<p>Methods from Quantum Information Theory are used to scrutinize quantum correlations encoded in the two-quark density matrix over light-cone momentum fractions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>x</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>x</mi><mn>2</mn></msub></math>. A nonperturbative three quark model light-cone wavefunction predicts significant nonclassical correlations associated with the “entanglement negativity” measure for asymmetric and small quark momentum fractions. We perform one step of QCD scale evolution of the entire density matrix, not just its diagonal (dPDF), by computing collinearly divergent corrections due to the emission of a gluon. Finally, we present first qualitative numerical results for single-step scale evolution of quantum entanglement correlations in double quark PDFs. At a higher <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>Q</mi><mn>2</mn></msup></math> scale, the nonclassical correlations manifest in the dPDF for nearly symmetric momentum fractions.</p>

United States

Adrian Dumitru

Eric Kolbusz

Quantum entanglement correlations in double quark PDFs

<p>Exclusive production of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub></math> axial vector quarkonia in photon-proton scattering at high energies requires a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi></math>-odd <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>t</mi></math>-channel exchange. In the limit of vanishing momentum transfer this occurs either via the exchange of a photon, the Primakoff process, where the spin of the proton does not change. For axial-vector meson production, as a consequence of the Landau-Yang theorem, the Primakoff cross section is finite as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>t</mi><mo stretchy="false">→</mo><mn>0</mn></math>. Alternatively, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi></math>-odd spin dependent Odderon can be exchanged, which involves a spin flip of the proton. The resulting cross section is related to the square of the collinear trigluon correlator or the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>k</mi><mo>⊥</mo></msub></math>-moment of the gluon Sivers function. Using two models for the gluon Sivers function from the literature we compute the ratio of Sivers to Primakoff cross sections and the angular coefficient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>λ</mi><mi>θ</mi></msub></math> governing the angular distribution of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>χ</mi><mrow><mi>c</mi><mn>1</mn></mrow></msub><mo stretchy="false">→</mo><mi>J</mi><mo>/</mo><mi>ψ</mi><mo>+</mo><mi>γ</mi></math> decay as functions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math>. We point out that these observables constrain the magnitude of the gluon Sivers function at small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> which could be accessed in electron-proton scattering and ultraperipheral proton-proton and nucleus-proton collisions.</p>

Croatia

Sanjin Benić

Poland

Leszek Motyka

Tomasz Stebel

Gluon Sivers function from forward exclusive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>c</mi><mn>1</mn></mrow></msub></mrow></math> photoproduction on unpolarized protons

The recently revised small-<math><mi>x</mi></math> helicity evolution [J. High Energy Phys. 07 (2022) 095], resumming the double-logarithmic factor, <math><mrow><msub><mrow><mi>α</mi></mrow><mrow><mi>s</mi></mrow></msub><msup><mrow><mi>ln</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mn>1</mn><mo>/</mo><mi>x</mi><mo>)</mo></mrow></math>, allows for the study of helicity distributions of quarks and gluons at small Bjorken <math><mi>x</mi></math>, corresponding to high center-of-mass energy. In this work, we calculate the moderate-<math><mi>x</mi></math> initial conditions in the regime, <math><msub><mi>α</mi><mi>s</mi></msub><msup><mi>ln</mi><mn>2</mn></msup><mo>(</mo><mn>1</mn><mo>/</mo><mi>x</mi><mo>)</mo><mo>∼</mo><mn>1</mn></math>, for the small-<math><mi>x</mi></math> helicity evolution using a light-front valence quark model of the proton, which provides additional physical information about the target. The perturbative emission and absorption of a gluon by the valence quarks are also included. The results, given in Eqs. (35), provide a new set of initial conditions with a significantly reduced number of free parameters than conventional models [Phys. Rev. D 108, 114007 (2023)]. Consequently, the predictive power of small-<math><mi>x</mi></math> helicity evolution is expected to improve once the initial conditions from this work are incorporated.

