Holography for Heavy Ions Collisions at LHC and NICA

This is a contribution for the Proceedings of 5th International Conference on New Frontiers in Physics (ICNFP 2016), held at Crete, 6-14 July 2016. Our goal is to obtain phenomenologically reliable insights for the physics of the quark-gluon plasma (QGP) from the holography. I briefly review how in the holographic setup one can describe the QGP formation in heavy ion collisions and how to get quantitatively the main characteristics of the QGP formation -- the total multiplicity and the thermalization time. To fit the experimental form of dependence of total multiplicity on energy, obtained at LHC, we have to deal with a special anisotropic holographic model, related with the Lifshitz-type background. Our conjecture is that this Lifshitz-type background with non-zero chemical potential can be used to describe future data expected from NICA. In particular, we present the results of calculations of the holographic confinement/deconfinement phase transition in the $(\mu,T)$ (chemical potential, temperature) plane in this anisotropic background and show the dependence of the transition line on the orientation of the quark pair. This dependence leads to a non-sharp character of physical confinement/deconfinement phase in the $(\mu,T)$-plane. We use the bottom-up soft wall approach incorporating quark confinement deforming factor and vector field providing the non-zero chemical potential. In this model we also estimate the holographic photon production.


Introduction
Quark Gluon Plasma (QGP) produced in heavy ion collisions (HIC) at RHIC and LHC is a new form of matter formed from quarks and gluons at high temperature [1,2]. It is a strong coupling fluid [3], or an "operator boiling fluid" [4]. This makes perturbative methods inapplicable to study properties of QGP and especially it formation. The lattice QCD, the powerful tool of non-perturbative study, is formulated in the Euclidean spacetime and cannot be applied to QGP formation, since this is time depending phenomena (cf. [5]). This gives a strong motivation to study the process of QGP formation in HIC through the gauge/string duality. The gauge/string duality (or AdS/CFT correspondence) is directly applicable only for a very special 4-dimensional theory, namely, N =4 SUSY Yang Mills theory [6]. This theory is conformally invariant on the quantum level and the conformal invariance is the main request for the AdS/CFT application. Of course, the real QCD is not conformally invariant, however, as we know from lattice calculations, the high-temperature QCD is. More precise, the deviation from conformality decreases as the temperature increases, see Fig.1.A. The agreement of the shear viscosity to entropy density ratio of the QGP formed at RHIC and LHC with its holographic calculation [7] strongly supports the idea to use holography to study physics of QGP formed in HIC [].
In Fig.1.B the QCD phase diagram is presented. The phase diagram of QCD is not well known either experimentally or theoretically. A commonly conjectured form of the phase diagram, temperature T vs quark chemical potential µ, is shown in Fig.1.B. The chemical potential µ is a measure of the imbalance between quarks and antiquarks in the system. The phase transition is not sharp and it is supposed to be the 1-st order.
Ordinary nuclear matter in this diagram is at µ = 310 MeV and T close to zero. If we increase the quark density, i.e. increase µ, keeping the temperature low, we go into a phase of more and more compressed nuclear such as matter neutron stars. Above the (blue on the on-line version of the paper) smeared line there is a transition to the quark-gluon plasma. At ultra-high densities one expects to find the phase of color-superconducting quark matter. In ultra-relativistic heavy ion collisions one studies this matter in the regime of extreme energy density. In Fig.1.B. the typical values of µ and T in heavy-ion collisions (RHIC and LHC) are shown by a filled (cyan) region near the T-axis. The regions expected to be available at NICA (Nuclotronbased Ion Collider fAcility) [9] and FAIR (Facility for Antiproton and Ion Research) are indicated by arrows.
The exact dual description of the real QCD is unknown, but holographic QCD models that fit perturbative (two loops β-function) and lattice QCD results (in particular, the quark confinement potential) have been proposed [11,12]. Using these models several static properties of QGP have been reproduced [8].
