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arXiv:2110.04881 · Physical review. D/Physical review. D.

Entanglement entropy production in deep inelastic scattering

Kun Zhang, Kun Hao, Dmitri E. Kharzeev, V. E. Korepin

Published Jan 4, 2022• 31 citations
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Abstract

Deep inelastic scattering (DIS) samples a part of the wave function of a hadron in the vicinity of the light cone. Lipatov constructed a spin chain which describes the amplitude of DIS in leading logarithmic approximation. Kharzeev and Levin proposed the entanglement entropy as an observable in DIS [Phys. Rev. D 95, 114008 (2017)], and suggested a relation between the entanglement entropy and parton distributions. Here we represent the DIS process as a local quench in Lipatov's spin chain and study the time evolution of the produced entanglement entropy. We show that the resulting entanglement entropy depends on time logarithmically, $\mathcal{S}(t)=1/3\mathrm{ln}(t/\ensuremath{\tau})$ with $\ensuremath{\tau}=1/m$ for $1/m\ensuremath{\le}t\ensuremath{\le}(mx{)}^{\ensuremath{-}1}$, where $m$ is the proton mass and $x$ is the Bjorken $x$. The central charge $c$ of Lipatov's spin chain is determined here to be $c=1$; using the proposed relation between the entanglement entropy and parton distributions, this corresponds to the gluon structure function growing at small $x$ as $xG(x)\ensuremath{\sim}1/{x}^{1/3}$.