Finland

Heikki Mäntysaari

Yossathorn Tawabutr

Polarized dipole scattering amplitudes meet the valence quark model

Exclusive <math><mi>C</mi><mo>=</mo><mo>+</mo><mn>1</mn></math> scalar, axial-vector, and tensor quarkonium production in high-energy electron-proton scattering requires a <math><mi>C</mi></math>-odd <math><mi>t</mi></math>-channel exchange of a photon or a three-gluon ladder. We derive the expressions for the corresponding amplitudes. The relative phase of the photon vs three-gluon exchange amplitudes is determined by the sign of the light-front matrix element of the eikonal color current operator <math><msup><mi>d</mi><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></msup><msup><mi>J</mi><mrow><mo>+</mo><mi>a</mi></mrow></msup><msup><mi>J</mi><mrow><mo>+</mo><mi>b</mi></mrow></msup><msup><mi>J</mi><mrow><mo>+</mo><mi>c</mi></mrow></msup></math> at moderate <math><mi>x</mi></math>, and is not affected by small-<math><mi>x</mi></math> QCD evolution. Model calculations predict constructive interference, which is particularly strong for momentum transfer <math><mrow><mo>|</mo><mi>t</mi><mo>|</mo><mo>∼</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><msup><mrow><mi>GeV</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math> where the cross section for <math><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub></math> production exceeds that for pure photon exchange by up to a factor of 4. Exclusive <math><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub></math> electroproduction at the high luminosity Electron-Ion Collider occurs with well measurable rates and measurements of these processes should find evidence for the perturbative odderon exchange. We also compute the total electroproduction cross section as a function of energy and provide first estimates of the number of <math><msub><mi>χ</mi><mrow><mi>c</mi><mi>J</mi></mrow></msub></math> events per month at the Electron-Ion Collider design luminosity.

Abhiram Kaushik

Photon-odderon interference in exclusive <math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math> charmonium production at the Electron-Ion Collider

Two-particle correlations in the proton on the light front are described by a mixed density matrix obtained by tracing over all other, unobserved, degrees of freedom. We quantify genuinely quantum quark azimuthal correlations in terms of the entanglement negativity measure of quantum information theory. While the two-quark state in color space is one of high-entropy and weak-quantum correlation, we find that a standard three-quark model wave function from the literature predicts an azimuthally correlated state of low-entropy and high-entanglement negativity. Low entropy is consistent with expectations for many colors (at fixed ’t Hooft coupling <math><msup><mi>g</mi><mn>2</mn></msup><msub><mi>N</mi><mi>c</mi></msub></math>) but high negativity indicates substantial two-particle quantum correlations at <math><msub><mi>N</mi><mi>c</mi></msub><mo>=</mo><mn>3</mn></math>. We show that suppressing quantum correlations associated with entanglement negativity strongly modifies quark-pair azimuthal moments <math><mo>⟨</mo><msup><mi>ζ</mi><mi>n</mi></msup><mo>⟩</mo></math>, <math><mrow><mi>ζ</mi><mo>=</mo><mi>exp</mi><mo>(</mo><mi>i</mi><mo>(</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>)</mo></mrow></math>, intrinsic to the proton state. We also describe how to account for the leading <math><mi>O</mi><mo>(</mo><msup><mi>g</mi><mn>2</mn></msup><mo>)</mo></math> correction to the density matrix from light cone perturbation theory which is due to the presence (or exchange) of a gluon in the proton. This correction increases the entropy and reduces the negativity of the density matrix for quark-pair azimuthal correlations. Hence, the entanglement negativity measure may provide novel insight into the structure of the proton state of QCD.

Quark pair angular correlations in the proton: Entropy versus entanglement negativity

