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arXiv:1805.12444 · Physical Review C

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

Xabier Feal, C. Pajares, Ricardo A. Vázquez

Published Jan 24, 2019• 40 citations
arXivHTMLPDF

Abstract

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 $p\text{\ensuremath{-}}p$ case. We relate these fluctuations to the temperature fluctuations by means of a Langevin equation for the white stochastic noise. We show that the multiplicity parton distribution for events with at least one hard parton collision is a $\mathrm{\ensuremath{\Gamma}}$ distribution. We use this result to compute the entanglement entropy, showing that the leading term is the logarithm of the number of partons, meaning that the $n$ microstates are equally probable and the entropy is maximal. There is another contribution related to the inverse of the normalized parton number fluctuation, which at very high energy changes the behavior from $lnn$ to $ln\sqrt{n}$.

DOI
10.1103/physrevc.99.015205

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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 p\ensuremath-pp\text{\ensuremath{-}}pp\ensuremath-p case. We relate these fluctuations to the temperature fluctuations by means of a Langevin equation for the white stochastic noise. We show that the multiplicity parton distribution for events with at least one hard parton collision is a \ensuremathΓ\mathrm{\ensuremath{\Gamma}}\ensuremathΓ distribution. We use this result to compute the entanglement entropy, showing that the leading term is the logarithm of the number of partons, meaning that the nnn microstates are equally probable and the entropy is maximal. There is another contribution related to the inverse of the normalized parton number fluctuation, which at very high energy changes the behavior from lnnlnnlnn to lnnln\sqrt{n}lnn​.
We present thermodynamic entropy calculations based on charged particle multiplicity data from proton-proton collisions measured by ALICE at the LHC in comparison to entanglement entropy calculations based on initial state parton distributions. The relative agreement of these distributions can be quantified by studying their higher order cumulants. The commonalities between the initial and final state entropy suggest that entanglement could be a possible source for the seemingly thermal and collective behavior in small systems.
These lectures advocate the idea that quantum entanglement provides a unifying foundation for both statistical physics and high-energy interactions. I argue that, at sufficiently long times or high energies, most quantum systems approach a Maximal Entanglement Limit (MEL) in which phases of quantum states become unobservable, reduced density matrices acquire a thermal form, and probabilistic descriptions emerge without invoking ergodicity or classical randomness. Within this framework, the emergence of probabilistic parton model, thermalization in the break-up of confining strings and in high-energy collisions, and the universal small xxx behavior of structure functions arise as direct consequences of entanglement and geometry of high-dimensional Hilbert space.
We calculate the entanglement entropy of a model proton wave function in coordinate space by integrating out degrees of freedom outside a small circular region A‾\overline{A}A of radius LLL, where LLL is much smaller than the size of the proton. The wave function provides a nonperturbative distribution of three valence quarks. In addition, we include the perturbative emission of a single gluon and calculate the entanglement entropy of gluons in A‾\overline{A}A. For both quarks and gluons, we obtain the same simple result: ${S}_{E}=\ensuremath{-}\ensuremath{\int}\frac{dx}{\mathrm{\ensuremath{\Delta}}x}{N}_{{L}^{2}}(x)\mathrm{log}[{N}_{{a}^{2}}(x)]$, where aaa is the UV cutoff in coordinate space and $\mathrm{\ensuremath{\Delta}}x$ is the longitudinal resolution scale. Here NS(x){N}_{S}(x)NS​(x) is the number of partons (of the appropriate species) with longitudinal momentum fraction xxx inside an area SSS. It is related to the standard parton distribution function by ${N}_{S}(x)=\frac{S}{{A}_{p}}\mathrm{\ensuremath{\Delta}}xF(x)$, where Ap{A}_{p}Ap​ denotes the transverse area of the proton.
We consider the following conundrum which attracted attention recently: on one hand the proton as a quantum object is in a pure state and is described by a completely coherent wave function with zero entropy. On the other hand in high energy experiments (DIS) when probed by a small external probe, it behaves like an incoherent ensemble of (quasi-free) partons. In this talk, we define the “entropy of ignorance,” which quantifies the entropy associated with the ability to perform only a partial set of measurements on a quantum system. For a parton model, the entropy of ignorance is equal to a Boltzmann entropy of a classical system of partons. We analyze a calculable model used for describing low x gluons in the Color Glass Condensate approach, which has similarities with the parton model of QCD. In this model we calculate the entropy of ignorance in the particle number basis as well as the entanglement entropy of the observable degrees of freedom. We find that the two are similar at high momenta, but differ by a factor of order unity at low momenta. This holds for the Renyi as well as von Neumann entropies. We conclude that entanglement does not seem to play an important role in the context of the parton model.
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