are co-planar, which are
convex. As usual, a $k$-gon (where $k\geq 3$) is a polygon with $k$ vertices.

###  1.1. Almost disjoint polygons

It was asked by Gil Kalai and independently by Günter Ziegler what the maximum
possible number of triangles spanned by $n$ points is, such that any two are
almost disjoint:

######  Definition 1 (Almost disjoint polygons).

Two polygons in $3$-space are said to be almost disjoint if the intersection
of their convex hulls is either empty or consists of one common vertex.

Let $f_{1}(n,k)$ be the maximum possible number of pairwise almost disjoint
$k$-gons on $n$ points in $3$-space. It is easy to see that $f_{1}(n,k)\geq
f_{1}(n,k+1)$ for all $k\geq 3$, by randomly forming a $k$-gon from each
$(k+1)$-gon.

Given a set of pairwise almost disjoint triangles on $n$ points, any given
point can be a vertex of at most $(n-1)/2$ triangles. It follows that

$f_{1}(n,k)\leq f_{1}(n,3)\leq\dfrac{n}{3}\dfrac{n-1}{2}<\dfrac{n^{2}}{6},$

for every $k\geq 3$.

Károlyi and Solymosi [2] constructed configurations showing that
$f_{1}(n,3)\geq cn^{3/2}$ for some universal constant $c>0$. Finding sharper
lower bounds seems like a very hard problem. In fact, it is not even known if
the genus of a polytope on $n$ vertices can have order ${n^{2}}$. If so, the
magnitude of $f_{1}(n,3)$ would be $n^{2}$. The best lower bound of the
largest genus is $n\log n$, due to a construction of McMullen, Schulz and
Wills [3]. For more details, we refer the interested readers to [5] where
Ziegler gives a simplified construction providing the same bound.

When $k=3$, a tight asymptotic bound can be obtained if we relax the
assumption a little by allowing the triangles to intersect also in an (entire)
edge. Let $f_{2}(n,k)$ be the maximum possible number of $k$-gons on $n$
points so that the convex hulls of any two of them are either disjoint, or
intersect in a vertex or an edge. Since we allow more intersections, we have
$f_{2}(n,k)\geq f_{1}(n,k)$ for all $n$ and $k\geq 3$. It is again clear that
$f_{2}(n,k)$ decreases with $k$, for all $n$.

######  Proposition 2.

As $n\to\infty$, there holds

$f_{2}(n,3)=\Theta(n^{2}).$

Its proof is given in Section 2.

###  1.2. Non-badly intersecting polygons

In this paper, we focus on an even more relaxed assumption on the
intersections of the $k$-gons.

######  Definition 3 (Badly intersecting polygons).

Two polygons in $3$-space are said to intersect badly if the intersection of
their convex hulls consists of at least one common vertex as well as an
interior point.

Figure 1. (left) A set of 4 quadrilaterals without bad intersections. (Right)
Two quadrilaterals that intersect badly.

We say that a collection of $k$-gons has no bad intersections if no two of
these polygons intersect badly. See Figure 1.

Let $f_{3}(n,k)$ be the maximum possible number of $k$-gons without bad
intersections on $n$ points in $3$-space. It is, once again, true that
$f_{3}(n,k)\geq f_{3}(n,k+1)$ and $f_{3}(n,k)\geq f_{2}(n,k)$, for all $k\geq
3$.

In such arrangements, two $k$-gons cannot share a diagonal and so
$f_{3}(n,k)=O(n^{2})$ for $k\geq 4$. In fact, the proof of Proposition 2
(first part) works here as well, and so

$f_{3}(n,k)<n^{2},$

for every $k\geq 3$.

This upper bound is actually sharp, in magnitude, for triangles and
quadrilaterals ($k=3,4$). One can give a construction of $\Omega(n^{2})$
quadrilaterals on $n$ points without bad intersections as follows: Let $n$ be
an even number and suppose we are given $n/2$ points $P_{1},\dots,P_{n/2}$ in
general position (no three points collinear) on a plane $\pi$. Fix any vector
$v$ not parallel to $\pi$. Then the $n$ points
$P_{1},\dots,P_{n/2},P_{1}+v,\dots,P_{n/2}+v$ are incident to
$\binom{n/2}{2}=(n^{2}-2n)/8$ desired quadrilaterals with vertices
$P_{i},P_{j},P_{j}+v,P_{i}+v$, where $1\leq i<j\leq n/2$. Figure 2 shows an
example when $n=8$.

