Abstract
The paper briefly presents history, status, and plans of the search for the critical structures -the onset of fireball, the onset of deconfinement, and the deconfinement critical point -in high energy nucleus-nucleus collisions. First, the basic ideas are introduced, the history of the observation of strongly interacting matter in heavy ion collisions is reviewed, and the path towards the quark-gluon plasma discovery is sketched. Then the status of the search for critical structures is discussed -the discovery of the onset of deconfinement, indications for the onset of fireball, and still inconclusive results concerning the deconfinement critical point. Finally, an attempt to formulate priorities for future measurements -charm quarks vs the onset of deconfinement and detailed study of the onset of fireball -closes the paper.

I.
INTRODUCTION AND VOCABULARY
One of the important issues of contemporary physics is the understanding of strong interactions and in particular the study of the properties of strongly interacting matter -a system of strongly interacting particles in equilibrium. The advent of the quark model of hadrons and the development of the commonly accepted theory of strong interactions, quantum chromodynamics (QCD), naturally led to expectations that matter at very high densities may exist in a state of quasi-free quarks and gluons, the quark-gluon plasma (QGP).
INTRODUCTION AND VOCABULARY
Does the QGP exist in nature? How does the transition proceed from a low-density state of strongly interacting matter, in which quarks and gluons are confined in hadrons, to the QGP?
INTRODUCTION AND VOCABULARY
Is it similar to the transition from liquid water to water vapour along a first order transition line ending in a second order critical point and followed by a cross over transition, see illustration plots in Fig. 1 ?
INTRODUCTION AND VOCABULARY
The study of high energy collisions of two atomic nuclei gives us the unique possibility to address these issues in well controlled laboratory experiments. This is because it is observed that a system of strongly interacting particles created in central heavy ion collisions is close to (at least local) equilibrium. How does the transition from a non-equilibrium system created in inelastic proton-proton interactions to the equilibrium system in central heavy ion collisions look like?
INTRODUCTION AND VOCABULARY
These questions have motivated broad experimental and theoretical efforts for about 50 years.
INTRODUCTION AND VOCABULARY
Systematic measurements of particle production properties in nucleus-nucleus (A+A) collisions at different collision energies and for different masses of colliding nuclei have been performed.
INTRODUCTION AND VOCABULARY
By changing collision energy and nuclear mass number one changes macroscopic parameters of the created system -its volume, energy, and net baryon number. This allows to move across the phase diagram and look for the theoretically predicted boundaries of equilibration and matter phases, see illustration plots in Fig. 1 . Consequently several physics phenomena might be observed when studying experimentally nuclear collisions at high energies. These are:
INTRODUCTION AND VOCABULARY
(i) onset of fireball -beginning of creation of large-volume ( 1 fm 3 ) strongly interacting matter, (ii) onset of deconfinement -beginning of QGP creation with increasing collision energy, (iii) deconfinement critical point -a hypothetical end point of the first order transition line to quark-gluon plasma that has properties of a second order phase transition.
INTRODUCTION AND VOCABULARY
These phenomena are expected to lead to rapid changes of hadron production propertiesthe critical structures -when changing collision energy and/or nuclear mass number of the colliding nuclei.
INTRODUCTION AND VOCABULARY
II.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
Strongly interacting matter. The equation of state defines the macroscopic properties of matter in equilibrium. It is a subject of statistical mechanics. The first step in this modelling is to clarify the types of particle species and inter-particle interactions. One should also choose an appropriate statistical ensemble which fixes the boundary conditions and conserves the corresponding global physical quantities, like energy and conserved charges. Strongly interacting matter at high energy density can be formed at the early stages of relativistic A+A collisions.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
As mentioned in the introduction the questions -(i) What types of particles should be considered as fundamental?
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
(ii) What are the composite objects?
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
(iii) What are the fundamental forces between the matter constituents?
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
(iv) What are the conserved charges? -should be addressed. Answers to these questions are changing with time as our knowledge about basic properties of elementary particles and their interactions increases.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
The first model of strongly interacting matter at high energy density was formulated in 1950 by Fermi [2] . It assumes that a system created in high energy proton-proton (p+p) interactions emits pions like black-body radiation, i.e., pions are treated as non-interacting particles, and the pion mass m π ∼ = 140 MeV is neglected compared to the high temperature of the system. The pressure p and energy density ε can then be represented by the following functions of the temperature T (the system of units with h/(2π) = c = k B = 1 will be used),
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
where σ = π 2 g/30 is the the so-called Stephan-Boltzmann constant, with g being the degeneracy factor (the number of spin and isospin states), and g = 3 counting the three isospin states π + , π 0 , π − .
