Thermal behavior and entanglement in Pb-Pb and p-p collisions

The thermalization of the particles produced in collisions of small size objects can be achieved by quantum entanglement of the partons of the initial state as it was analyzed recently in proton-proton collisions. We extend such study to Pb-Pb collisions and to different multiplicities of proton-proton collisions. We observe that, in all cases, the effective temperature is approximately proportional to the hard scale of the collision. We show that such relation between the thermalization temperature and the hard scale can be explained as a consequence of the clustering of the color sources. The fluctuations on the number of parton states decreases with multiplicity in Pb-Pb collisions as far as the width of the transverse momentum distributions decreases, contrary to the p-p case. We relate these fluctuations to the temperature time fluctuations by means of a Langevin equation for the white noise due to the quench of a hard parton collision.


I. INTRODUCTION
The presence of an exponential shape in the transverse momentum distribution (TMD) of the produced particles in collisions of small size objects together with the approximate thermal abundances of the hadron yields constitutes an indicative sign of thermalization.
This thermalization, however, can not be achieved under the usual mechanism, namely final state interactions in the form of several secondary collisions.
The emergence of this phenomenon has been recently studied [1][2][3][4], showing that thermalization can be obtained during the rapid quench induced by the collision due to the high degree of entanglement inside the partonic wave functions of the colliding protons. Thus, the effective temperature obtained from the TMD of the particles produced in the collision depends on the momentum transfer, that is, it constitutes an ultraviolet cut-off on the quan-arXiv:1805.12444v1 [hep-ph] 31 May 2018 tum modes resolved by the collision. In diffractive processes with a rapidity gap, the entire wave function of the proton is involved and no entanglement entropy arises. Consequently, we expect no thermal radiation as it has been observed.
In this paper we further explore the relation between parton entanglement and thermalization by studying p-p and Pb-Pb collisions at different multiplicities. In the second case we expect an interplay between thermalization and final state interactions leading to some differences with p-p collisions concerning the entanglement and thermalization.
We show that the TMD of both collisions at different multiplicities can be fitted by the sum of an exponential plus a power like function, characterized by a thermal like temperature T th and a temperature scale T h respectively. For any fixed multiplicity and in all collisions the relation 4T th ≈ T h is satisfied. The power index n describing the hard spec- The organization of the paper is as follows. In section II we introduce the entanglement of the partonic state following reference [1] and we analyze the TMD of p-p and Pb-Pb collisions at different multiplicities. In section III we discuss the obtained results remarking the similarities and differences of p-p and Pb-Pb collisions in connection with thermalization and entanglement. We briefly discuss the clustering of color sources in connection with the TMD in section IV and in the section V we introduce the Langevin and Fokker-Planck equations to study the time temperature fluctuations. Finally in section VI the conclusions are presented.

MOMENTUM DISTRIBUTIONS
A hard process with momentum transfer Q probes only the region of space H of transverse size 1/Q. Let us denote by S the region of space complementary of H. The proton is described by the wave function of a suitably chosen orthonormal sets of states |Ψ H n and |Ψ S n localized in the domains H and S. In the parton model this full orthonormal set of states is given by the Fock states with different number n of partons. The state (1) can not be separated into a product |ϕ H ⊗ |ϕ S and therefore |Ψ HS is entangled. The density matrix of the mixed state probed in region H is where |α n | 2 = p n is the probability of having a state with n partons, independently of whether their interaction is hard or soft. The Von Neumann entropy of this state is given by We can consider that a high momentum partonic configuration of the proton when the collision takes place undergoes a rapid quench due to the QCD interaction. The onset τ of this hard interaction is given by the hardness scale Q, τ ∼ 1/Q. Since τ is small the quench creates a highly excited multi-particle state. The produced particles have thermallike exponential spectrum with an effective temperature T ≈ (2πτ ) −1 ≈ Q/2π. Thus, the thermal spectrum can be originated due to the event horizon formed by the acceleration of the color field [5]- [8]. On the other hand, the comparison with LHC data on hadron multiplicity distributions [2] indicates that the produced Boltzmann entropy is close to the entanglement entropy of equation (3).
In reference [1], the thermal component of charged hadron transverse momentum distribution in p-p collisions at √ s = 13 TeV is parameterized as [9][10][11] 1 N ev where T th is the effective temperature and m t = m 2 + p 2 t is the transverse mass. The hard scattering, meanwhile, is parameterized as, where the temperature T h and the index n are parameters determined from the fit to the experimental data. The value T th = 0.17 GeV was found [1], agreeing with the one expected from the extrapolation of the relation obtained at lower energies. Similarly the hard scale T h is given by the relation At √ s = 13 TeV, the values found for the hard scale are T h = 0.72GeV and n = 3.1. We notice that from Equations (6) and (7) one finds independently of the energy. The ratio of the particular values obtained in the fit [1] are close to this values.
In order to study the dependence on the multiplicity of T th and T rmh we have used the transverse momentum distribution of K 0 S produced in p-p collisions at √ s = 7TeV in the range up to p t ≤ 10 GeV/c [13]. We use K 0 S instead of π or charged particles because we have not found published data covering a broad range of soft and hard regions at different multiplicities. In Figures (1), (2)  n in p-p and Pb-Pb collisions as a function of the charged particle production. n decreases with multiplicity for p-p collisions and, on the contrary, increases for Pb-Pb collisions. For p-p collisions at √ s =13 TeV the value obtained for the case of charged particles is n =3.1, slightly smaller than the value of Figure (7). The values of n are larger for Pb-Pb than for p-p collisions as expected due to the jet quenching and correspondingly to high p t particle suppression.

