^{3}.

The thermalization of the particles produced in collisions of small objects can be achieved by quantum entanglement of the partons of the initial state as 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 a relation between the thermalization temperature and the hard scale can be explained as a consequence of the clustering of the color sources. The fluctuations of the number of parton states decrease with multiplicity in Pb-Pb collisions as long as the width of the transverse-momentum distribution decreases, contrary to the

The presence of an exponential shape in the transverse-momentum distribution (TMD) of the produced particles in collisions of small objects together with the approximate thermal abundances of the hadron yields constitutes an indicative sign of thermalization. This thermalization, however, cannot 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

In this article we further explore the relation between parton entanglement and thermalization by studying

We show that the TMDs of both collisions at different multiplicities can be fitted by the sum of an exponential plus a powerlike function, characterized by a thermal-like temperature,

The organization of the article is as follows. In Sec.

A hard process with momentum transfer

In Ref.

To study the dependence on the multiplicity of

Normalized differential

We have extended the study to Pb-Pb collisions at different multiplicities by fitting the ALICE Collaboration TMD data for charged particles

Normalized differential charged-particle production in Pb-Pb collisions at

Variation of

In Fig.

Power index

Our results for

Equations

The ratio

Multiparticle production is currently described in terms of color sources (strings) stretched between the projectile and the target. These strings decay by the Schwinger

We notice that according to Eq.

The Gibbs distribution in energy,

It could be thought that any non-Gaussian distribution may only come from nonthermalized systems, in particular, the obtained power-tail distribution or the

Let us consider the Langevin equation of a variable

Let us consider the probability

This selection procedure of the events satisfying certain

The origin of the nonextensive thermodynamics related to Eq.

In terms of the reduced matrix density

We use Eq.

Entanglement entropy

It could be thought that because

The differences between

Entanglement entropy difference

The dependence of

The analysis of the dependence on the multiplicity of the LHC

The cluster-size distribution is a

Also, we have shown that the multiplicity parton distribution for events with at least one hard parton is the

We are grateful for a grant from the María de Maeztu Unit of Excellence of Spain and the support of Xunta de Galicia under Project No. ED431C2017. This work has been partially carried out under Project No. FPA2017-83814-P of Ministerio de Ciencia, Innovación y Universidades (Spain).