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
Hadrons and resonances. The study of particle production in high energy collisions started in the 1950s with discoveries of the lightest hadrons -π, K, and Λ -in cosmic-ray experiments. Soon after with the rapid advent of particle accelerators new particles were discovered almost day-by-day. The main feature of the strong interactions appears to be creation of new and new types of particle species -hadrons and resonances -when increasing the collision energy. A huge number (several hundreds) of different hadron and resonance species are known today. The simplest statistical model treats the hadron matter, i.e. a system of strongly interacting particles at not too high energy density, as a mixture of ideal gases of different hadron-resonance species.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
Hadron-resonance gas. In the grand canonical ensemble the pressure function is then written as
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
where g i is the degeneracy factor of the i th particle and the normalized function the Boltzmann approximation. The chemical potential for the i th hadron is given by
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
with b i = 0, ±1, s i = 0, ±1, ±2, ±3, and q i = 0, ±1, ±2 being the corresponding baryonic number, strangeness, and electric charge of the ith hadron. Hadrons composed of charmed and beauty quarks are rather heavy and thus rare in the hadron-resonance gas, and their contribution to the thermodynamical functions are often neglected. Chemical potentials are denoted as µ ≡ (µ B , µ S , µ Q ) and correspond to the conservation of net-baryon number, strangeness, and electric charge in the hadron-resonance gas. The entropy density s, net-charge densities n i (with i = B, Q, S), and energy density ε are calculated from the pressure function (2) according to the standard thermodynamic identities:
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
Note that only one chemical potential µ B is considered as independent variable in fits of the model to particle multiplicities produced in A+A reactions. Two others, µ S and µ Q , should be found, at each pair of T and µ B , from the requirements that the net-strangeness density equals to zero, n S = 0, and the ratio of the net-electric charge density, n Q , to n B equals to the ratio of the number of protons, Z, to the number of all nucleons, A (protons and neutrons) in the colliding nuclei, n Q /n B = Z/A. Equations (2-4) define the ideal hadron-resonance gas model. In spite of evident simplifications this model rather successfully fits the rich data on mean multiplicities of hadrons measured in central A+A collisions at high energies.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
Hagedorn model. Is there an upper limit for the masses of mesonic and baryonic resonances? In 1965 Hagedorn formulated a statistical model assuming an exponentially increasing spectrum of hadron-resonance states at large masses [3] :
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
where C, a, and T H are the model parameters. At that time the number of experimentally detected hadron-resonance states was much smaller than it is today. Nevertheless, Hagedorn made the brave assumption that these m-states interpolate the low-mass spectrum and extend to m → ∞, and that their density at large m behaves as in Eq. (5) . The pressure function at µ = 0 then becomes
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
with the function φ m (T ) behaving at m/T 1 as
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
The result (7) can be found from Eq. the properties of QCD matter and came to a qualitatively similar conclusion: QCD matter at high temperature is best described by quark and gluon degrees of freedom and the name quark-gluon plasma was coined [8, 9] .
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
Questions concerning QGP properties and properties of its transition to matter consisting of hadrons, have been considered since the late 1970s (see, e.g., Ref. [10] ). The Hagedorn model was still rather popular at that time due to its successful phenomenological applications. For example, the temperature parameter T ch found from fitting the data on hadron multiplicities in p+p interactions and A+A collisions at high energies (the so-called chemical freeze-out temperature) was found to be close to the limiting Hagedorn temperature,
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
Hagedorn and Rafelski [11] as well as Gorenstein, Petrov, and Zinovjev [12] suggested that the upper limit of the hadron temperature, the Hagedorn temperature T H , is not the limiting temperature but the transition temperature to the QGP, T C = T H ≈ 150 MeV. The Hagedorn fireball was then interpreted as the quark-gluon bag formed in the early stage of the collision.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
It also had an exponential mass spectrum (5) like in the Hagedorn model, but was not a point like object. The average volume of the quark-gluon bag increases linearly with its mass. This causes the excluded volume effects in the system of bags and leads to the transition to the high temperature QGP phase. Note that the first QCD-inspired estimate of the transition temperature to the QGP gave T C ≈ 500 MeV [8] , the most recent QCD-based estimates obtain [13] .