III. DISCUSSION
Our results for p-p collisions show that for each multiplicity the effective thermal temperature obtained from the TMD of the produced particles can be viewed as a rapid quench of the entangled partonic state. The behavior of T th and T h as a function of the multiplicity is very similar, holding the relation T h ≈ 4T th for all the studied multiplicities. This fact adds TeV.
evidence to the cases studied in reference [1]. It is remarkable that the same relation holds for Pb-Pb collisions, realizing the different values of T th and T h compared with the p-p case.
The increase of T th with multiplicity in p-p collisions is larger than in the Pb-Pb case as it was expected, as far as T th is nothing but p t and experimentally the LHC data [12] has shown a larger increase with multiplicity in p-p than in Pb-Pb collisions.
Equations (4) and (5) can be obtained in the framework of clustering of color sources [14][15][16] as we show in section IV. In this approach, T th ≈ T h /π √ 2 and the cluster size distribution is a gamma function [17][18][19][20] which coincides with the stationary solution of the Fokker-Planck equation derived from the Langevin equation corresponding to a white noise due to the quench of a hard parton collision [21,22]. In this way the fluctuation in the cluster size are related to the time temperature fluctuations.
The ratio R of the integral under the power law curve (hard component) and the integral over the total (hard + thermal components) is plotted in Figure ( the area S n of the cluster. Thus Q 2 n = ( n 1 Q i ) 2 , and given that the individual string colors can be arbitrarily oriented in color space, the average Q i · Q j is zero, so Q 2 n = n Q 2 1 . As Q n depends also on the area, we have Q n = nS n /S 1 Q 1 , where S 1 is the area of the individual string. The mean multiplicity and the mean transverse momentum are proportional to the color charge and to the color field respectively which in the limit of high density, ξ = N s S 1 /S, becomes where N s is the number of color sources and F (ξ) is an universal factor The weight function is the gamma function because the process of increasing the centrality or energy of the collision can be regarded as a transformation of the color field located in the sites of the surface area, implying a transformation of the cluster size distribution of the type This renormalization group type of transformation have been studied long time ago in probability theory showing that the only stable distributions under such transformations are the generalized gamma function. We take the simplest case, namely, the gamma function with Introducing eq. 15 into equation 13 we obtain the distribution [17] f which takes the form of the parameterization used in the eq. 5 with which grows with the density ξ and thus with the energy and centrality as it is observed in the analysis of p-p and Pb-Pb collisions. In the last case, if the fits include larger p t values it is observed a flattening of the dependence of T h with multiplicity due to jet quenching effects. At low p t , equation 18 behaves as independently of n. In this low p t regime, there are other effects, like fluctuations of the color field, which should be taken into account. In fact, assuming that such fluctuations are Gaussian, we have [6,7] 2 where we defined x 2 h = πT 2 h . The equation 21 expresses the thermal behavior with In other words, the thermal temperature is given by the fluctuations of the hard temperature which are proportional to this hard temperature.
We notice that according to equation 18 fig. 7 is related to the critical percolation point. Notice that n is decreasing with multiplicity in p-p collisions but we expect that above a given multiplicity n should start to grow in the same way as in Pb-Pb collisions.

V. TIME EVOLUTION
The scale of the TMD is u = T 2 h which obeys the Langevin equation where τ is a characteristic damping time and ζ(t) is a white Gaussian noise, with mean ζ(t) = 0 and a correlator corresponding to fast changes such as expected in the quench induced by a hard collision, where D is the variance and φ a constant.
The associated Fokker Planck equation for the probability to have the temperature T at time t under the above noise is given by [21,22] ∂f (T, t) ∂t being The stationary solution of equations (23,25) is just the gamma distribution on the variable with This distribution coincides with the cluster size distribution of equation 15. In this way, the temperature fluctuations given by the inverse of the index n are related to the product τ D which have to do with the time evolution.

VI. CONCLUSIONS
The analysis of the dependence on the multiplicity of the LHC p-p and Pb-Pb data confirms the picture of thermalization induced by quantum entanglement. In all the analyzed data, the effective thermalization temperature obtained from data is proportional to the hard scale of the collision T given by the average momentum transfer. The coefficient of proportionality is universal, independent of the considered collision, even though T th and T h are different in each collision type. Thermal and hard temperatures increase with multiplicity in both collision scenarios, and this rise reproduces the known correlation of p t and dN ch /dη for p-p and Pb-Pb collisions. In the framework of clustering of color sources the proportionality between T th and T h is understood, being T th /T h = π √ 2. The n parameter of the hard distribution decreases with multiplicity for p − p collisions and increases for Pb-Pb collisions. This fact means that the normalized transverse momentum fluctuations behave quite different with multiplicity in p-p and Pb-Pb collisions. This behavior is naturally explained by the clustering of color sources. The change in the behavior of n is related to the formation of a large cluster of the initial color sources (partons), which marks the percolation phase transition.
The cluster size distribution is a gamma function which is also the stationary solution of the Fokker-Planck equation associated to the Langevin equation for a white Gaussian noise due to the quench of a hard partonic collision.

VII. ACKNOWLEDGMENTS
We thank the grant María de Maeztu Unit of Excelence of Spain and the support of Xunta de Galicia under the project ED431C2017. This paper has been partially done under the project FPA2014-58293-C2-1-P of MINECO (Spain).