The description of the QGP formation in HIC is a difficult subject, since it -2 - supposes to study a complicated real time phenomena -thermalization. We also do not know much from experiments about the details of the QGP formation in HIC, one can just estimates the time of QGP formation as well as the total multiplicity (there are arguments that the main part of particles is produced during the QGP formation) [13,14]. The QGP formation has been the subject of the active studies within holographic approach in last years (see [15][16][17] and refs therein). Initially this problem was considered in AdS background [18][19][20][21][22][23][24] and the total multiplicity within this approach was estimated as For the improved holographic background the estimation was [25] M IHQCD ∼ s 0. 22 (1 + log corrections), (1. 2) The experimental multiplicity dependence on energy [13,14] is and it has been shown in [26] that the dependence (1.3) requires an unstable background.
In [27] it has been shown that the model that reproduces the Cornell potential also gives a correct energy dependence of multiplicities if we assume that the multiplicity is related with the dual entropy produced during a limited time period. However in this consideration there is a limitation on the possible energy of colliding shock walls [27]. Since in this consideration we have used a more or less general isotropic background reproducing AdS at UV and confinement at IR, we can think that the assumption about the isotropic background prevents to reproduce (1.3) at high energy.
-3 -  [4][5][6][7] and Au-Au [8][9][10][11][12] collisions (see text) as a function of p sNN. Measurements for inelastic pp collisions and pp collisions as a function of p s are also shown [26][27][28] along with those from non-single diffractive p-A and d-A collisions [29,30]. The s-dependence, proportional to s 0.155 NN for AA collisions is indicated by a solid line: similarly a dashed line shows an s 0.103 NN dependence in pp collisions. The shaded bands show the uncertainties on the extracted power-law dependencies. The central Pb-Pb measurements from CMS and ATLAS at 2.76 TeV have been shifted horizontally for clarity. b = 0.155 ± 0.004. It is a much stronger s-dependence than for proton-proton collisions, where a value of b = 0.103 ± 0.002 is obtained from a fit to the same function [28]. The fit results are plotted with their uncertainties shown as shaded bands. The result at p sNN = 5.02 TeV confirms the trend established by lower energy data since b is not significantly different when the new point is excluded from the fit. It can also be seen in the figure that the values of 2 hNparti hdNch/dhi measured by ALICE for p-Pb [25] and PHOBOS for d-Au [11] collisions fall on the curve for proton-proton collisions, indicating that the strong rise in AA is not solely related to the multiple collisions undergone by the participants since the proton in p-A collisions also encounters multiple nucleons.
The centrality dependence of 2 hNparti hdNch/dhi is shown in Figure 2. The point-to-point centralitydependent uncertaintes are indicated by error bars whereas the shaded bands show the correlated contributions. The statistical uncertainties are negligible. The data are plotted as a function of hNparti and a strong dependence is observed, with 2 hNparti hdNch/dhi decreasing by a factor 1.8 from the most central collisions, large hNparti, to the most peripheral, small hNparti. There appears to be a smooth trend towards the value measured in minimum bias p-Pb collisions [25]. The data measured at p sNN = 2.76 TeV [4,26] are also shown, scaled by a factor 1.2, which is calculated from the observed s 0.155 dependence of the results in the most central collisions, and which describes well the increase for all centralities. Given In [31] we have considered a special anisotropic backgrounds, parametrized by a paranmeter ν and have got and therefore to get an estimation (1.3) we take ν = 4.45. Note, that ν = 4 gives M ν=4 ∼ s 0.167 , that is rather closed to (1.3). ν = 1 corresponds to the AdS case. Let us note, that the idea to use anisotropic background in the holographic approach to QGP is not new. There were several proposals to deal with anisotropic backgrounds, see for example [38] and refs therein. Up to couple years ego, it has been believed that the matter was in the pre-equilibrium period up to 1 fm/c and then the QGP appears and it is isotropic. However now it is believed that the QGP is created after a very short time after collision, τ therm ∼ 0.1f m/c, and it is anisotropic ("anisotropic" means a spatial anisotropy) for a short time τ after collision, 0 < τ therm < τ < τ iso , and the time of local isotropization is about τ iso ∼ 2f m/c [39].

Thermalization
Suppose we deal with a correlator of a time depending quantity O(t), < O(t) >. If at large t the system exhibits the behaviour one says that the system goes to the stationary state and this state is a thermal state, or in other words the system thermalizes. The simples thermalization process can be described by interaction with thermostat. For one oscillator interacting with thermostat one can show the thermalization explicitly.

