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arXiv 2202.01803hep-ph
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中间态$x$光锥上质子的夸克胶子纠缠

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

Adrian Dumitru, Eric Kolbusz

备注：
v2: minor revisions; v3: corrected two expressions, final version
发表信息：
Phys. Rev. D 105, 074030 (2022)
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摘要：

在具有$N_c$颜色的QCD中，质子的反对称价夸克色空间单重态$\sim ε_{i_1\cdots i_{N_c}} |i_1,\cdots, i_{N_c}\rangle$对应于单色自由度的约化密度矩阵$ρ_{ij}=(1/N_c) δ_{ij}$。它的简并谱、$λ_i=1/N_c$、纯度$\mathrm{tr}~ρ^2 = 1/N_c$和von~Neumann熵$S_\mathrm{vN}=\log(N_c)$都表明了最大的色纠缠。另一方面，对于$N_c\to\infty$，质子的空间波函数分解为由平均场（E. Witten，Nucl.Phys.B160（1979）p.57），其中没有空间自由度的纠缠。在$N_c=3$上，使用Brodsky和Schlumpf的简单三夸克模型光阵面波函数进行了模型计算，预测了空间自由度的百分比能级纠缠。利用光锥微扰理论，我们还导出了四部分子$|qqqg\rangle$ Fock态的密度矩阵。通过对夸克的追踪，我们构造了胶子自由度的约化密度矩阵，它对胶子与源的纠缠进行了编码。我们的表达式提供了密度矩阵的依赖性软截止$x$胶子光锥动量，共线和紫外监管机构。在简单近似下得到的数值结果表明，胶子（含$x_g < \langle x_q\rangle$）的纠缠比夸克在三夸克Fock态的纠缠强。

英文摘要：

In QCD with $N_c$ colors the anti-symmetric valence quark color space singlet state $\sim ε_{i_1\cdots i_{N_c}} |i_1,\cdots, i_{N_c}\rangle$ of the proton corresponds to the reduced density matrix $ρ_{ij}=(1/N_c) δ_{ij}$ for a single color degree of freedom. Its degenerate spectrum of eigenvalues, $λ_i=1/N_c$, the purity $\mathrm{tr}~ρ^2 = 1/N_c$, 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.

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