Holographic phase diagram of quark-gluon plasma formed in heavy-ion collisions

We use a holographic dual model for the heavy-ion collision to obtain the phase diagram of the quark-gluon plasma (QGP) formed at a very early stage just after the collision. In this dual model, colliding ions are described by the charged gravitational shock waves. Points on the phase diagram correspond to the QGP or hadronic matter with given temperatures and chemical potentials. The phase of the QGP in dual terms is related to the case where the collision of shock waves leads to the formation of a trapped surface. Hadronic matter and other confined states correspond to the absence of a trapped surface after collision. In the dual language, the multiplicity of the ion collision process is estimated as the area of the trapped surface. We show that a nonzero chemical potential reduces the multiplicity. To plot the phase diagram, we use two different dual models of colliding ions, the pointlike and the wall shock waves, and find that the results agree qualitatively.


Introduction
For the last decade, since the publication of the fascinating papers [1][2][3], it has been realized that supersymmetric and nonsupersymmetric theories in the strong coupling limit could in principle be quite close in their properties [4]. The AdS/CFT correspondence, which appeared as a formal duality between the N =4 super Yang-Mills theory and quantum gravity in the AdS background, has become a powerful tool for studying various properties of real physical systems in the strong-coupling limit [5]. An important branch of these investigations is the analysis of the quark-gluon plasma (QGP) from the standpoint of AdS

