Partonic Entropy of the Proton in QCD

• Partonic Entropy is a measure of quantum entanglement among gluons and sea quarks, defined through the logarithmic scaling of parton densities.
• It integrates theoretical frameworks like von Neumann and Page entropy with saturation models to connect QCD predictions to DIS observables.
• The approach highlights distinctions between protons and nuclei by revealing differences in entropy due to shadowing and geometric scaling effects.

The partonic entropy of the proton quantifies the degree of entanglement and multiplicity within QCD degrees of freedom—primarily gluons and sea quarks—resolved in deep inelastic scattering (DIS) at small Bjorken-$x$. In this regime, QCD evolution leads to a maximally entangled proton wavefunction, characterized by equiprobable microstates and rapid growth in the number of resolved partons. Multiple theoretical frameworks, including the von Neumann entropy, Page entropy, and saturation models anchored in the color dipole picture, yield consistent formulas for the partonic entropy, which can be confronted with experimental observables such as hadron multiplicity distributions. The entropy scales logarithmically with the effective parton density, saturates under nonlinear QCD effects, and manifests observable differences in protons versus nuclei.

1. Theoretical Definition: Entanglement Entropy in the Proton

The proton is modeled as a pure light-cone quantum state, with its Fock space partitioned into two subsystems: region %%%%1%%%% (probed by the virtual photon at scale $Q^2$ in DIS) and region $B$ (the remainder of the proton). Tracing out $B$ yields a reduced density matrix,

$\rho_A = \mathrm{Tr}_B\, \rho_{AB}$

whose von Neumann entropy defines the entanglement entropy,

$S_E = -\mathrm{Tr}(\rho_A \ln \rho_A)\,.$

At sufficiently small $x$ (large rapidity $Y = \ln(1/x)$), nonlinear QCD evolution ensures equipartition among $n$-parton microstates. In this maximally entangled state, $p_n \approx 1/N_{\rm max}$ for $n=N_{\rm max}$, leading to

$S_E \to \ln N_{\rm max}$

where $N_{\rm max}$ is the average number of partons resolved at $(x, Q^2)$ (Ramos et al., 2020, Kharzeev et al., 2017).

2. Analytical Expression: Relation to Parton Densities

The average number of resolved partons, $N_{\mathrm{parton}}$, is given by the inclusive sum of gluons and sea quarks,

$$N_{\mathrm{parton}}(x, Q^2) = x G(x, Q^2) + x \Sigma(x, Q^2)$$

where $x G$ is the gluon PDF and $x \Sigma$ is the sea-quark singlet PDF. In the small-$x$ regime,

$S_E(x, Q^2) = \ln[N_{\mathrm{parton}}(x, Q^2)]$

This formula is phenomenologically successful, aligning closely with measured charged-hadron entropy in DIS from H1 (Hentschinski et al., 2021, Sheikhi et al., 23 Nov 2025, Hentschinski et al., 2022, Kharzeev et al., 2021, Kou et al., 2022). For pure gluonic descriptions,

$S_E(x, Q^2) = \ln[x G(x, Q^2)]$

while inclusion of sea quarks is essential for quantitative agreement with experimentally accessible kinematics.

3. Saturation Models: The Color-Dipole Approach

In the color dipole framework, especially the Golec–Biernat–Wüsthoff (GBW) model, the unintegrated gluon density is parametrized as

$$\mathcal{F}(x, k_T^2) = \frac{3 \sigma_0}{4\pi^2 \alpha_s} \frac{k_T^2}{Q_s^2(x)} e^{-k_T^2/Q_s^2(x)}\,,$$

with saturation scale $Q_s^2(x) = (x_0/x)^\lambda$ and phenomenological parameters fixed by DIS data. Integration up to $Q^2$ yields

$$x G(x, Q^2) = \frac{3 \sigma_0}{4\pi^2 \alpha_s} Q_s^2(x) \left[ 1 - \left( 1 + Q^2/Q_s^2(x) \right) e^{-Q^2/Q_s^2(x)} \right]$$

Consequently,

$$S_E(x, Q^2) \approx \ln \left\{ \frac{3 \sigma_0}{4\pi^2 \alpha_s} Q_s^2(x) [1 - (1 + Q^2/Q_s^2(x)) e^{-Q^2/Q_s^2(x)}] \right\}$$

This entropy grows with $Y = \ln(1/x)$ as $S_E \sim \lambda \ln(1/x)$ at fixed $Q^2$ for $x \ll 10^{-3}$ (Ramos et al., 2020).

4. Quantum Information-Theoretic Perspective: Page Entropy and Generalizations

Page's random pure-state framework yields the average entanglement entropy as

$S = \ln m - \frac{1}{2}$

where $m$ is the dimension of the subsystem ($m \sim x G + x\Sigma$). This formula encodes a quantum correction $-1/2$ compared to the classical Boltzmann entropy, and exhibits near-maximal entanglement in the small-$x$ limit (Kou et al., 2022). The Page curve analog clarifies the transition from linear to nonlinear QCD dynamics (saturation, CGC onset) when $m \sim n$ (probing and unprobed degrees of freedom).

5. Numerical Behavior, Uncertainties, and Experimental Validation

Empirically, $S_E(x, Q^2)$ rises monotonically with decreasing $x$ and increasing $Q^2$ up to a saturation regime where the QCD evolution slows entropy growth. For $Q^2 = 2$ GeV$^2$ and $x = 10^{-3} \to 10^{-5}$, the entropy ranges $S_E \approx 2.5$ to $4.0$ (Ramos et al., 2020). Comparisons to H1 multiplicity-derived entropy show excellent agreement across multiple models (GBW, BK, BFKL-evolved PDFs, and DGLAP fits with small-$x$ resummation) (Hentschinski et al., 2021, Sheikhi et al., 23 Nov 2025, Hentschinski et al., 2022). Uncertainties derive from PDF parametrizations, treatment of saturation, choice of $\alpha_s$, and binning procedures.

6. Proton versus Nuclear Targets: Shadowing and Scaling

For nuclei, geometric scaling modifies the saturation scale, $Q_{s,A}^2 \propto A^{4/9} Q_s^2$, and the effective cross-section. The per-nucleon entropy

$$S_A/A = \ln \left[ \frac{ x G_A(x, Q^2) }{A} \right ]$$

is suppressed by shadowing relative to the free proton

$S_A/A < S_E^p$

in agreement with both Page-style analyses and independent numerical extractions (Ramos et al., 2020, Kou et al., 2022). Physically, nuclear shadowing reduces the available phase space for entangled partonic states.

7. Physical Interpretation and Open Directions

Partonic entanglement entropy in the proton quantifies the quantum correlations between the measured (resolved by the photon probe) and unmeasured components of the QCD wavefunction. In the small-$x$ limit, nonlinear QCD evolution ensures that all Fock-space microstates become equiprobable, driving maximal entropy in the resolved subsystem. This entropy links directly to observable hadron multiplicity distributions in DIS, allows discrimination of saturation effects, and enables tests of quantum-information bounds (e.g., Bekenstein entropy limits). Extensions to more differential observables, higher orders in nonlinear evolution, spatial entanglement mapping, and systematic comparisons to future EIC data remain active areas of research (Ramos et al., 2020, Hentschinski et al., 2021, Kou et al., 2022, Wang, 2022, Golec-Biernat, 28 Dec 2025, Sheikhi et al., 23 Nov 2025).