Thermalalization as a black hole creation in AdS
According to the holographic scenario the thermalization process in d-dimensional space-time can be understood as horizon formation in a d+1 dimensional gravitational theory. To initiate the process of BH formation one has to perturb the initial AdS metric. According to AdS/CFT correspondence this deformation should be related with the deformation of the energy-momentum tensor of the boundary theory. So the idea is to make some perturbation of AdS metric that near the boundary mimics the heavy ions collisions and see what happens, see Fig.2.B. One can consider several deformations of AdS metric: • a deformation by adding colliding gravitational shock waves.
• we can drop of a shell of matter with vanishing rest mass ("null dust"), • we can study the toy 3-dim model with colliding ultra relativistic particles It seems natural to think about an ultrarelativistic nucleus as a shock wave in 4-dimensional space-time. One can also assume that this shock has a profile function. According to AdS/CFT correspondence, this energy-momentum tensor corresponds to a shock wave metric in AdS 5 of the following form is the profile function, L is the scale parameter. For φ(x ⊥ , z, L) = 0 the metric (2.3) is nothing but the Poincare metric for AdS 5 . For two colliding nuclei corresponds the following metric in AdS 5 It is interesting to note that gaussian regularization of the point shock wave profile function φ(x ⊥ , 0) exactly corresponds to the Woods-Saxon profile for the nuclear density. This correspondence gives us L equal to 4.3 fermi for gold and L equal to 4.4 fermi for lead. It is natural to ask the question: can we guarantee the black hole formation under collisions of these shock waves. If the answer is "yes", then we take the entropy of this formed black hole as a multiplicity of particles production. This idea has been explored in several papers [18][19][20][21][22][23][24][25]. The idea to relate multiplicities with the thermodynamic characteristic of the media produced as a result of a collision of particles comes back to Landau and Fermi [28][29][30].

O. Andreev and V. Zakharov hep-ph/0604204
S~tV ove to consider thermalization of rectangular Wilson loops in the Lifshitz-Vaidya nd (2.12)-(2.14), which describes collapsing geometry in the Lifshitz-like spaceproceed in a similar manner as in the static case studying three possible configof spatial Wilson loops. atial Wilson loops on the xy 1 plane ectangular strip infinite along the y 1 -direction c. 3 we start from the spatial rectangular Wilson loop on the xy 1 plane with the on that one side is infinite along the y 1 -direction and has finite size along the x-(see 3.1). Here we suppose the dependence v = v(x), z = z(x). The Nambu-Goto kes the form similar to (3.6) incide with the equations for the AdS case for ⌫ = 1.
s the length of the Wilson loop along the x-direction.

Wilson loop in a time-dependent background
Now we move to consider thermalization of rectangular Wilson loops in the Lifshitz-Vaidya background (2.12)-(2.14), which describes collapsing geometry in the Lifshitz-like spacetime. We proceed in a similar manner as in the static case studying three possible configurations of spatial Wilson loops.

Rectangular strip infinite along the y 1 -direction
As in Sec. 3 we start from the spatial rectangular Wilson loop on the xy 1 plane with the assumption that one side is infinite along the y 1 -direction and has finite size along the xdirection (see 3.1). Here we suppose the dependence v = v(x), z = z(x). The Nambu-Goto action takes the form similar to (3.6) which coincide with the equations for the AdS case for ⌫ = 1.
Eqs. (4.2)-(4.3) obey the following boundary conditions where`is the length of the Wilson loop along the x-direction.
ceed in a similar manner as in the static case studying three possible configatial Wilson loops.
l Wilson loops on the xy 1 plane angular strip infinite along the y 1 -direction e start from the spatial rectangular Wilson loop on the xy 1 plane with the hat one side is infinite along the y 1 -direction and has finite size along the x-3.1). Here we suppose the dependence v = v(x), z = z(x). The Nambu-Goto he form similar to (3.6) e with the equations for the AdS case for ⌫ = 1. )-(4.3) obey the following boundary conditions  We note that the dynamical system governed by (4.1) has the following integral of motion where we denote Taking into account (4.7)-(4.8) one can represent (4.1) in the following form