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holography (see, e.g., [6,7]). These applications of the AdS/CFT correspondence to the strongly coupled QGP have been mostly related to equilibrium properties of the plasma or to its kinetics/hydrodynamics near equilibrium. A particular application of the AdS/CFT correspondence to the strongly coupled QGP is the analysis of the thermalization of matter and early entropy production immediately after the collision of relativistic heavy ions. The RHIC experiments have shown that a QGP forms at a very early stage just after the heavy-ion collision, i.e., a rapid thermalization occurs, and the QGP produced in the RHIC is believed to be strongly coupled, as is evidenced by its rapid equilibration. Strong collective flows and strong jet quenching are well reproduced by hydrodynamics [8][9][10]. This obviously requires calculating the strongly coupled field theory in a nonequilibrium process.
Gubser, Yarom, and Pufu recently proposed the gravitational shock wave in AdS 5 as a possible holographic dual for the heavy ion and related the area of the trapped surface (TS) formed in a collision of such waves to the entropy of matter formed after the collision of heavy ions [13]. Early papers mentioning an analogy between colliding heavy ions and colliding gravitational shock waves in AdS space include [14]- [18]. This AdS-holographic model was also used to find the stress-energy tensor of the QGP formed by ion collision. According to the AdS/CFT dictionary, this stress-energy tensor is dual to the space-time metric after the shock-wave collision [18]. Analytic calculations that involve the analogy between colliding heavy ions and colliding gravitational shock waves have also been extensively studied recently [19]- [27].
The main result in [13,28], confirmed by numerical calculations reported in [29,30], is that in the limit of a very large collision energy E, the multiplicity (the entropy S) increases as where C is a numerical factor (see section 2.1.4). Alvarez-Gaume et al. considered the central collision of shock waves sourced by a nontrivial matter distribution in the transverse space and found a critical phenomenon occurring as the shock wave reaches some dilution limit [31]. This criticality may be related to the criticality found in [29], where the numerical results show the existence of a simple scaling relation between the critical impact parameter and the energy of colliding waves.
The size of colliding nuclei is introduced via the distance of those objects from the boundary along the holographic coordinate z.
Shuryak and Lin proposed a model of an infinite homogenous wall [29], which was analyzed in [29,32]. The advantage of this model is the essential simplicity of the calculations. But the legitimacy of these calculations requires some additional examination (see our discussion in section 2.2).
In heavy-ion collisions, the energy per nucleus is not the only important variable. Associating different nuclei with different kinds of shock waves can be tried. There are several proposals in the literature on this subject. For example, a holographic model with the UV part of the bulk geometry cut off was proposed in [33]. The formation of TSs in head-on collisions of charged shock waves in the (A)dS background was considered in [34], where it JHEP05(2012)117 was shown that the formation of TSs on the past light cone is only possible when the charge is below a certain critical value, a situation similar to the collision of two ultrarelativistic charges in the Minkowski space-time [35]. This critical value depends on the energy of colliding particles and the value of a cosmological constant. The formation of TSs in head-on collisions of shock waves in gravitational theories with more complicated bulk dynamics, in particular, with the Einstein-dilaton dynamics, claimed to describe a holographic physics that is closer to QCD than the purely AdS theory [33,36,37], was recently considered by Kiritsis and Taliotis [38], 1 who found that the multiplicity increases as which is rather close to the experimental data.
Here, we propose to incorporate the study of collisions of charged gravitational shock waves [34] into the description of colliding nuclei with a nonzero baryon chemical potential. In the holographic context, the chemical potential of a strongly coupled QGP on the gravity side is related to the temporal component A t of the U(1) gauge field [40]- [47]. The asymptotic value of this gauge field component in the bulk is interpreted as the chemical potential in the gauge theory We use the same identification (1.3) for colliding ions. It would be interesting to calculate for the off-center collision of charged gravitational waves or generally smeared charged shock waves. Postponing this problem for further investigations, we here consider the head-on collision of charged point shock waves and charged wall shock waves. This provides the holographic picture for the QGP phase diagram in the first moment after collisions of heavy ions. These phase diagrams of the chemical potential (charge) µ versus the temperature (energy) T are displayed in figures 5 and 11. The colored lines separate the TS phase from the phase with no TS. We note that the obtained diagrams differ from the phase diagram for the equilibrium QGP (see figure 1 in section 2.1.2). This paper is organized as follows. In section 2, we set up the problem, describe the role of black holes (BHs) in the AdS/CFT description of a strongly coupled QGP in section 2.1.1, describe the QGP chemical potential in the AdS/CFT correspondence in section 2.1.2, recall the main facts about shock waves in AdS 5 related to the TS formation in section 2.1.3, and describe the dual conjecture proposed in [13] in detail in section 2.1.4. In section 2.2, we pay special attention to the regularization problem that appears in the wall shock wave approach. In section 3, we present the phase diagram of chemical potential versus temperature for the QGP formed in the heavy-ion collisions using the holographic approach with the central collision of charged shock waves. In section 4, we calculate the same problem using the regularized version of the charged wall shock waves. In section 5, we summarize our calculations and also discuss further directions related to the holographic description of the QGP formed in heavy-ion collisions. The idea of using the AdS/CFT correspondence to describe the QGP is based on the possibility of establishing a one-to-one correspondence between phenomenological/thermodynamic plasma parameters (T , E, P , and µ) and the parameters characterizing AdS 5 deformations. In the dual gravity setting, the source of temperature and entropy are attributed to the gravitational horizons. The relation between the energy density and temperature typical for the BH in the AdS according to [48,49] is In the phenomenological model of a QGP, such as the Landau or Bjorken hydrodynamic models [50,51], the plasma is characterized by a space-time profile of the energy-momentum tensor T µν (x ρ ), µ, ν, ρ = 0, . . . , 3. This state has its counterpart on the gravity side as a modification of the geometry of the original AdS 5 metric. This follows the general AdS/CFT line: operators in the gauge theory correspond to fields in SUGRA. In the case of the energy-momentum tensor, the corresponding field is just the five-dimensional metric. It is convenient to parameterize the corresponding five-dimensional geometry as which is the five-dimensional Fefferman-Graham metric [52]. The flat case g µν = η µν parameterizes AdS 5 in Poincaré coordinates. The conformal boundary of the space-time is at z = 0 and g µν (x ρ , z) = η µν + z 4 g (4) µν (x ρ ) + . . . .