Figure 2. A set of $\binom{8/2}{2}=6$ quadrilaterals without bad
intersections.

When $k=6$, we show that the number of hexagons without bad intersections is
$o(n^{2})$, under an extra assumption on the ‘fatness’ of the hexagons defined
below.

######  Definition 4 (Fat hexagons).

Let $c\geq 1$ and $0<\alpha<\pi/2$. A hexagon is $(c,\alpha)$-fat if

  1. (1)

the ratio of any two sides is bounded between $1/c$ and $c$; and

  2. (2)

it has three non-neighbour vertices having interior angles between $\alpha$
and $\pi-\alpha$.

Our main tool is the Triangle Removal Lemma of Ruzsa and Szemerédi, which
states that for any $\varepsilon>0$, there exists $\delta>0$ such that any
graph on $n$ vertices with at least $\varepsilon{n^{2}}$ pairwise edge-
disjoint triangles, has at least $\delta{n^{3}}$ triangles in total. See [4]
for the original formulation of this result. The precise statement of our main
theorem is as follows.

######  Theorem 5.

For any $c\geq 1$ and $0<\alpha<\pi/2$ numbers there is a function
$F_{(c,\alpha)}(n),$

$\frac{F_{(c,\alpha)}(n)}{n^{2}}\rightarrow 0\quad\text{ as }\quad
n\rightarrow\infty,$

such that any family of $(c,\alpha)$-fat hexagons in $3$-space on $n$ points
without bad intersections has size at most $F_{(c,\alpha)}(n)$.

We prove Theorem 5 in Section 3. We believe that the statement holds for any
set of hexagons or even for pentagons.

##  2\. Proof of Proposition 2

We first show that $f_{2}(n,3)<n^{2}$. Given such a set of triangles on $n$
points. Pick any of these $n$ points, say $P$, and project the remaining $n-1$
points in $3$-space onto a sphere $S$ centred at $P$. We want to upper bound
the number of triangles incident with the point $P$. For each of such
triangles, say $PQR$, we project the line segment between $Q$ and $R$ onto the
sphere $S$. These geodesic segments, together with the projected points, form
a graph $G$ on $S$ having at most $n-1$ vertices. Here, we subdivide an edge
(geodesic segment) if there are vertices lying on it. We note that if there
were multiple edges on $G$, then their corresponding triangles would lie on
the same plane and intersect in an interior point, as shown in Figure 3.
Hence, the number of edges in $G$ is at least the number of triangles incident
to $P$. As illustrated in Figure 4, the graph $G$ is planar, and so it has at
most $3(n-1)-6=3n-9$ edges. Hence,

$f_{2}(n,3)\leq\dfrac{n}{3}(3n-9)<n^{2}.$

Now, we show that $f_{2}(n,3)\geq(n-1)^{2}/4$ whenever $n$ is odd. To see
this, we are using the well-known Christmas tree arrangement. Let there be $m$
points on a circle centred at the origin on the $xy$-plane and $m+1$ points on
$z$-axis, as in Figure 5 (left). So we have a total of $n=2m+1$ points in
$3$-space. We consider the $m^{2}=(n-1)^{2}/4$ triangles with one vertex on
the circle, and the other two vertices being a consecutive pair of points
chosen on the $z$-axis. See Figure 5 (right). It is easy to see that if two
triangles are not disjoint, they intersect in either a vertex or an edge.

Figure 3. The graph $G$ does not have multiple edges, or there are some
triangles intersecting in an unwanted way. Figure 4. If there were two
triangles both incident to $P$ so that the projections of their corresponding
line segments intersect internally, then these two triangles would intersect
in an unwanted way. Figure 5. (left) There are $m=5$ points on the $xy$-plane
and $m+1=6$ points on the $z$-axis. (right) There are $m^{2}=25$ triangles on
$2m+1=11$ points.

