Quark and gluon entanglement in the proton on the light cone at intermediate $x$

TL;DR

This paper investigates the entanglement of quark and gluon degrees of freedom within the proton using light-front wave functions and perturbation theory, revealing insights into how entanglement varies with gluon momentum fraction.

Contribution

It introduces a detailed analysis of parton entanglement in the proton, including derivation of density matrices and dependence on momentum cutoffs, highlighting differences between quark and gluon entanglement.

Findings

Gluons exhibit stronger entanglement than quarks at low momentum fractions.

Density matrices depend on soft cutoff and regulators, affecting entanglement measures.

Numerical results suggest significant gluon entanglement at small x.

Abstract

In QCD with $N_c$ colors the anti-symmetric valence quark color space singlet state $\sim \epsilon_{i_1\cdots i_{N_c}} |i_1,\cdots, i_{N_c}\rangle$ of the proton corresponds to the reduced density matrix $\rho_{ij}=(1/N_c) \delta_{ij}$ for a single color degree of freedom. Its degenerate spectrum of eigenvalues, $\lambda_i=1/N_c$, the purity $\mathrm{tr}~\rho^2 = 1/N_c$, and the von~Neumann entropy $S_\mathrm{vN}=\log(N_c)$ all indicate maximal entanglement of color. On the other hand, for $N_c\to\infty$ the spatial wave function of the proton factorizes into valence quark wave functions determined by a mean field (E. Witten, Nucl. Phys. B 160 (1979) p. 57) where there is no entanglement of spatial degrees of freedom. A model calculation at $N_c=3$ 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 $|qqqg\rangle$ 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 $x$ 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 $x_g < \langle x_q\rangle$) than for quarks in the three quark Fock state.