(2.3)
The AdS/CFT duality leads to the relation 4) where N c is a number of colors (see [53] for a brief review). Applying the AdS/CFT correspondence to the hydrodynamic description of the QGP is based on the fact that the energy-momentum tensor can be obtained directly from the expansion of the BH in AdS 5 metric (2.3) corresponding to the simple hydrodynamic model (2.5) The AdS 5 BH in the Fefferman-Graham coordinates has form (2.2) with the following nonzero components of g µν (x ρ , z) (see [6,7] and the references therein): (2.6)

The chemical potential in a QGP via the AdS/CFT correspondence
The Reissner-Nordström (RN) metric in the AdS space has the form where Λ is a cosmological constant, Λ/3 ≡ 1/a 2 , M and Q are related to the Arnowitt-Deser-Misner mass m and the charge σ, and σ is the charge of the electromagnetic field (purely electric) with only one nonzero component and R + is the largest real root of g(R). The thermodynamics of the charged BH is described by the grand canonical potential (free energy) W = I/β, the Hawking temperature T = 1/β, and the chemical potential [54][55][56][57] given by where R + is the outer horizon, g(R + ) = 0, and I is given by the value of the action at (2.9) and (2.11). The relation to the first law of thermodynamics, dE = T dS + µ dQ, is obtained under the identifications We note that just the asymptotic value of a single gauge field component gives the chemical potential [40]- [47] µ = lim r→∞ A t (r). (2.14) The QGP is characterized at least by two parameters: the temperature and the chemical potential. Generally speaking, quantum field theories can have nonzero chemical potentials for any or all of their Noether charges. In the AdS/CFT context, two different types of chemical potential are considered: related to the R-charge and to the baryon number. The baryon number charge can only occur when we have a theory containing fundamental flavors. Introducing flavors into the gauge theory via a D7 brane leads to the appearance of a U(N f ) global flavor symmetry. The flavor group contains U(1) B , i.e., a baryon number symmetry, and a chemical potential µ B is added for this baryon number [41]. To calculate the free energy, we must calculate the Dirac-Born-Infeld action for a D7 brane. We note that there is a divergence in the formal definition, and we must hence go through the renormalization process (see, e.g., the lectures in [58] and also see [59]).
The R-charge chemical potential appears for SUSY models [47]. In the N =1 case, there is a U(1) R-symmetry group. In the extended SUSY case, for example, in the N =2 case, the quark mass term breaks the R-symmetry.
A typical phase diagram of chemical potential versus temperature is presented in figure 1 (the diagram is taken from [40]). In the phase diagram, µ q = µ B /N c , µ q is the quark chemical potential,M ∝ m q is a mass scale defined asM = 2M q / √ λ, and λ = g 2 Y M N c .

Shock waves in AdS 5
Shock waves propagating in the AdS space have the form where u and v are light-cone coordinates and x ⊥ is the coordinate transverse to the direction of motion of the shock wave and to the z direction. This metric is sourced by the stressenergy-momentum tensor T M N with only one nonzero component T SW uu , and the Einstein equation of motion reduces to and Different forms of the shock waves correspond to different forms of the source where q is the chordal distance In this case ρ, related to J uu as

23) and the Einstein equation of motion becomes
The point shock wave shape F p is given by the solution of (2.17) with and has the form It has the form The shape of the charged point shock wave is a sum of two components (see appendix A for the calculation details)

The GYP dual conjecture
Gubser, Yarom, and Pufu (GYP) proposed the following holographic picture for colliding nuclei dual to QCD [13]: • the bulk dual of the boundary nuclei is the shock waves of form (2.15) propagating in the AdS space; • the bulk dual of two colliding nuclei in the bulk is the line element for two identical shock waves propagating towards one another in the AdS space, • when the shock waves collide in the bulk, a BH should form, signifying the formation of a QGP.
The TS technic [66,67] is usually used to estimate the BH formation. 2 A TS is a surface whose null normals all propagate inward [69,70]. There is no rigorous proof that the TS formation in the asymptotically AdS space-time provides the BH formation, but there is a common belief that TSs must lie behind an event horizon and that a lower bound on the entropy S AdS of the BH is given by the TS area A trapped ,