Thermalization time
Thermalization time depends on the physical quantities that are expected to thermalized. This issue has been discuss in details in our talk [40] and is not discussed here, see also [41,42] for estimations of thermalization time of different variables in the Lifshitz-type background.
3 Physics in the Lifshitz-type background

Lifshitz-type background
In [31] we have studied collisions of shock waves in the anisotropic background where ν is the critical exponent. After collision the black hole is produced and the stationary black hole metric in this background is [32] where the blackening function is Motivated by [33][34][35] we deform this background as 1 where the blackening function takes into account a non-zero chemical potential and b(z) is a factor that provides the quark confinement in the isotropic case [36] b(z) = exp( cz 2 2 ) (3.6) and the zero-component of the vector field is given by According the AdS/CFT dictionary, the boundary value A 0 (0) is related to the quark chemical potential: The vanishing of A 0 (z) at the horizon, A 0 (z h ) = 0, sets a relation between the chemical potential and the charge of the black hole: The temperature T is defined by the relation . We can also present the dependence of T on the chemical see Fig.4. F means the fundamental representation. Following the holographic approach [43][44][45] the expectation value of the Wilson loop in the fundamental representation calculated on the gravity side reads as:

Temporal
where C = T × X in a contour on the boundary, S xt is the minimal Nambu-Goto action of the string hanging from the contour C in the bulk where h αβ is the induced metric of the world-sheet h αβ = g M N ∂ α X M ∂ β X N , α, β = 0, 1, g M N is the background metric, M, N = 0, . . . , 4, X M = X M (σ 0 , σ 1 ) specify the string worldsheet with the worldsheet coordinates σ 1 , σ 2 . We parametrize the worldsheet as X 0 ≡ t = σ 0 and X 1 ≡ x = σ 1 and consider the static configuration X 4 ≡ z = z(x). The action S (xt) is: where means d dx . If we take z = 0 in (3.15) we get the "potential" Note that the form of the action (3.15) is the same as for the isotropic case, and information about anisotropy is stored only in the form of blackening function f (3.5). We consider the symmetrical parameterization z(± ) = 0 and z(0) = z * , z (0) = 0. The distance between the two endpoints of the string can be represented as We consider the case when z * < z h i 0 , where z h i 0 is the smallest of the horizons, i.e. if f (z h i ) = 0, then z h i 0 < z h i . Therefore f (z) > 0 for 0 < z < z * .
To get the energy of the string we subtract the mass of the two free quark [44,45] πα .  • There is no extremal point in the interval 0 < z < z h i 0 . This case corresponds to the deconfinement phase and dependence of on z * has the form as presented in Fig.7.a). We see that there are two branches indicated by blue and red color, the same can be obtained by two different z * , except z * = z * 0 where reaches its maximal value 0 , i.e. = (z * ) monotonically increases from (0) = 0 to (z * 0 ) = 0 (the first branch shown by the blue color) and = (z * ) monotonically decreases from (z * 0 ) = 0 to (z h 0 ) = 0 (the second branch). In Fig.7.a) z h 0 = z h . The plot in Fig.7.b) shows the values of the energy between quarks located along x-direction as function of z * for two branches. The corresponding values of the energy as functions are presented in Fig.7.c). As shown in this plot, the values of energy corresponding to the second branch are larger than those for the first one.
• There are two extremal points in the interval 0 < z < z h i 0 , z min = z 0 and z max = z 0 and the potential is a decreasing function only on the intervals 0 < z < z min and z max < z < z h i 0 , (3.19) so we can guarantee that V (z)/V (z * ) < 1 only in the region (3.19).
This case corresponds to the confinement phase and the dependence of L on z * is presented in Fig.8.a). Suppose that V (z) has a local minimum at z = z 0 . In -10 -  In Fig.9 we present the same quantities as in Fig.7 and Fig.8 at the same parameters except that in Fig.7 and Fig.8 ν = 1 (isotropic case) and in Fig.9 ν = 4 (anisotropic case, longitudinal orientation of the Wilson loop). The confinement solution is realized on the first branch discussed above. Let us estimate the string tension for this configuration. Suppose that (3.20) We can also calculate the energy near z * = z 0 using (3.18). We have Therefore we get