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
Many physicists started to speculate that the QGP could be formed in A+A collisions at sufficiently high energies in which one expects that strongly interacting matter of high energy density will be created. Therefore the QGP might be discovered in laboratory experiments.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
The first experiments. In parallel to the theoretical ideas and models, experimental studies of A+A collisions were initiated in 1970 at the Synchrophasotron (JINR Dubna) [18, 19] and in 1975 at the Bevelac (LBL Berkeley) [20] . Several effects were observed which could be attributed to a collective behaviour of the created system of hadrons. These are anisotropic and radial flow of particles [21] [22] [23] , enhanced production of strange particles [24] and suppressed production of pions [25] . They could only be explained by assuming that strongly interacting matter is produced in the studied collisions [26, 27] . In what follows we use the term fireball as the notation for a large volume ( 1 fm 3 ) system consisting of strongly interacting particles close to at least local equilibrium. They can be either hadrons and resonances or quarks and gluons.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
Initiating the hunt for QGP. The two findings, (i) theoretical: QCD matter at sufficiently high temperature is in the state of a QGP;
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
(ii) experimental: strongly interacting matter is produced in heavy ion collisions at energies of several GeV per nucleon;
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
led activists of the field [28] to the important decision to collide heavy ions at the maximum possible energy with the aim to discover the QGP. In the 1980s the maximum possible energy for heavy ion collisions was available at CERN, Geneva. This is why heavy ion physics entered (i) suppressed production of charmonium states, in particular J/ψ mesons [30] , (ii) enhanced production of strange and multi-strange hadrons from the QGP [31] , (iii) characteristic radiation of photons and dilepton pairs from the QGP [9] .
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
Measurements at the CERN SPS. The search for the QGP at the CERN SPS was performed in two steps: An estimate of the energy density in A+A collisions can be obtained from measurement of the transverse energy production and the size of the collision system. Already in S collisions with heavy nuclei (see Fig. 3 ) it was found that values above 1 GeV/fm 3 were reached (NA34 [32, 33] , NA35 [34] , NA49 [35] ). Moreover the fireball showed effective temperature increasing linearly with particle mass, a characteristic of collective radial expansion (see Fig. 4 (left)). Also mean multiplicities of produced hadrons are well reproduced by the statistical model [36] (see Fig. 4 (right)). Thus conditions in collisions of heavy nuclei at the top energy of the CERN SPS are promising for the production of the QGP. The QGP discovery. Results on central collisions of medium size and heavy nuclei from the QGP search programme at the CERN SPS appeared to be consistent with the predictions for the QGP:
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
(i) The relative yield of J/ψ mesons is significantly suppressed compared to that in p+p and p+A interactions (NA38 [38] , NA50 [39] , NA60 [45] ) as expected for the J/ψ melting in a QGP (see Fig. 5 (left)).
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
(ii) The relative strangeness yield is consistent with the yield expected for the equilibrium QGP. Moreover, it is significantly enhanced compared to that in p+p and p+A interactions (see Fig. 5 (center )) (NA35 [40] , NA49 [41] ). Even larger enhancement is measured for the relative yield of multi-strange hyperons (see Fig. 5 (right)), (WA97 [46] , NA57 [42] ).
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
Note that QGP formation was not expected in p+p and p+A collisions. Inverse slope parameter (effective temperature) of the transverse mass distribution versus particle mass measured by WA97, NA44 and NA49 [37] . Right: Mean hadron multiplicities measured by NA49 compared to the statistical model fit [36] . Pb+Pb collisions at the top CERN SPS energy.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
(iii) Spectra of directly produced dimouns (virtual photons) and photons emerge above a dominant background at large mass respectively transverse momentum and show a thermal contribution with an effective temperature of about 200 MeV. This is significantly larger than the expected transition temperature to QGP (see Fig. 6 ) (NA60 [44] and WA98 [43] ).
