Deep inelastic scattering as a probe of entanglement: confronting experimental data
DDeep inelastic scattering as a probe of entanglement:confronting experimental data
Dmitri E. Kharzeev
1, 2, ∗ and Eugene Levin
3, 4, † Center for Nuclear Theory, Department of Physics and Astronomy,Stony Brook University, New York 11794-3800, USA Department of Physics and RIKEN-BNL Research Center,Brookhaven National Laboratory, Upton, New York 11973-5000, USA Department of Particle Physics, School of Physics and Astronomy, Tel Aviv University, Tel Aviv, 69978, Israel Departamento de F´ısica, Universidad T ´ e cnica Federico Santa Mar´ıa andCentro Cient´ıfico-Tecnol ´ o gico de Valpara´ıso, Casilla 110-V, Valparaiso, Chile (Dated: February 22, 2021)Parton distributions can be defined in terms of the entropy of entanglement between the spatialregion probed by deep inelastic scattering (DIS) and the rest of the proton. This approach leadsto a simple relation S = ln[ xG ( x )] between the gluon structure function xG ( x ) and the entropy ofthe produced hadronic state S ; it is valid at sufficiently small Bjorken x , where gluons dominateand the proton becomes a maximally entangled state. Recently, the H1 Collaboration analyzedthe entropy of the hadronic state in DIS, and studied its relation to the gluon structure function;poor agreement with the predicted relation was found. Here we show that the data from the H1Collaboration in fact agree well with the prediction based on entanglement, once two importanteffects are taken into account: i) because the hadron multiplicity N in the H1 measurement is notlarge, ∼ /N corrections to the predicted relation have to be included; and ii) since the measuredhadrons are mostly in the current fragmentation region, the relevant structure function is not thegluon but the sea quark one. PACS numbers: 13.60.Hb, 12.38.Cy
In our paper [1] (see also [2, 3]) we computed the von Neumann entropy of the system of partons resolved by deepinelastic scattering (DIS) at a given Bjorken x and momentum transfer q = − Q . We then proposed to interpretit as the entropy of entanglement between the spatial region probed by deep inelastic scattering and the rest of theproton. We found that in the small x , large rapidity Y regime, all partonic micro-states have equal probabilities – theproton is composed by an exponentially large number N of micro-states that occur with equal and small probabilities1 /N . This yields a simple relation between the entanglement entropy and the multiplicity of partons (dominated bygluons at small x ): S ( x ) = ln[ N ] = ln[ xG ( x )] (1)where xG ( x, Q ) is the gluon structure function . Assuming that the multiplicity of produced hadrons is proportionalto the multiplicity of partons (“local parton-hadron duality” [4, 5]), eq. (1) imposes a relation between the partonstructure function (extracted from the inclusive cross section of DIS) and the entropy of produced hadrons that canbe directly tested in experiment. The comparison to the experimental data on hadron multiplicity distributions fromCMS Collaboration at the LHC provided encouraging results [1, 2].However, recent dedicated experimental analysis performed by the H1 collaboration [6] shows a disagreement withEq. (1) (see Fig. 12 in Ref. [6] and the dotted curves in our Fig. 1). In this letter we demonstrate that the H1 datain fact are in a good agreement with our approach, once two important effects are taken into account. First, sincethe experimental multiplicities are not large, we need to take into account corrections of the order of 1 /N to Eq. (1).Second, the H1 data are concentrated in the current fragmentation region (see also Refs.[6–9]), where the dominantpartons are (sea) quarks and antiquarks – so the relevant structure functions are the quark and antiquark ones, seeFig. 