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 <math><mover><mi>A</mi><mo>¯</mo></mover></math> of radius <math><mi>L</mi></math>, where <math><mi>L</mi></math> 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 <math><mover><mi>A</mi><mo>¯</mo></mover></math>. For both quarks and gluons, we obtain the same simple result: <math><mrow><msub><mrow><mi>S</mi></mrow><mrow><mi>E</mi></mrow></msub><mo>=</mo><mo>−</mo><mo>∫</mo><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mi>Δ</mi><mi>x</mi></mrow></mfrac><msub><mrow><mi>N</mi></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mi>log</mi><mo>[</mo><msub><mrow><mi>N</mi></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>]</mo></mrow></math>, where <math><mi>a</mi></math> is the UV cutoff in coordinate space and <math><mi>Δ</mi><mi>x</mi></math> is the longitudinal resolution scale. Here <math><msub><mi>N</mi><mi>S</mi></msub><mo>(</mo><mi>x</mi><mo>)</mo></math> is the number of partons (of the appropriate species) with longitudinal momentum fraction <math><mi>x</mi></math> inside an area <math><mi>S</mi></math>. It is related to the standard parton distribution function by <math><mrow><msub><mrow><mi>N</mi></mrow><mrow><mi>S</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>S</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub></mrow></mfrac><mi>Δ</mi><mi>x</mi><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math>, where <math><msub><mi>A</mi><mi>p</mi></msub></math> denotes the transverse area of the proton.

Alex Kovner

Vladimir V. Skokov

Entanglement entropy of the proton in coordinate space

The forward scattering amplitude of a small dipole at high energies is given in the mean field approximation by the Balitsky-Kovchegov (BK) evolution equation. It requires an initial condition <math><mi>N</mi><mo>(</mo><mi>r</mi><mo>;</mo><msub><mi>x</mi><mn>0</mn></msub><mo>)</mo></math> describing the scattering of a dipole with size <math><mi>r</mi></math> off the target that is probed at momentum fraction <math><msub><mi>x</mi><mn>0</mn></msub></math>. Rather than using ad hoc parametrizations tuned to high-energy data at <math><mi>x</mi><mo>≪</mo><msub><mi>x</mi><mn>0</mn></msub></math>, here we attempt to construct an initial scattering amplitude that is consistent with low-energy, large-<math><mi>x</mi></math> properties of the proton. We start from a nonperturbative three quark light-cone model wave function from the literature. We add <math><mi>O</mi><mo>(</mo><mi>g</mi><mo>)</mo></math> corrections due to the emission of a gluon, and <math><mi>O</mi><mo>(</mo><msup><mi>g</mi><mn>2</mn></msup><mo>)</mo></math> virtual corrections due to the exchange of a gluon, computed in light-cone perturbation theory with exact kinematics. We provide numerical data as well as analytic parametrizations of the resulting <math><mi>N</mi><mo>(</mo><mi>r</mi><mo>;</mo><msub><mi>x</mi><mn>0</mn></msub><mo>)</mo></math> for <math><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>0.01</mn><mi>–</mi><mn>0.05</mn></mrow></math>. Solving the BK equation in the leading logarithmic approximation towards lower <math><mi>x</mi></math>, we obtain a fair description of the charm cross section in deeply inelastic scattering measured at HERA by fitting one parameter, the coupling constant <math><msub><mi>α</mi><mi>s</mi></msub><mo>≃</mo><mn>0.2</mn></math>. However, without the option to tune the initial amplitude at <math><msub><mi>x</mi><mn>0</mn></msub></math>, the fit of the high precision data results in <math><msup><mi>χ</mi><mn>2</mn></msup><mo>/</mo><msub><mi>N</mi><mi>dof</mi></msub><mo>=</mo><mn>2.3</mn></math> at <math><msub><mi>N</mi><mi>dof</mi></msub><mo>=</mo><mn>38</mn></math>, providing clear statistical evidence for the need of systematic improvement, e.g., of the photon wave function, evolution equation, and initial condition.

Risto Paatelainen

High-energy dipole scattering amplitude from evolution of low-energy proton light-cone wave functions