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The relations between the bulk parameters G 5 , L, and E and the QGP phenomenological parameters must be fixed to make the proposed duality prescription more precise. According to [33], one of these relations is The arguments supporting (2.39) are as follows. Lattice calculations for the QGP in [71] showed that ET 4 is a slowly varying quantity and Just to match BH equation of state (2.1) to (2.40), GYP assumed (2.39) (see [13]). It is important that an identification of the total energy of each nucleus with the energy of the corresponding shock wave is assumed here. We can modify this identification and assume that only a part of the gravitational shock wave energy is related to the total energy of the nucleus. The AdS/CFT dual relation (2.4) between the expectation value of the gauge theory stress tensor and the AdS 5 metric deformation by the shock wave was used in [13] to fix the dimensionless parameter EL: For the point shock wave Φ p given by (2.27), we obtain the stress tensor in the boundary field theory (2.42) The right-hand side of (2.42) depends on the total energy E and L, and L has the meaning of the root-mean-square radius of the transverse energy distribution. It was assumed in [13] that L is equal to the root-mean-square transverse radius of the nucleons, which in accordance with a Woods-Saxon profile for the nuclear density [72,73] is of the order of a few fm. In particular, it is equal to L ≈ 4.3 fm for Au and L ≈ 4.4 fm for Pb. However in principle L here is an arbitrary parameter, that can be fixed to fit the experimental data. The RHIC collides Au nuclei (A=197) at √ s N N = 200 GeV. This means that each nucleus has the energy E = 100 GeV per nucleon, for a total of about E = E beam = 19.7 TeV for each nucleus. The LHC will collide Pb nuclei (A=208) at √ s N N = 5.5 TeV, which means E = E beam = 570 TeV. The estimates in [13] for the dimensionless values EL for Au-Au and Pb-Pb collisions are

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We note that tuning the scale L or z 0 of the bulk colliding object to the size of the nucleus or to the "saturation scale" Q s in the "color glass" models was proposed in [29].
The calculations in [13] show that in the limit of a very large collision energy E, the entropy increases as E 2/3 , Considering off-center collisions of gravitational shock waves in the AdS space do not change the scaling E 2/3 . But a critical impact parameter, beyond which the TS does not exist, was observed in [29] (cf. the result in [31]). Experimental indications for a similar critical impact parameter in real collisions had been noted [29]. The relation of the total multiplicity S QGP (given by experimental data) to the entropy S AdS produced in the gravitational wave collision in AdS 5 has some subtleties [33]. Phenomenological considerations [13,74,75] lead to estimating the total multiplicity S QGP by the number N ch of charged particles times a factor ∼ 7.5, (2.46) The TS analysis does not give the produced entropy, but it provides a lower bound, Taking into account that in the calculations in [13], the gravitational shock wave energy was identified with the energy of colliding ions and L was identified with the nucleus size, we can introduce proportionality constants between these quantities and obtain where all proportionality factors are included in the overall factor M. We can take M to fit the experimental data at some point. But the scaling S trapped ∝ s 1/3 N N implied by (2.45) differs from the observed scaling, which is closer to the dependence S ∝ s 1/4 N N , which is predicted by the Landau model [50] (see figure 2). Obviously, we can avoid a conflict between [13] and the experiment if E < E max , but if E can be arbitrary large, then the conflict arises.
In figure 2, we plot the dependence of entropy bound (2.45) on the energy together with the curve schematically representing the realistic curve that fits the experimental data [76]. It can be seen that by changing the coefficient M, we can avoid the conflict only for energy up to some E max . We chose the overall coefficient of the numerical plot to fit the RHIC data [76], which are indicated by dots in figure 2.
In the above estimate, the energy of each shock wave is identified with the energy of colliding beams. As was noted in [28], the fit to the data can be improved by identifying the energy of each shock wave with the fraction of the energy of the nucleus carried by a nucleus participating in the collision. This gives an extra parameter for fitting the data. But a conflict still arises at high energies. It was proposed in [28] to solve this problem by removing a UV part of the AdS bulk. Shock waves corresponding to the BH with a nonzero dilaton field [36,37] were considered in [38], where it was shown that the lower bound on N ch scales is closer to s