DOI
10.1103/physrevd.105.014002

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Deep inelastic scattering (DIS) samples a part of the wave function of a hadron in the vicinity of the light cone. Lipatov constructed a spin chain which describes the amplitude of DIS in leading logarithmic approximation. Kharzeev and Levin proposed the entanglement entropy as an observable in DIS [Phys. Rev. D 95, 114008 (2017)], and suggested a relation between the entanglement entropy and parton distributions. Here we represent the DIS process as a local quench in Lipatov's spin chain and study the time evolution of the produced entanglement entropy. We show that the resulting entanglement entropy depends on time logarithmically, S(t)=1/3ln(t/\ensuremathτ)\mathcal{S}(t)=1/3\mathrm{ln}(t/\ensuremath{\tau})S(t)=1/3ln(t/\ensuremathτ) with \ensuremathτ=1/m\ensuremath{\tau}=1/m\ensuremathτ=1/m for 1/m\ensuremath≤t\ensuremath≤(mx)\ensuremath−11/m\ensuremath{\le}t\ensuremath{\le}(mx{)}^{\ensuremath{-}1}1/m\ensuremath≤t\ensuremath≤(mx)\ensuremath−1, where mmm is the proton mass and xxx is the Bjorken xxx. The central charge ccc of Lipatov's spin chain is determined here to be c=1c=1c=1; using the proposed relation between the entanglement entropy and parton distributions, this corresponds to the gluon structure function growing at small xxx as xG(x)\ensuremath∼1/x1/3xG(x)\ensuremath{\sim}1/{x}^{1/3}xG(x)\ensuremath∼1/x1/3.
Relying on previous results, that link entanglement entropy and parton distribution functions in deep inelastic scattering, and focusing on the small Bjorken scaling region we present here indications that gluon shadowing might indeed be explained as due to a depletion of the entanglement entropy, between observed and unobserved degrees of freedom, per nucleon within a nucleus, with respect to the free nucleon. We apply to gluon shadowing the general Page approach to the calculation of the entanglement entropy in bipartite systems, giving physical motivations of the results.
We study entanglement entropy in Deep Inelastic Scattering (DIS) using the dipole formulation of the high-energy limit of QCD. We argue that a reduced density matrix arises in low xxx DIS due to a trace over unobserved color degrees of freedom and we obtain entanglement entropy in terms of dipole multiplicities, directly from the von Neumann entropy. Dipole multiplicities are obtained from a solution to low xxx evolution equations, which we solve numerically. Unlike previous studies, we take into account both transverse-size and azimuthal-angle dependence in the dipole evolution kernel. We study both the exact solution of the equation as well as its double leading-logarithmic approximation (DLLA). We find that for the same initial dipole size, the DLLA solution generates a larger entropy. Finally, we calculate the dipole multiplicities and entanglement entropy and compare our results to the Shannon entropy of hadron multiplicities, as measured by the H1 collaboration.
The non-equilibrium dynamics of disordered many-body quantum systems after a global quantum quench unveils important insights about the competition between interactions and disorder, yielding in particular an insightful perspective on many body localization (MBL). Still, the experimentally relevant effect of bond randomness in long-range interacting spin chains on the quantum quench dynamics have so far not been investigated. In this letter, we examine the entanglement entropy growth after a global quench in a quantum spin chain with randomly placed spins and long-range tunable interactions decaying with distance with power ααα. Using a dynamical version of the strong disorder renormalization group (SDRG) we find for α>αcα>α_cα>αc​ that the entanglement entropy grows logarithmically with time and becomes smaller with larger ααα as S(t)=Spln⁡(t)/(2α)S(t) = S_p \ln(t)/(2α)S(t)=Sp​ln(t)/(2α). Here, Sp=2ln⁡2−1S_p= 2 \ln2 -1Sp​=2ln2−1. We use numerical exact diagonalization (ED) simulations to verify our results for system sizes up to $ N\sim 16$ spins, yielding good agreement for sufficiently large α>αc≈1.8α> α_c \approx 1.8α>αc​≈1.8. For α<αcα<α_cα<αc​, we find that the entanglement entropy grows as a power-law with time, S(t)∼tγ(α)S(t)\sim t^{γ(α)}S(t)∼tγ(α) with 0<γ(α)<10<γ(α)<10<γ(α)<1 a decaying function of the interaction exponent ααα.
The nonequilibrium dynamics of disordered many-body quantum systems after a quantum quench unveils important insights about the competition between interactions and disorder, yielding, in particular, an interesting perspective toward the understanding of many-body localization. Still, the experimentally relevant effect of bond randomness in long-range interacting spin chains on their dynamical properties have so far not been investigated. In this Letter, we examine the entanglement entropy growth after a global quench in a quantum spin chain with randomly placed spins and long-range tunable interactions decaying with distance with power $\ensuremath{\alpha}$. Using a dynamical version of the strong disorder renormalization group we find for $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}$ that the entanglement entropy grows logarithmically with time and becomes smaller with larger $\ensuremath{\alpha}$ as $S(t)={S}_{p}ln(t)/(2\ensuremath{\alpha})$. Here, ${S}_{p}=2ln2\ensuremath{-}1$. We present results of numerical exact diagonalization calculations for system sizes up to $N\ensuremath{\sim}16$ spins, in good agreement with the analytical results for sufficiently large $\ensuremath{\alpha}>{\ensuremath{\alpha}}_{c}\ensuremath{\approx}1.8$. For $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}$, we find that the entanglement entropy grows as a power law with time, $S(t)\ensuremath{\sim}{t}^{\ensuremath{\gamma}(\ensuremath{\alpha})}$ with $0<\ensuremath{\gamma}(\ensuremath{\alpha})<1$ a decaying function of the interaction exponent $\ensuremath{\alpha}$.
We show that the hierarchy of quantum correlations produced in deep-inelastic scattering (DIS) can serve as a novel probe of the proton's nonperturbative structure. Specifically, we show that quantum entanglement, discord, steering, and magic provide nontrivial and complementary sensitivities to the nucleon's parton distribution functions (PDFs), particularly those encoding the transverse-spin polarization of the interacting quark. This connection leads to a unique probe of the proton's parton-level tensor charges with implications for beyond Standard Model (BSM) physics searches. We propose how quantum information measures can be utilized for precision studies of hadron structure at DIS experiments like the upcoming Electron-Ion Collider (EIC).
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.