Entanglement, Trace Anomaly and Confinement in QCD

Abstract

We formulate confinement in QCD as an entropic surface phenomenon. Quark and gluon quantum information is localized on a transverse entangling two-sphere of radius $R_{EE}$; at this radius the QCD vacuum -- partitioned by a hadron into interior and exterior regions -- reaches its maximal entanglement entropy. Lattice-QCD determinations of the scalar (trace) gravitational form factors fix both $R_{EE}$ and the transverse trace-anomaly density $ρ_h(R_{EE})$, yielding a parameter-free slope $c_h = 8π^2 R_{EE}^2\,ρ_h(R_{EE})$ and a mechanical entropy $S_{EE}(y)=c_h\,y$ that grows linearly with rapidity $y$. The entropy gradient $\partial_R S_{EE}$ changes sign at $R_{EE}$: it pushes colored degrees of freedom outward for $r<R_{EE}$ and pulls them inward for $r>R_{EE}$, thereby localizing them on the codimension-2 entangling two-sphere $Σ_\perp = S^2_{R_{EE}}$ (which, in the infinite-momentum frame, projects onto the transverse plane), the 'information wall'. This provides a high-energy (large-$y$) entropic confinement diagnostic that complements -- rather than replaces -- Wilson's area-law criterion, which probes long-distance dynamics near the rest frame ($y\to 0$). Imposing unitarity on an entropic ansatz for the amplitude yields $σ(s)\propto y^δ$. World data favor $δ=2$ for elastic $pp\,(p\bar p)$ scattering and heavy-quark photoproduction, whereas $φ$ photoproduction favors a softer $δ=0.387$. All extracted cross sections remain well below the Froissart--Martin bound. These results provide a confinement criterion quantified directly from non-perturbative QCD inputs, unifying the trace anomaly, entanglement entropy, and high-energy scattering within a single quantitative framework.

Figures (6)

• Figure 1: Mechanical entanglement entropy per nucleon, $S_{\!EE}/(A\,y)$, versus entangling radius $R$. The light band spans $2\!\le\!A\!\le\!500$; dashed curves highlight representative nuclei. All curves level off at $S_{\!EE}/(A\,y)\!\simeq\!8.94$, signaling rapid saturation with $A$.
• Figure 2: Radial derivative $\partial_R S_{\!EE}$ for the systems in Fig. \ref{['fig:SEE_nuclei']}. Positive values ($R<R_{\!EE}$) represent an outward entropic pressure; negative values ($R>R_{\!EE}$) pull inward. The zero crossing defines the “information wall’’ at $R_{\!EE}$.
• Figure 3: Visual comparison of proton radii using lattice gravitational form factor \ref{['eq:EMT_nucl']} from Hackett:2023rif, see Table \ref{['tab:gff_params']}: maximum-entropy radius $R_{\!EE}$, mechanical core $R_{q,g}$\ref{['eq:Rcore-analytic']} (from the node of the 3D Breit-frame pressure density, see Fig. \ref{['fig:pressure']}), rms scalar radii $r_s$\ref{['eq:radii-analytic']}, rms energy radii $r_{00}$\ref{['eq:radii-analytic']}, and the experimental rms charge radius $r_c$ from PRad Xiong:2019umf. All rms radii shown are computed separately for the quark and for the gluon sectors using 3D Breit-frame densities and Eq. \ref{['eq:radii-analytic']} in the Supplemental Material; no $q\!+\!g$ sum is taken. However, $R_{\!EE}$ is computed from $q+g$ 2D (impact-parameter space) trace anomaly density Eq. \ref{['anomlyDensity']}.
• Figure 4: Cross sections $\sigma$ are plotted versus the rapidity $y=\mathop{\mathrm{arcosh}}\nolimits\gamma=\ln\!\bigl(\gamma+\sqrt{\gamma^{2}-1}\bigr)$, with $\gamma_{\text{beam}}\equiv E_{\gamma}/E_{\gamma,\mathrm{thr}}(V)$ for photoproduction and $\gamma_{\text{CM}}\equiv \sqrt{s}/(2m_{N})$ for elastic $pp$ scattering. Panels display (a) $\phi$ photoproduction, (b) $J/\psi$ photoproduction, (c) $\Upsilon(1S)$ photoproduction, and (d) elastic $pp$ scattering. The solid blue curves show the scaling ansatz $\sigma \propto y^{\delta}$ with $\delta_\phi=0.387$ in (a) and $\delta=2$ in (b)–(d). The shallower $y^{0.39}$ rise for $\phi$ indicates enhanced infrared sensitivity of light–quark dynamics, whereas heavy quarkonia and elastic $pp$ retain the full $y^{2}$ growth expected from the high-energy eikonal picture. Data sources: $\phi$ from Refs. Egloff:1979mgBusenitz:1989gq; $J/\psi$ and $\Upsilon$ from the compilation in Ref. Mamo:2019mka; elastic $pp$ from Refs. TOTEM:2018hkiParticleDataGroup:2022pthATLAS:2022mgx.
• Figure S1: Region $A$ is bounded by the entangling surface $\Sigma_{\perp}$ (the inner circle is a 1D illustration of the 2D surface); its complement is region $B$. The arrow denotes the radial coordinate $r$ from the center of $A$ to $\Sigma_{\perp}$ localized at $r=R=R_{\!EE}$.