Remarks about the regularization of TS calculations in the case of wallon-wall collisions
A much simpler dual description of the colliding nuclei using a wall-on-wall collision in the bulk was proposed in [29]. The Einstein equation for the profile of the wall shock wave [29] has the form To find a TS that can be formed in the collision of two wall shock waves, we must find a solution of Einstein equation (2.49) satisfying two conditions. It is convenient to write these conditions in terms of the function ψ(z) related to φ by They have the forms where z a and z b are assumed to be the boundaries of the TS [29]. But as is seen below, strictly speaking, we cannot call the solution of (2.49) with boundary conditions (2.51) and (2.52) the TS, because this surface is assumed to be smooth and compact by definition, while the solution in [29] is nonsmooth and noncompact. We therefore call the solution found in [29] a quasi-TS. We recall the construction presented in [29], where the solution of Einstein equation (2.49) was written such that property (2.51) is automatically satisfied. This solution JHEP05(2012)117 .
We first note that solution (2.53) is nonsmooth and can be decomposed as The nonsmooth part of it is (2.58) To smooth the solution, we must therefore smooth the function Ξ. We can do this by regularizing the Heaviside step function and considering the regularized functionsΥ 1 andΥ 2 , For derivatives, we have We present the derivatives of the functions Υ 1 and Υ 2 and also of the smoothed functionsΥ 1 andΥ 2 in figure 3. For R = 10 4 (see below), the differences between the derivatives dΥ i /dz and their approximations given by (2.63) and (2.64), are of the order 10 −3 fm 3 only in the interval z ∈ [z ′ 0 , z ′′ 0 ], where z ′ 0 = 4.293 fm and z ′′ 0 = 4.307 fm. Indeed, in our consideration (spread case), the largest value of z a is 4.260706906 fm, and the smallest value of z b is 4.340400579 fm. At the points z ′ 0 = 4.260706906 fm and z ′′ 0 = 4.340400579 fm, the quantity ∆ 1 is less then 5 · 10 −6 fm 3 . At the points z ′ 0 = 0.6948439783 fm and z ′′ 0 = 1018.393720 fm, the quantity ∆ 1 is less then 2 · 10 −12 fm 3 .
We present a schematic picture of the root locations and the region where |∆ i (z)| 10 −3 in figure 4. It can be seen that the difference ∆ i is inessential in the root locations and that using approximations (2.63) and (2.64) is therefore acceptable. The regularized version of the function ψ is We must now impose conditions (2.52) on the regularized functions, and findz a andz a from these conditions. But these calculations are difficult. Instead of findingz a from condition (2.68), we propose using a regularization that does not change z a JHEP05 (2012)117   000  000  000  000  000 000  000  000  000  000 000  000  000  000  000 000  000  000  000  000 000  000  000  000  000 000   111  111  111  111  111 111  111  111  111  111 111  111  111  111  111 111  111  111  111  111 111  111  111  111  111   found from formal conditions (2.52). We can verify that the formal z a in fact also satisfies the regularized condition if the regularization is sufficiently smooth. We therefore take z a and substitute it in the left-hand side of regularized condition (2.68). We define We can calculate F a,reg . The deviation of F a,reg from unity shows how the regularization changes conditions (2.52). In the following table, we present the results of calculating F a,reg for a wide range of the theory parameter. We choose the parameter R as minimally needed to make δ 1 and δ 2 negligible at energies in the range 10 −4 < E < 10 2 TeV. Using direct numerical calculations, we choose R = 10 4 . We perform numerical calculations at R = 10 4 and obtain the following It follows from the table that obviously F a ≈ 1 and F b ≈ −1.
As above, strictly speaking, we cannot regard an infinite surface as a TS of any kind. Nevertheless, we can assume that the transverse size of colliding objects is finite but very large, and boundary conditions therefore do not affect the gravitational interaction processes of the inner parts of sources. If we are interested only in the specific area of the

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formed TS with respect to the unit of shock wave area, then we can define it as , (2.70) and the approximate equality holds because the boundary effects are negligible. As often happens, we can obtain answers for finite physical systems by calculating for infinite nonphysical objects.

