Thermal behavior and entanglement in Pb-Pb and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mtext>−</mml:mtext><mml:mi>p</mml:mi></mml:mrow></mml:math> collisions

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 $ln\sqrt{n}$.

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