Quantum entanglement between partons in a strongly coupled quantum field theory

Abstract

We perform a first-principles, non-perturbative investigation of quantum entanglement between partonic constituents in a strongly coupled 3+1-dimensional scalar Yukawa theory, using light-front Hamiltonian methods with controlled Fock-space truncations. By explicitly constructing reduced density matrices for (mock) nucleon, pion, and anti-nucleon subsystems from light-front wave functions, we compute key entanglement witnesses, including von Neumann entropy, mutual information, and linear entropy, in both quenched (no sea pairs) and unquenched frameworks. We find that the entanglement entropy is closely related to the Shannon entropy of the transverse momentum dependent distribution, establishing a link between quantum information and parton structure. In contrast, the unquenched theory reveals genuinely non-classical correlations: the entanglement entropy cannot be reduced to any Shannon entropy of normalized parton distributions, demonstrating that the full hadronic wave function encodes quantum information beyond classical probabilities. Our findings highlight the role of entanglement as a fundamental probe of non-perturbative dynamics in relativistic quantum field theory and lay the groundwork for extending these concepts to QCD and future collider phenomenology.

Figures (10)

• Figure 1: Numerical convergence of the Fock sector expansion for two representative observables in the quenched scalar Yukawa theory at coupling $\alpha = 2.0$: (Left) electromagnetic form factor $F(Q^2)$; (Right) PDF $f_\pi(x)$. The PV mass is set to be 15 GeV. The left panel uses units of $\mathrm{GeV}^2$ on the horizontal axis; the PDF is evaluated at factorization scale $\mu_F = \infty$. The left panel is adapted from Ref. Karmanov:2016yzu, and the right panel is adapted from Ref. Duan:2024dhy.
• Figure 2: Comparison of nucleon PDFs in the physical nucleon: quenched PDF $f^{\text{Que}}_N(x)$; one-parton PDF $\bar{f}_N(x)$; normalized unquenched PDF $\hat{f}_N(x)$ and valence PDF $f^v_N(x)$. Results are based on a three-body Fock truncation (see texts) with a coupling (Left): $\alpha = 1.0$; (Right): $\alpha = 2.0$. The $\delta(1-x)$ contribution from the one-body sector is omitted for clarity.
• Figure 3: The entanglement entropy $S_E \equiv S_N = S_\pi$ in the quenched theory as a function of the coupling $\alpha$. Since $S_E$ contains a logarithmic divergence $\log P^+_0V$, we fix $P^+_0V = 10^3$ and $8\times 10^4 \,\mathrm{GeV}^{-2}$. The "continuous" result uses Eq. \ref{['eqn:EE_for_sykw']}; the "discrete" result uses Eq. \ref{['eqn:EE_for_sykw_discrete']} with box regularization (see Sec. \ref{['sec:density_matrix']}) and $P^+_0 = 1\,\mathrm{GeV}$. A transverse UV cutoff $\mu_F$ is applied in the discrete sums, as labeled. All quantities are in GeV units.
• Figure 4: Distribution of entanglement entropy in transverse and longitudinal momentum for $\alpha = 1.0$ and $2.0$, based on quenched three-body truncation with $P^+_0V = 10^3\,\mathrm{GeV}^{-2}$. The transverse peak near $0.1\,\mathrm{GeV}$, close to the pion mass scale. Note that radial phase-space factor $k_\perp$ is not included in the plot. The $\delta(1-x)$ one-body contribution is omitted for visibility.
• Figure 5: Linear entropy as a function of $\alpha$ in the quenched three-body theory. In the continuum limit, $S_L \to 1 - Z^2$, providing a finite entanglement measure.