HOLOGRAPHIC THERMALIZATION IN A QUARK CONFINING BACKGROUND
D. S. Ageev* I. Ya. Aref'eva**
Steklov Mathematical Institute, Russian Academy of Sciences 119991, Moscow, Russia
Received October 1, 2014
We study holographic thermalization of a strongly coupled theory inspired by two colliding shock waves in a vacuum confining background. Holographic thermalization means a black hole formation, in fact, a trapped surface formation. As the vacuum confining background, we considered the well-know bottom-up AdS/QCD model that provides the Cornell potential and reproduces the QCD ¿¡-function. We perturb the vacuum background by colliding domain shock waves that are assumed to be holographically dual to heavy ions collisions. Our main physical assumption is that we can make a restriction on the time of trapped surface formation, which results in a natural limitation on the size of the domain where the trapped surface is produced. This limits the intermediate domain where the main part of the entropy is produced. In this domain, we can use an intermediate vacuum background as an approximation to the full confining background. We find that the dependence of the multiplicity on energy for the intermediate background has an asymptotic expansion whose first term depends on energy as which is very similar to the experimental dependence of particle multiplicities on the colliding ion energy obtained from the RHIC and LHC. However, this first term, at the energies where the approximation of the confining metric by the intermediate background works, does not saturate the exact answer, and we have to take the nonleading terms into account.
Cwitribvtiwi for the JETP special issue in honor of V. A. Rubakov's 60th birthday
QCD, which is the currently accepted theory of strong interactions, still has the well-known problems with describing a strong-coupling phenomena. The physics of heavy-ion collisions, in particular, a quark-gluon plasma (QGP) formation, involves real-time strongly coupled phenomena, which makes these phenomena difficult to study within the standard QCD methods. In the recent years, a powerful approach to QGP is explored, based on a holographic duality between the strong-coupling quantum field in ii-dimensio-nal Minkowski space and classical gravity in (d + ^-dimensional anti de Sitter (AdS) space fl 3]. In particular, there is considerable progress in the holographic description of equilibrium QGP . The holographic approach is also applied to nonequilibrium QGP. Within this holographic approach, thermalization is described
as a process of formation of a black hole in the AdS space.
The AdS/'CFT (conformal-field theory) correspondence is based on string theory and perfectly works for the A/' = 4 SUSY Yang Mills theory, while the dual description of real QCD is unknown. Much effort has been invested into the search for holographic QCD from string theory (see, in particular, [5 7]). This approach is known as the "top-down" approach. An other approach, known as the "bottom-up" approach, is supposed to propose a suitable holographic QCD models from experimental data and lattice results [8 15]. The main idea of this approach is to use natural prescriptions of the general AdS/'CFT correspondence to try to recover nonperturbative QCD phenomena, in particular, nonperturbative vacuum phenomena, finite-temperature, high-density, and nonzero chemical potential phenomena.
The 5-dimensional metrics that reproduce the Cornell potential flC], as well as p-meson spectrum, etc., have been proposed [10, 13, 14]. A so-called improved holographic QCD (IHQCD) that is able to reproduce
the QCD /i-function has boon constructod . Ther-nial deformations of these backgrounds are intensively studied in the last years (see  for a review).
The problem of QGP formation is the subject of intensive study within holographic approach in last years (see [IT, 18] and the references therein). There is considerable progress in the understanding of the therma-lization process from the gravity side as BH formation. Initially, this process has been considered starting from the AdS background [19 25]. However, the pure AdS background is unable to describe the vacuum QCD with quark confinement, nor is it able to reproduce the QCD /i-function. There are backgrounds that solve one, or even two of these problems. The first was solved in  (see also [13, 14]), where a special version of a soft wall was proposed, and the /i-function was reproduced from IHQCD [12, 15].
To describe thermalization, it is natural to study deformations of these backgrounds. Suitable deformations of IHQCD by shock waves have been studied in [26, 27], and it has been shown that without additional assumptions, the IHQCD metric does not reproduce the experimental multiplicity dependence on energy. It was noted in  that holographic realization of the experimental multiplicity requires an unstable background.
The goal of this paper is to close this gap and to show that the model that reproduces the Cornell potential can at the same time be used as a gravity background to give the correct energy dependence of multiplicities produced in a finite time. As a bonus of our approach, we obtain a reasonable estimation of the thermalization time.