2. Of course, in DIS one always probes the quarks – but at very small x and moderate Q , the sea quark andgluon structure functions are proportional to each other. This is not the case in the current fragmentation region inH1 kinematics.In Refs.[1, 3] it is shown that in QCD cascade the multiplicity distribution has the following form: Note that this relation is a quantum analog of the Boltzmann formula underlying statistical physics. a r X i v : . [ h e p - ph ] F e b ■ ■ ■ ■ HERAPDF, Σ sea = ( u ˜+ d ˜+ s ) HERAPDF, Σ sea = ( u ˜+ d ˜) HERAPDF, S gluon ■ ��� � < � � < �� ��� � × - × - × - < x Bj > S s ea ■ ■ ■ ■ HERAPDF, Σ sea = ( u ˜+ d ˜+ s ) HERAPDF, Σ sea = ( u ˜+ d ˜) HERAPDF, S gluon ■ H1: 10 < Q <
20 GeV × - × - × - < x Bj > S s ea ■ ■ ■ ■ HERAPDF, Σ sea = ( u ˜+ d ˜+ s ) HERAPDF, Σ sea = ( u ˜+ d ˜) HERAPDF, S gluon ) ■ H1: 20 < Q <
40 GeV × - × - × - < x Bj > S s ea ■ ■ ■ ■ HERAPDF, Σ sea = ( u ˜+ d ˜+ s ) HERAPDF, Σ sea = ( u ˜+ d ˜) HERAPDF, S gluon ■ H1: 40 < Q <
100 GeV × - × - × - × - < x Bj > S s ea FIG. 1: Comparison of the experimental data of the H1 collaboration [6] on the entropy of produced hadrons in DIS [6] withour theoretical predictions, for which we use the sea quark distributions from the NNLO fit to the combined H1-ZEUS data. p n ( N ) = 1 N (cid:18) − N (cid:19) n − (2)where N is the average parton multiplicity. The distribution (2) leads to the following von Neumann entropy: S = − (cid:88) p n ln p n = ln( N −
1) + N ln (cid:18) N − (cid:19) (3)One can see that at large N we obtain S (cid:39) ln N , but corrections are sizable when N ≤
10 (see Fig. 3). It shouldbe noted that the distribution of Eq. (2) describes quite well the experimental hadron multiplicity distributions inproton-proton collisions (see Refs. [1–3]).For comparison with the H1 experimental data [6], we first assume, following [1], that the hadron multiplicityis equal to the number of partons. This assumption is based on “parton liberation” picture [5] and on the ”localparton-hadron duality” [4]. In the current fragmentation region the partons that interact with the virtual photon arethe sea quarks (see Fig. 2). For high energies (small x ) the sea quark multiplicities are proportional to the gluon one,leading to Eq. (1). However, in the region of x corresponding to the current fragmentation region (corresponding tothe H1 experimental data [6]), the difference between xG ( x ) and Σ sea ( x ) = 2 x (cid:0) ¯ u ( x ) + ¯ d ( x ) + s ( x ) (cid:1) turns out to bequite large. In Fig. 1 we use the sea quark distribution from NNLO fit [10] to the combined H1 and ZEUS data [11].One can see that our approach in fact describes the H1 data quite well – this is the first test of the relation betweenentanglement and the parton model in DIS enabled by the H1 analysis. We stress that once the data in the target current fragmentationproton gluon productionhad sean ~ x Ghad FIG. 2: DIS at small x . � ���� ����� = �� � � � � � � � ������������������� � � FIG. 3: Entropy versus multiplicity N from Eq. (1) and Eq. (3). fragmentation region becomes available at the Electron-Ion Collider, one should be able to use xG ( x, Q ) in therelation (1), as it has been done in Refs. [1–3, 12, 13]. Acknowledgements:
We thank Kong Tu, Thomas Ullrich and our colleagues at BNL, Stony Brook University,Tel Aviv University and UTFSM for stimulating discussions. We are very grateful to Aharon Levy and KatarzynaWichmann for help in finding and extracting the parton distributions in the NNLO fit to HERA data. This work wassupported in part by the U.S. Department of Energy under Contracts No. DE-FG88ER40388 and DE-SC0012704,BSF grant 2012124, ANID PIA/APOYO AFB180002 (Chile) and Fondecyt (Chile) grants 1180118. ∗ Electronic address: [email protected] † Electronic address: [email protected], [email protected][1] D. E. Kharzeev and E. M. Levin, “Deep inelastic scattering as a probe of entanglement,”
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