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EPJ Web of Conferences

Signals of initial state quantum entanglement in relativistic particle collision

R. Bellwied, A. Hutson

Published Jan 1, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.1051/epjconf/202531607005

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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.
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 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 are interested in the entanglement entropy of the final state of two particles in an elastic scattering. One can expect that the entanglement is yielded by the interaction between the particles. Within an S-matrix theory, by using the partial wave expansion of the S-matrix elements for two-particle states, we calculate a density matrix in a high energy elastic scattering, and obtain the entanglement entropy formula, which is described by physical observables. In the derivation of this formula, there appear divergences due to the infinite volume of two-particle Hilbert space. In order to avoid such divergences, we suggest a volume-regularization. Then we concretely evaluate the entanglement entropy for proton-proton scattering by using experimental data.
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.
Entanglement of the two scattered particles is expected to occur in elastic collisions, even at high energy where they are in competition with inelastic ones. We study how to evaluate quantitatively the corresponding entanglement entropy SEE{S}_{\mathrm{EE}}SEE​. For this sake, we regularize the divergences occurring in the formal derivation of SEE{S}_{\mathrm{EE}}SEE​ using a regularization procedure acting on the two-particle Hilbert space of final states. A quantitative application is performed in proton-proton collisions at collider energies, comparing the results of SEE{S}_{\mathrm{EE}}SEE​ with two different cutoffs and with a volume-regularization obtained by a prescription fixing the finite two-body Hilbert space volume. A significant entanglement is found which persists even at the highest available energies.
We present universal relations between entanglement entropy, which quantifies the quantum correlation between subsystems, and the cross section, which is the primary observable for high-energy particle scattering, by employing a careful formulation of wave packets for the incoming particles. For 2-to-2 elastic scattering with no initial entanglement and subdividing the system along particle labels, we show that both the Rényi and Tsallis entropies in the final states are directly proportional to the elastic cross section in units of the transverse size for the initial wave packets, which is then interpreted as the elastic scattering probability. The relations do not depend on the underlying dynamics of the quantum field theory and are valid to all orders in coupling strengths. Furthermore, computing quantum correlations between momentum and nonkinematic data leads to entanglement entropies expressed as various semi-inclusive elastic cross sections. Our result gives rise to a novel “area law” for entanglement entropy in a two-body system.