This paper is organized as follows. In Sec. 2.1, we recall the confining metrics that reproduce the Cornell potential. In Sec. 2.2, we recall the previous results concerning the dependence of multiplicities on energy. In Sec. 2.3, we present the main formula for the size of trapped surfaces formed in collision of domain walls. In Sec. 3, we consider a special metric that is far away from the confining metrics, but gives a suitable entropy. We also note that a restriction of the size of the trapped surface permits determining the thermalization time. In Sec. 4, we show that the confining metric in  can be approximated at intermediate values of the holographic coordinate j by the metric considered in Sec. 3. As a result, for the entropy produced during a short time Tierm ~ 0.25 fm1^, this gives an asymptotic expansion with the leading term that depends on energy as E1/3. The same is true for the metric in .
2. SETUP 2.1. Confining backgrounds
It is well known that the AdS space does not reproduce the quark confinement. To reproduce quark confinement, in particular, the appropriate glueball spectrum, Polchinski and Strassler  imposed a cut-off in the AdS space, a "hard wall model". Another modification of the AdS space, a "soft wall models" , is related to the dilaton. In the bottom-up approach, the metric is usually taken to be
d*2 = b2(z)(-dt2+dz
where b2(z) is some function usually taken to be the AdS metric in the UV zone (this leads to the Coulomb potential in the UV) and is a deformed AdS metric in the IR. The deformation in the IR should be taken in such a way that the quark antiquark potential exhibit confinement.
The experimental model of the potential that is used to fit lattice and experimental data  is usually taken to be the Cornell potential. In principle, this potential should reproduce the quarkonium spectrum, interpolating between one-gluon exchange in the UV and linear confinement in the IR.
The model proposed in  uses the warp factor
haz = exp
u = 0.42 GoV2
In , it was shown that this factor reproduces the static interquark potential obtained from SU(3) lattice calculations .
In [141. the modification
a = 0.34 GoV
exp (—az2/2) [(î/iî — 2)/î/iï]C( ce = 1, zIR = 2.54 GoV"1
was considered. This modification is in fact very close to the model in  for 0 < î < 2 fin and reproduces the Cornell potential and the /i-function.
In this paper, we consider modification (4.1) (see below) of factor (2.2), which also fits the Cornell potential well.
The experimental data for multiplicities in heavy-ion collisions at the RHIC and LHC indicate that 
11 Here and below the light velocity c = 1.
M,,,. X E0 3
The multiplicities obtained for the simplest holographic calculation in a conformal background with the AdSs metric [19 25],
M „>,(/•:) x (2.5)
are in fact worse than the Landau bound
MLandau(E) ^E1'2. (2.6)
To improve the energy dependence of multiplicities, Iviritsis and Taliotis  proposed to use modifications of the 6-factor. They considered 6-factors corresponding to conformal and nonconformal backgrounds. More precisely, they considered collision of holographic pointlike sources in dilaton models and obtained estimations for a variety of models (depending on the dilaton potential)
Ma^1/3 oc E(3a+3>/(3a+2\ M"<-1/3 * E'3a+1)/3a.
We note that a perturbative QCD-inspired UV cut-off was also used in . This modification provides logarithmic corrections. Following , where an energy-dependent cut-off in the high-energy limit was proposed, Iviritsis and Taliotis  have shown that this cut-off reduces the powers in (2.7) and (2.8) as
Ma<.1/Z x £,'2/[3(1-q)
Later, in , we confirmed the results in (2.7) and (2.8) by considering the domain-wall collision models that generalized the Lin Shuryak model [30, 31] to nonconformal cases. In , we also noted that the model with the 6-factor b(z) = Lcff/z gives a more realistic bound
MPh-diiaion(E) x E1/3,
which is closer to (2.4). But the price for this modification is the phantom kinetic term for the dilaton. We note that we have not performed any UV cut-off in this model to obtain estimation (2.11).
2.3. Trapped surface for domain-wall shock waves
The equation for the domain-wall wave profile <f>w(z) in the space with a b- fact or is
is a dimensionless variable, is the 5-dimensional gravitational constant, E is an energy, and
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