##  3\. Proof of Theorem 5

Let $\varepsilon>0$. Given $\varepsilon{n^{2}}$ many $(c,\alpha)$-fat hexagons
on $n$ vertices in $3$-space. We show that two of these hexagons intersect
badly, when $n$ is large enough. We may assume, in particular, that no two
hexagons share a diagonal.

To reduce the dimension of the ambient space, we project these hexagons onto a
random plane such that no two vertices share the same projection and that a
positive fraction of the hexagons is $(c^{\prime},\alpha^{\prime})$-fat.
Indeed, if we project a $(c,\alpha)$-fat hexagon $H$ onto a plane making an
angle at most $\theta<\pi/2$ with the plane containing $H$, it is
straightforward to show that the projected hexagon is
$(c^{\prime},\alpha^{\prime})$-fat, where

$c^{\prime}=\dfrac{c}{\cos\theta}\quad\quad\text{and}\quad\quad\alpha^{\prime}=\cos^{-1}\left(\dfrac{\cos\alpha+\sin^{2}\theta}{\cos^{2}\theta}\right).$

The existence of badly intersecting hexagons relies on a similar-slope
property. This can be described quantitatively by the difference of two angles
of inclination. To this end, let $\phi>0$ be the smallness of such difference
which is to be determined later.

We choose from the $(c^{\prime},\alpha^{\prime})$-fat projected hexagons the
most popular family consisting of $\varepsilon^{\prime}n^{2}$ hexagons, which
have inscribed triangles of similar shapes and orientations.

More precisely, let us enumerate by any order the projected hexagons as
$\\{H_{i}\\}$ and label their vertices as
$A_{i},B_{i},C_{i},D_{i},E_{i},F_{i}$, oriented counter-clockwise, where
$B_{i},D_{i},F_{i}$ are the three non-neighbour vertices having angles between
$\alpha^{\prime}$ and $\pi-\alpha^{\prime}$.

There exists a positive fraction of these hexagons so that for any $i,j$, the
inclined angles of the diagonals $A_{i}C_{i}$ and $A_{j}C_{j}$ differ by at
most $\phi$. Similarly the same property holds true for the diagonals
$C_{i}E_{i}$ and $E_{i}A_{i}$ in yet a sub-collection of
$\varepsilon^{\prime}n^{2}$ hexagons.

We define $G$ to be the graph whose vertices are the $n$ projected points and
whose edges are from the triangles formed by the vertices $A_{i},C_{i},E_{i}$
chosen above. Then, as we assumed that no two hexagons share a diagonal, $G$
contains $\varepsilon^{\prime}n^{2}$ edge-disjoint triangles. An application
of the Triangle Removal Lemma yields, when $n$ is large enough, a triangle $T$
whose edges come from three different hexagons, say $H_{1}$, $H_{2}$ and
$H_{3}$. For each $i=1,2,3$, let $T_{i}$ be the triangle $A_{i}C_{i}E_{i}$.

We are ready to study the intersection properties of these three hexagons in
the $3$-space. In other words, we now ‘unproject’ the $n$ points.

Two of the triangles, say $T_{1}$ and $T_{2}$, lie on the same side of $T$ and
let $T_{1}$ be the triangle making a larger angle with $T$. Then, as shown in
Figure 6, the hexagon $H_{2}$ intersects badly with the triangle $T_{1}$, and
hence with the hexagon $H_{1}$, as long as the three non-neighbour vertices
$B_{1},D_{1},F_{1}$ lie outside of the triangle $T$ on the plane of
projection, which is guaranteed if we choose

$\phi<\tan^{-1}\left(\dfrac{\sin\alpha^{\prime}}{c^{\prime}+\cos\alpha^{\prime}}\right),$

the right hand side being a lower bound of the six angles $B_{1}A_{1}C_{1}$
etc. under the $(c^{\prime},\alpha^{\prime})$-fatness assumption. This
completes the proof.

Figure 6. The triangle $T_{1}=AC_{1}E_{1}$ and the hexagon
$H_{2}=AB_{2}C_{2}D_{2}E_{2}F_{2}$ intersect badly. Here the triangle $T$ is
$AC_{2}E_{1}$.

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