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
Standard model of heavy-ion collisions. These and other results established the standard picture of heavy-ion collisions [47] :
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
(i) High density strongly interacting matter is created at the early stage of heavy-ion collisions. Starting at SPS collision energies it is in the QGP phase. [38, 39] (NA38, NA50). Center:
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
Comparison of K + /π + yield ratio in p+p, p+A and A+A collisions [40, 41] (NA35, NA49). Right:
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
Comparison of the mid-rapidity ratios of of hyperon production to number of wounded nucleons in p+Be, p+Pb and Pb+Pb collisions [42] (NA57). Top CERN SPS energy.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
(iv) The hadronic matter after hadronization is still dense enough to modify the hadron composition and continue expansion.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
(v) At sufficiently low densities the hadron interaction rate drops to zero (freeze-out). resonances decay and long-lived hadrons freely fly away e.g. towards particle detectors at the CERN SPS.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
Conclusion from the QGP-search. These major achievements were compiled by the heavy-ion community [48] and led to the CERN press release -on is also an important step forward in the understanding of the early evolution of the universe.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
We now have evidence of a new state of matter where quarks and gluons are not confined. There Clearly, in order to distinguish between different effects and verify the J/ψ signal of the QGP creation systematic data on open charm production is needed.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
Strangeness enhancement. A fast equilibration of ss pairs was predicted as a QGP signature [31, 58] . This is mainly because of the small mass of the strange quark, m s ∼ 100 MeV compared to the QGP temperature: T ≥ T C > m s ≈ 100 MeV. The estimated strangeness equilibration time was found to similar to the life time of the QGP phase in heavy-ion collisions at high energies. In fact the strangeness yield measured in A+A collisions at the top SPS energy and above corresponds to the QGP equilibrium yield, for recent review see Ref. [58] . Moreover, it was estimated that the strangeness equilibration time in the confined matter is about 10 times longer than the life time of the hadronic phase in A+A collisions. This is because masses of strange hadrons, starting form the lightest one, the kaon (m K ∼ 500 MeV) are much larger than the maximum temperature of the hadron-resonance gas T ≤ T C ≈ 150 MeV. Thus a small yield of strangeness was expected for reactions in which the QGP was not expected, p+p and p+A interactions and A+A collisions at low energies. Consequently the enhanced production of strangeness was predicted as the next QGP signal [59] .
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
The strangeness enhancement is quantified by comparing a strange-hadron to pion ratio in A+A collisions with that in p+p interactions. In particular a double ratio is calculated:
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
where . . . AA and . . . pp denote the event averages of K + and π + yields in, respectively, A+A collisions and p+p interactions at the same center of mass collision energy √ s N N of the nucleon pair. Ratios of different strange hadrons to pions were considered, e.g., K + K / π , Λ / π , . . ., Ω / π , and then analyzed by forming the double ratios R, similar to the one given by Eq. (8) . The confrontation of this expectation with the data was for the first time possible in 1988 when results from the SPS and the AGS became available.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
NA35 reported [40] that in central S+S collisions at 200A GeV the strangeness to pion ratio (8) is indeed about two times higher than in nucleon-nucleon interactions at the same energy per nucleon. But an even larger enhancement (R = 14 − 5) was measured at the AGS at 2A − 10A GeV [60, 61] demonstrating that strangeness enhancement increases with decreasing collision energy. Moreover, the enhancement factor (8) should evidently go to infinity at the threshold energy of strange hadron production in nucleon-nucleon interactions. Note also that the strangeness neutrality introduced to statistical models using the canonical ensemble leads to a suppression of the relative yield of strange particles in systems with a low multiplicity of strangeness carriers [62] , e.g. p+p interactions at SPS energies. In any case, the AGS measurements indicating a strangeness enhancement larger than that at the CERN SPS show clearly difficulties in interpreting the strangeness enhancement as the QGP signal.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
New strategy. Difficulties in interpretation of the QGP signatures forced scientists to rethink the QGP-hunt strategy. The emerging new strategy was similar to the one followed by physics studying molecular liquids and gases. In these essentially simpler and familiar cases it is also sometimes difficult to distinguish the properties of a dense gas from those of (ii) softening of collective flow of hadrons [65] [66] [67] [68] , which should be observed in hadron distributions in transverse [65] and longitudinal momenta [68] and azimuth angle [66, 67] . Discovery of the onset of deconfinement. Results on the collision energy dependence of hadron production in central Pb+Pb collisions from the onset-of-deconfinement search programme at the CERN SPS [70, 71] appeared to be consistent with the predicted signals (for review see Ref. [72] ):
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