Holographic QGP phase diagram for the central heavy-ion collisions
In this section, we construct the phase diagram for a TS formed in the central collision of two identical pointlike charged shock waves [34]. The profile of pointlike charged shock waves in the AdS space is given by (2.30) with (2.29) and (2.32). The existence of the TS in the central collision of two pointlike charged shock waves means the existence of a real solution q 0 of the equation (see [34] for the details) The left-hand side of (3.1) can be written as The numerator N (L, E,Q 2 , q) contains just one term that depends onQ 2 . This dependence is linear with a positive coefficient: The denominator in (3.2) does not take infinite values. To find solutions of (3.1) for the shape function given by (2.30), we can graph the function −N (a,M , q) ≡ −(512a 3 q 5 + 1280a 3 q 4 − 96M πaq 2 + 1024a 3 q 3 − 96M πaq + 256a 3 q 2 ) and see where this function can be equal to a given value 15Q π /a. To find the maximum allowedQ 2 at which solution of (3.1) still exists, we find the maximum of the function N for a fixed energy, dN (a,M , q) dq q=qmax = 0, (3.4) and the value a 15π N (a,M , q) q=qmax definesQ 2 max . We recall that we work in physical units and use notation (2.34) and (2.33):M = 4G 5 E/3π andQ 2 = 4G 5 Q 2 n /3π. We present the calculation results in figure 5.

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E, T ev Q n , f m 1/2 Figure 5. The allowed zone for the TS formation is under the line in the diagram. The plot is constructed using formulas in [34].
To estimate corrections to the GYP multiplicity due to a nonzero chemical potential, we use formula (3.17) in [34]. In notation (2.29) and (2.32) used here, the formula has the form (3.5) In figure 6, we show the entropy A AdS 5 for Q n = 0 and Q n = 0. The blue line represents Q n = 0. The red line represents Q n = 2 · 10 6 . We see that the deviation from the GYP multiplicity is essential for small energies and is almost negligible for large energies.

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4 Holographic QGP phase diagram in the wall-on-wall dual model of heavy-ion collisions

Charged wall as a dual model for a heavy ion with a nonzero chemical potential
We note that the form of the J WP uu in (2.49) can be obtained by spreading out the energymomentum tensor of an ultrarelativistic point, i.e, J uu in form (2.22) with ρ(q) given by (2.28), over the transverse surface. The Einstein equation for the charged wall (membrane) has the form where J WP uu is given by (2.49) and we assume that J WQ uu (Q, z) can be obtained by similarly spreading the energy-momentum tensor of the ultrarelativistic charged point T pQ uu over the transverse surface. In the preceding calculations, where the superscript pQ means the electromagnetic part of the energy-momentum tensor of the charged point particle and Dx ⊥ means that we integrate over the induced metrics on the orthogonal surface M. For this, we take where ρ pQ is given by (2.31) and we integrate over all transverse coordinates according to our prescription (4.2), The result is We see a divergence at z = z 0 , as should be for the energy-momentum tensor of a charged plane. We introduce regularization by adding the ǫ factor in the denominator.