In QCD with <math><msub><mi>N</mi><mi>c</mi></msub></math> colors the antisymmetric valence quark color space singlet state <math><mo>∼</mo><msub><mi>ε</mi><mrow><msub><mi>i</mi><mn>1</mn></msub><mo>⋯</mo><msub><mi>i</mi><msub><mi>N</mi><mi>c</mi></msub></msub></mrow></msub><mo>|</mo><msub><mi>i</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>i</mi><msub><mi>N</mi><mi>c</mi></msub></msub><mo>⟩</mo></math> of the proton corresponds to the reduced density matrix <math><msub><mi>ρ</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mo>(</mo><mn>1</mn><mo>/</mo><msub><mi>N</mi><mi>c</mi></msub><mo>)</mo><msub><mi>δ</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></math> for a single color degree of freedom. Its degenerate spectrum of eigenvalues, <math><msub><mi>λ</mi><mi>i</mi></msub><mo>=</mo><mn>1</mn><mo>/</mo><msub><mi>N</mi><mi>c</mi></msub></math>, the purity <math><mi>tr</mi><msup><mi>ρ</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn><mo>/</mo><msub><mi>N</mi><mi>c</mi></msub></math>, and the von Neumann entropy <math><msub><mi>S</mi><mi>vN</mi></msub><mo>=</mo><mi>log</mi><mo>(</mo><msub><mi>N</mi><mi>c</mi></msub><mo>)</mo></math> all indicate maximal entanglement of color. On the other hand, for <math><msub><mi>N</mi><mi>c</mi></msub><mo>→</mo><mi>∞</mi></math> the spatial wave function of the proton factorizes into valence quark wave functions determined by a mean field [E. Witten, Nucl. Phys. B160, 57 (1979)] where there is no entanglement of spatial degrees of freedom. A model calculation at <math><msub><mi>N</mi><mi>c</mi></msub><mo>=</mo><mn>3</mn></math> using a simple three quark model light front wave function by Brodsky and Schlumpf predicts percent level entanglement of spatial degrees of freedom. Using light cone perturbation theory, we also derive the density matrix associated with the four parton <math><mrow><mo>|</mo><mi>q</mi><mi>q</mi><mi>q</mi><mi>g</mi><mo>⟩</mo></mrow></math> Fock state. Tracing out the quarks, we construct the reduced density matrix for the degrees of freedom of the gluon, which encodes its entanglement with the sources. Our expressions provide the dependence of the density matrix on the soft cutoff <math><mi>x</mi></math> for the gluon light cone momentum, and on the collinear and ultraviolet regulators. Numerical results obtained in a simple approximation indicate stronger entanglement for the gluon (with <math><mrow><msub><mrow><mi>x</mi></mrow><mrow><mi>g</mi></mrow></msub><mo>&lt;</mo><mo>⟨</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>⟩</mo></mrow></math>) than for quarks in the three quark Fock state.

Quark and gluon entanglement in the proton on the light cone at intermediate <math><mi>x</mi></math>

The three point correlation function of color charge densities is evaluated explicitly in light -one gauge for a proton on the light cone. This includes both <math><mi>C</mi></math>-conjugation even and odd contributions. We account for perturbative corrections to the three-quark light -cone wave function due to the emission of an internal gluon which is not required to be soft. We verify the Ward identity as well as the cancellation of UV divergences in the sum of all diagrams so that the correlator is independent of the renormalization scale. It does, however, exhibit the well-known soft and collinear singularities. The expressions derived here provide the <math><mi>C</mi></math>-odd contribution to the initial conditions for high-energy evolution of the dipole scattering amplitude to small <math><mi>x</mi></math>. Finally, we also present a numerical model estimate of the impact parameter dependence of quantum color charge three-point correlations in the proton at moderately small <math><mi>x</mi></math>.

Cubic color charge correlator in a proton made of three quarks and a gluon

Color charge correlators provide fundamental information about the proton structure. In this Letter, we evaluate numerically two-point color charge correlations in a proton on the light cone including the next-to-leading order corrections due to emission or exchange of a perturbative gluon. The non-perturbative valence quark structure of the proton is modelled in a way consistent with high-x proton structure data. Our results show that the correlator exhibits startlingly non-trivial behavior at large momentum transfer or central impact parameters, and that the color charge correlation depends not only on the impact parameter but also on the relative transverse momentum of the two gluon probes and their relative angle. Furthermore, from the two-point color charge correlator, we compute the dipole scattering amplitude. Its azimuthal dependence differs significantly from a impact parameter dependent McLerran-Venugopalan model based on geometry. Our results also provide initial conditions for Balitsky-Kovchegov evolution of the dipole scattering amplitude. These initial conditions depend not only on the impact parameter and dipole size vectors, but also on their relative angle and on the light-cone momentum fraction x in the target.