(i) The average number of pions per wounded nucleon, N π / W , in low energy A+A collisions is smaller that this value in p+p reactions. This relation is however changed to the opposite at collision energies larger than ≈ 30A GeV, the so-called kink structure.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
(ii) The collision energy dependence of the K + AA / π + AA ratio shows the so-called horn structure. Following a fast rise the ratio passes through a maximum in the SPS range, at approximately 30A GeV, and then decreases and settles to a plateau value at higher energies. This plateau was found to continue up to the RHIC and LHC energies.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
(iii) The collision energy dependence of the inverse slope parameter of the transverse mass spectra, T * , of charged kaons shows the so-called step structure. Following a fast rise the T * parameter passes through a stationary region (or even a weak minimum for K − ), which starts at the low SPS energies, approximately 30A GeV, and then enters a domain of a steady increase above the top SPS energy. Figure 7 shows examples of the most recent plots [69] illustrating the observation of the onset-of-deconfinement signals. As seen data from the RHIC BES I programme (2010) (2011) (2012) (2013) (2014) and LHC (see Ref. [69] for references to original experimental papers) confirm the NA49 results and their interpretation.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
Two comments are appropriate here. The strangeness to pion ratio, e.g., K + AA / π + AA , strongly increases with collision energy in the hadron phase. This happens because m K /T 1,
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
whereas m π /T ∼ = 1. Thus, a much stronger increase with increasing temperature is expected for the multiplicities of heavy strange hadrons than that of pions. The strangeness to pion ratio reaches its maximum inside the hadron phase at the onset of the deconfinement. The plateau-like behavior at high collision energies reflects the approximately constant value of the strangeness to entropy ratio in the QGP. It equals to the ratio of the degeneracy factor of strange quarks, g s = 7 8 · 2 · 2 · 3 = 10.5 ,
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
to the total degeneracy factor of the quark-gluon constituents in the QGP,
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
These degeneracy factors count 2 spin states of quarks and gluons, 3 flavor quark states, 8 colour states of gluons and 3 colour states of quarks, one more factor 2 appears due to antiquarks (the factor 7/8 is due to the Fermi statistics of quarks). The strangeness to entropy ratio in the HRG at the largest hadron temperature T ∼ = T H ∼ = 150 MeV appears to be larger than this ratio in the QGP which is approximately constant at all QGP temperatures T ≥ T H . Therefore, the transition region from hadron matter to the QGP reveals itself as the suppression of strangeness yield relative to pion yield.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
The second comment concerns the inverse slope parameter T * of the transverse mass (m T =
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
The parameter T * is sensitive to both the thermal and collective motion transverse to the collision axis and behaves as
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
where T is the temperature and v T is the transverse collective (hydrodynamic) velocity of the hadronic matter at the kinetic freeze-out. The parameter T * increases strongly with collision energy up to the energy ≈ 30A GeV. This is because an increasing collision energy leads to an increase of both terms in Eq. The goal is to search for the deconfinement critical point and study system size dependence of the onset of deconfinement. Possibilities to perform these measurements at the CERN SPS, FAIR SIS300 and RHIC were discussed. The event initiated a series of the Critical point and onset of deconfinement workshops. (ii) enhanced fluctuations of (pion multiplicity)-(transverse momentum) [80] ,
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
The signals were expected to have a maximum in the parameter space of collision energy and nuclear mass number of colliding nuclei -the hill of fluctuations [81] . This motivated NA61/ SHINE to perform a two dimensional scan at the CERN SPS [82] in these two parameters, which are well controlled in laboratory experiments.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
Measurements at SPS and RHIC. The systematic search for the d-CP of strongly interacting matter was started in 2009 with the NA61/SHINE data taking on p+p interactions at six beam momenta in the range from 13A GeV/c to 158A GeV/c. In the following years data on Be+Be, Ar+Sc, Xe+La and Pb+Pb collisions were recorded, see Fig. 9 (left) for an overview.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
In 2010 the beam energy scan (BES-I and BES-II) with Au+Au collisions started at the BNL RHIC [83] . Search for the deconfinement critical point has been the most important goal of this programme. Above the collision energy of √ s N N = 7.7 GeV (≈ 30A GeV/c) the scan was conducted in the collider mode, whereas below in the fixed target mode. The location of the recorded data in the phase diagram are shown in Fig. 9 (right) .
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
Status of the d-CP search. Many experimental results have already been obtained within the d-CP search programmes at SPS and RHIC, for a recent review see Ref. [84] . Five of them were considered as possible indications of the d-CP and are presented and discussed in the following.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
(i) A maximum of fluctuations is expected in a scan of the phase diagram (see Fig. 8 (left) ).