Charged wall-on-wall collision as a dual model for heavy-ion collisions with a nonzero chemical potential
To find the TS formation condition in the wall-wall collision, we must solve the Einstein equation , (4.10) with the boundary conditions where z a and z b are the boundaries of the TS and ψ is related to φ(z) as We seek a solution of the Einstein equation with a charged source in the form of the sum of the "neutral" solution and a correction proportional to Q 2 : where φ n denotes the solution in the neutral case. As in the neutral case, it is convenient to consider the domains z < z 0 and z > z 0 separately, and we have Solutions of (4.16) and (4.17) can be represented as , z 0 > z, (4.18) , z > z 0 , (4.19)

(4.23)
We note that the constructed solution automatically satisfies the condition ψ(z a ) = ψ(z b ) = 0. Condition (4.12) gives (4.25) These equations do not have analytic solutions, and we treat them numerically. The roots of system (4.24), (4.25) cannot be found analytically, because these equations are equivalent to polynomial equations in z a and z b of a high degree (> 4). We therefore take z 0 = L and numerically analyze the system To show the movement of the roots of (4.26) and (4.27), we assume that z b for a given Q is already known and represent the function F a (z a , z b ) as a function of z a in figure 7. Similarly, assuming that z a is already known, we represent the function F b (z a , z b ) as a function of z b in figure 8.
In figure 9, we show the charge flows of the roots. Different lines correspond to different energies. We see that the flows go to z 0 and reach the line z = z 0 for Q = Q cr . In figure 10, we draw the corresponding flow for physical parameters.

Comparison of results
It is interesting to compare the phase diagrams of the energy (temperature) E versus the charge (chemical potential) Q corresponding to the pointlike charge and the spread charge. The results of these calculations are collected in the table below and presented in figure 11. It can be seen that these two phase diagrams are almost the same. It is evident from figure 11 that the red and blue lines intersect. We present the intersection in the natural and logarithmic scales in figure 12.

The square TS calculation
Following [29], we calculate the lower bound of the entropy as the "TS area" per unit square of the wall 3 using the formula In the absence of transverse dependence, we ignore x 2 ⊥ in (4.28). Equation (4.29) measures the entropy per transverse area.
The TS decreases as the charge increases. The corresponding graphical representations are in figure 13.
In figure 14, we show the entropy per volume given by (4.29) as a function of energy for different Q. This plot is similar to the plot presented in figure 6. It can be seen that the influence of the chemical potential on the multiplicity is essential for small energies and is almost negligible for large energies.
In the work [28] it has been suggested to place UV and IR cut-offs in order to reduce the multiplicities. Later, in the work [38] it was realized that most of the entropy comes from the UV part of the trapped surface (small z) and hence the IR cut-off is not very important. In the case of the wall shock wave collisions the main contribution comes from the region located near the wall (z = z 0 ). Performing the IR cut-off by removing z larger than z b (E) we do not change the multiplicity. This is evident from the formula (4.29). But the UV cut-off can affect the multiplicity. If we remove z smaller then z a (E) we do change the multiplicity significantly, see figure 9.
As to the point source shock waves the explicit formulae are more complicated but one can expect the similar results.

Remarks about the regularization
The regularized version of the the function ψ is where ψ a (z) and ψ b (z) define the function ψ without regularization: Figure 14. Entropy per volume as a function of energy for different Q: red line, Q n = 0 fm 1/2 ; blue line, Q n = 2000 √ π fm 1/2 ; black line, Q n = 5000 √ π fm 1/2 . and We must now impose conditions (2.52) on the regularized functions But it is difficult to findz a from condition (4.34). Instead of findingz a from condition (4.34), we propose to use a regularization that does not change z a found from formal conditions (2.52). We can verify that the formal z a in fact also satisfies the regularized condition if the regularization is sufficiently smooth. We therefore take z a and substitute it in the left-hand side of regularized condition (4.34). We define We can calculate F a,reg and F b,reg . In the following