Energy between quarks located along y-direction in the quark confinement background (3.4)
The Nambu-Goto action S yt of the string hanging from the contour C, being the rectangular with sites along the time direction and one of transversal directions in the bulk, is this formula has an explicit dependence on the anisotropy and is in agreement with the general formula obtained in [38]. The corresponding "potential" is, see Fig.10, and for L y we have  Figure 10. The potential V y (z) = V (z, c, q, ν, z h ) given by (3.26) with b(z) given by (3.6) with c = 2 and f (z) = f (z, q, ν, z h ) given by (3.5) with ν = 4 and z h = 2. The red line shows V y (z) for q = 0. We see that the corresponding potential has one minimum and one maximum. The blue lines show V y (z) for q = 0.15, 0.2 and q = 0.23. The corresponding potentials have minimum and maximum. The green line shows V y (z) for q = 0.25 and second zeros of the corresponding function f (z) is at z h 2 = 1.89 < z h . The magenta line shows V (z) for q = 0.3. The corresponding potential has no minimum and maximum and the second zero of the corresponding function f (z) = f (z, q, 1, z h ) is at z h 2 = 1.61. The brown line shows V y (z) for q = 0.35 and the corresponding potential has no minimum and maximum on 0 < z < z h 2 , where z h 2 = 1.40 is the second zero of the corresponding functions f (z). The critical charge q cr | c=2,z h =2 = 0.3 To get for the energy of the string between two quarks stretched in the y-direction we perform the subtraction of the two quark mass that corresponds the subtraction of the action on the sheet starting at the boundary z = 0 and ending at the horizon -13 - Note that the expression for E y , (3.28), in term of the potential V is the same as E x , (3.28), the only difference is in the form of the potentials. Supposing that V y (z) has a minimum at z = z 0 , we can estimate the contributions near this minimum, we have and we get the answer similar to (3.24),  Figure 11. The same quantities at the same parameters as in Fig.9 except that now we deal with the transversal orientation of the Wilson loop, ν = 4. We see that both c = 1.2 and c = 2 correspond to the confinement phase: a) and d) L = L(z s ), b) and e) W = W (z s ) (note that in this case we make a different subtraction), c) and f) W = W (L) .
In Fig.11 we present the same quantities as in Fig.7 and Fig.8 at the same parameters except that in Fig.7 and Fig.8 ν = 1 (isotropic case) and in Fig.11 ν = 4 (anisotropic case, transversal orientation of the Wilson loop). We see that for the transversal orientation at the same parameters as for the isotropic case we can get different solutions: for the isotropic case, as well as for the anisotropic longitudinal case, c = 1.2 corresponds to the deconfined phase, meanwhile this value of c in the -14 - anisotropic transversal case corresponds to the confined phase. This observation indicates that the phase diagram should be essential depends on the orientation in the case of the anisotropic background.

Holographic anisotropic QCD phase diagrams
In Fig.12 we present the phase diagram for isotropic and anisotropic cases for longitudinal and transversal orientations of the quark pair. Holographic isotropic QCD phase diagram has been studied previously in bottom-up approaches [35,48,49]. We see that for zero chemical potential the deconfinement occurs for the low temperature in the anisotropic case, and for near zero temperature the deconfinement occurs for the larger chemical potential in the anisotropic casel. Fig.12 shows the dependence of the transition line on the orientation. Since quarks can be arbitraly oriented in respect to the collision line, this dependence on the orientation leads to a broadening of the line separating confinement and deconfinement phases in the (µ, T )-plane. It would be interesting to study this diagram for the anisotropic lattice. Decreasing of anisotropy decreases the broadening of the phase transition boundary.
Let us remind that in isotropic case the confinement/deconfinment diagram has been studied in lattice QCD. Experimentally the phase boundary between hadronic matter and the quark -gluon plasma in relativistic heavy ion collisions is probing using the HRM [52,53] -15 -