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
Measurements of (pion multiplicity)-(transverse momentum) fluctuations from NA61/ SHINE shown in Fig. 8 (right) do not show such a feature [74] .
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
(ii) The energy dependence of fluctuations of conserved quantities such as the net baryon number is predicted to be sensitive to the presence of the d-CP. This holds in particular for higher moments. The scaled third and fourth moments of the net-proton multiplicity distribution in Au+Au collisions from the STAR experiment is plotted in Fig. 10 (iv) The ratio of yields of light nuclei production can be related to nucleon number fluctuations [87] . The measurements from STAR in central Au+Au collisions show strong collision energy dependence and peak at √ s N N ≈ 20 − 30 GeV [88] . These results are presented in Fig. 12 . Such behaviour is not reproduced by model calculations without a d-CP and may thus be attributable to a critical point. (v) The energy and centrality dependence of short-range two-pion correlations as parameterized by source radius parameters determined from Bose-Einstein correlation analysis was used to search for indications of the d-CP [89, 90] . The result for the difference R 2 out -R 2 side in Au+Au collisions at RHIC is shown in Fig. 13 . A finite size scaling analysis of these results led to an estimate of the position of a d-CP at T ≈ 165 MeV and µ B ≈ 95 MeV. The effect of the n-CP on fluctuations of conserved charges, baryon number (B), electric charge (Q), and strangeness (S), was studied in Refs. [91, 92] within the HRG model with van der Waals interactions between baryons and between anti-baryons. The second, third, and fourth order cumulants (susceptibilities) are calculated in the grand canonical ensemble from the pressure function by taking the derivatives over the corresponding chemical potentials:
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
where i stands for B, Q, S and n is the moment order.
STRONGLY INTERACTING MATTER IN HEAVY ION COLLISIONS
The obtained results show that the n-CP may significantly impact event-by-event fluctuations in A+A collisions even at high energies. Thus, the nuclear-CP should be taken into account in future searches for the deconfinement-CP. 
C. Indication of the onset of fireball
Predictions of reference models on system-size dependence. There are two models often used to obtain reference predictions concerning the system-size dependence of hadron production properties [94] -the Wounded Nucleon Model (WNM) [95] and the Statistical Model (SM) [62] . For the K + /π + ratio at the CERN SPS energies they read:
C. Indication of the onset of fireball
(i) The WNM prediction: the K + /π + ratio is independent of the system size (number of wounded nucleons).
C. Indication of the onset of fireball
(ii) The SM prediction: in the canonical formulation incorporating global quantum number conservation the K + /π + ratio increases monotonically with the system size and approaches the limit given by the grand canonical approximation of the model. The rate of this increase is the fastest for small systems. Unexpected result of measurements. Measurements of the system size dependence of hadron production properties at different collision energies were carried out by NA61/SHINE, for detail see Sec. IV A. Figures 15 and 16 show the unexpected result [96, 97] . The K + /π + ratio in Fig. 15 and the scaled variance of the multiplicity distribution at 150A GeV/c in Fig. 16 are nuclei. However, its predictions disagree with the data on p+p to Be+Be collisions.
C. Indication of the onset of fireball
The rapid change of hadron production properties when moving from Be+Be to Ar+Sc collisions is interpreted and referred to as the onset of fireball. From Fig. 15 (right) follows that the increase of the K + /π + ratio depends on the collision energy. On the other hand, the scaled variance ω of the multiplicity distribution shows only weak collision energy dependence (see Fig. 16 (right) ). The physics behind the onset of fireball is under discussion [98] . Whereas one does not observe the formation of fireball in the collisions of light nuclei at the SPS energies, the large size system of strongly interacting matter is probably formed at LHC energies even in p+p high multiplicity events.
C. Indication of the onset of fireball
In summary, the scans in collision energy and nuclear mass number of colliding nuclei performed at SPS and RHIC indicate four domains of hadron production separated by two thresholds: the onset of deconfinement and the onset of fireball. The sketch presented in (ii) A detailed study of the onset of fireball and its collision energy dependence in the onset of deconfinement region. The goal is to understand the underlying physics of this phenomenon, for details see Ref. [99] . This requires a two dimensional scan in the nuclear mass number of the colliding nuclei and in collision energy performed with small steps in nuclear mass number.
C. Indication of the onset of fireball
Conclusive results from the data recorded by NA61/SHINE and RHIC BES-II are needed to plan future measurements for the deconfinement-CP search. 

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