Summary
We have used a holographic dual model for a heavy-ion collision to construct the phase diagram of the QGP formed at a very early stage just after the collision. In this dual model, colliding ions are described by charged gravitational shock waves. Points on the phase diagram correspond to the QGP or hadronic matter with given temperatures and chemical potentials. The QGP phase in dual terms is related to the case where the collision of shock waves leads to the formation of a TS. Hadronic matter and other confined states correspond to the absence of a TS after the collision. The multiplicity of the ion collision process was estimated in the dual language as the TS area. We showed that a nonzero chemical potential reduces the multiplicity. To plot the phase diagram, we used two different dual models of colliding ions. The first model uses the point shock waves, and the second uses the wall shock waves. We found that the results agree qualitatively.
We paid special attention to the regularization procedure for the calculations for wall shock waves. On one hand, these calculations are essentially simpler technically. On the other hand, this approach, strictly speaking, is incorrect and requires a regularization. We showed that a natural regularization does exist. Moreover, the proposed regularization does not make the calculations more complicated compared with the naive (direct) calculations. This opens new possibility for simple calculations for wall shock waves carrying nontrivial matter charges.

Further directions
Head-on collisions of charged point shock waves have only two parameters. In the dual language, they correspond to the energy and the chemical potential per nucleus. Off-center collisions are also specified by the impact parameter, and the change of this parameter can be associated with a dual change from "nonthermal" peripheral to "thermal" central collisions [29]. But this is still an oversimplification of the problem. The physics of heavyion collisions in the RHIC is richer, and as indicated in [29,77], the rapid equilibration and hydrodynamic behavior experimentally observed at the RHIC for collisions of two heavy ions such as Au-Au does not occur for deuteron-Au collisions at the same rapidity. It is perhaps too naive to believe that the simplest shock wave related by a boost to the Schwarzschild BH in the AdS space can mimic the nuclear matter in the colliders. But this JHEP05(2012)117 simple shock wave in fact reproduces the interaction of a relativistic quark with gravity and can therefore be regarded as a simplest candidate for mimicking nuclear matter in the holographic conjecture. We can try to associate different nuclei with different forms of shock waves. In this context, we recall that the form of the shock wave follows from the eikonal approximation of the gravity-quark interaction in five dimensions [39,78,79]. The presence of an electromagnetic field or other fields and also any improvements of the eikonal approximation certainly changes the form of the shock waves, and it would be interesting to see the holographic consequences of this consideration.
The lower bound obtained for N ch scales as s 1/3 N N , which is a faster energy dependence than the s 1/4 N N scaling predicted by the Landau model [50] and largely satisfied by the data. If we have an a priori restriction on the allowed energy, then we can fit constants to ensure that the experimental data are above the AdS bound. We note that taking the chemical potential into account allows increasing the permitted energy. But we cannot expect too much from the chemical potential corrections. The relevant chemical potential for the baryon number is not expected to be large, i.e., µ B ∼ 30M eV or µ B /T ∼ 0.15 for recent experiments at the RHIC [80], and any effects will therefore be limited. Nevertheless, as mentioned in the text, the relation between the chemical potential value and the fivedimensional charge value is at our disposal, and we can assume that they have a huge ratio.
It would also be interesting to try to use plane gravitational waves in AdS 5 to describe nonperturbative stages in the gauge theories and to use collisions of these waves to describe the QGP formed in heavy-ion collisions. In the planar case, the Chandrasekhar-Ferrari-Xanthopoulos duality between colliding plane gravitational waves and the Kerr BH solution was used as a model of BH formation [81]. It would be interesting to generalize this duality to the AdS case. This may yield a new insight into the possible dependence of multiplicities on the rapidity. In the plane coordinates, metric (A.1) is where ds 2 0 is the AdS metric and the perturbation ds 2 p has the form g DD = −a 2 Z 2 0 + Z 4 0 + Z 2 D Z 2 0 + a 2 Z 2 D , (A.12) (A.14) Performing a boost in the Z 1 direction, we write the first-order deformation of the metric in the form (A.20) To obtain the limit as γ → ∞ in the AdS case as in the absence of a charge, we can apply the lemma on distributions in [60,85]: (A.21) For the shape function F D,AdS (M ,Q 2 , Z), we obtain the formula Here, a 2 = 3/Λ. We note that there are more subtleties in the case of the dS space [34,86].
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