Direct photons and electric conductivity
The thermal-photon production from the QGP plays an essential role, since photons after they are produced in HIC almost do not interact with the QGP and, therefore, they give us the local information in heavy ion collisions. The photon-emission rate is related to the retarded correlator of currents in momentum space [54] so that in the thermal equilibrium it is given by the light-like retarded correlator as where Γ denotes the number of photons emitted per unit time per unit volume and n b (|k|) denotes the thermal distribution function for bosons. The imaginary part of the retarded correlator is also related with the spectral function and due to the Kubo formula with the conductivity tensor The spectral function can be evaluated by holography using the flow method related with the membrane paradigm [55] . Below we shortly remind this prescription, see also [56,57]. One starts from the Maxwell action in the given background where it is assumed also the presence of the dilaton field φ, and denote by N the normalization constant including the 5-dimensional gravitational constant G 5 etc. The gauge field A µ (t, x, r) it is taken to be A µ (t, x, z) satisfies the Maxwell equation, and A µ (r, ω, k) satisfies the boundary conditions at the boundary lim One can consider for simplicity the case k = k x , k y 1 = k y 2 = 0. Let us parametrize the metric as In our model p(z) = z 2−2/ν . The Maxwell action (with the dilaton) has the form and Assume that the fluctuating vector field depends only on x 0 , x 3 , r-coordinates, one can write the E.O.M. in term of E L and E ⊥ here E L = kA 0 + wA 3 and E ⊥,i = wA ⊥,i , i = 1, 2.
The boundary terms come from that terms the total action (4.10) which have the z-derivative and we get S boundary = S boundary,1 + S boundary,2 (4.15) where S boundary,1 and S boundary,2 are In the analogy with the isotropic case we introduce -17 -which satisfy the following equations From these eqs follows that if ζ ⊥ = ∞ and ζ L = ∞ then on the horizon should be Since for w = 0 the derivative ζ ⊥ = 0 and we have 1/z h and it dependence on the temperature is read from (3.10) and approximately In what follows we use this approximation. Therefore, the dependence of the electric conductivity on the temperature and the chemical potential is given by Form this formula we see, that increasing the anisotropy we increase the electric conductivity at hight temperatures, see Fig.13. We also see that there is a critical temperature T 0 = T 0 (ν, q) such that for T < T 0 the conductivity decreases as we increase ν.
Here we have to note that equation (4.26) is valued only for T > T app , see Fig.14. In Fig.14 and Fig.15 we plot the electric conductivity as function of T for ν = 1 and ν = 4 for different values of q and the constant c specifying the factor b. We see that the confining factor b(z) does not change the qualitative picture much.
Note, that in the end of this section few comments are in order. Almost all theoretical predictions underestimate the direct-photon spectra. Several attempts [59,60], including the effects from strong magnetic fields [61], have been undertaken to fit theoretical predictions to LHC experimental data. As it is stressed in [57] direct photons calculations from holography [54,57,62,63] have to be complemented by -18 - the medium evolution and the photons production from other phases. This also concerns to our consideration. Here we have just presented a preliminary estimation of the role of the chemical potential, the Lifshitz type anisotropy and the confining factor on the photon emission rate. To give predictions for direct photons that can be observed at NICA one has to perform the study similar to [57]. We have seen that at hight temperature the anisotropy of the Lifshitz type increases the photon production and the chemical potential also increases it, meanwhile for temperature less than a critical one, they act in the opposite direction.

Conclusion
As a conclusion let me summarize. Motivated by the fact that to fit the experimental form of dependence of total multiplicity on energy we have to deal with a special anisotropic holographic model, related with the Lifshitz-like background, we have estimated the holographic confinement/deconfinement phase transition in the (µ, T ) -20 -plane in this anizotropic background. We have found the dependence of the transition line on the orientation of the quark pair. This dependence leads to a non-sharp character of physical confinement/deconfinement phase in the (µ, T )-plane. This calculation seems relevant in the context of the future project NICA and FAIR.
We are also going to estimate non-zero magnetic effects in the the Lifshitzlike background, in particular, by analogy with [64] one can estimate the crossover temperature in a magnetic field. As in isotropic case [65] it is interesting to find Debye screening mass near deconfinement in the anisotropic background.