Radius-Flow Entanglement in Hadron States and Gravitational Form Factors

Abstract

We propose a lattice-ready entanglement observable for QCD hadrons: the vacuum-subtracted radius flow of the ball Rényi entropy, $\mathfrak{s}_n(R;h)\equiv R\,\partial_RΔS_n(B_R;h)$, defined via the Euclidean replica cut-and-glue construction in a rest-frame momentum-projected one-hadron state, with spin averaging performed at the level of the final flow. In the continuum, varying $R$ at fixed shape is equivalent to a Weyl rescaling, so the flow is trace selected and admits a surface-plus-remainder organization on the entangling sphere. We use this to formulate a lattice stability test of boundary dominance: fit the measured flow on local $R$ windows to a low-curvature remainder plus a small template basis built from hadronic gravitational form factors (GFFs). The two endpoint templates are the spin-0/trace shape $\mathfrak{t}_h^{(0)}(R)=R^3ρ_S(R)$ constructed from $A^S(t)$ and a spin-2/TT proxy $\mathfrak{t}_h^{(2)}(R)=R^3ρ_A(R)$ constructed from $A(t)$, together with the mixed family $\mathfrak{t}_h^{\rm mix}(R;c_0,c_2)=c_0\mathfrak{t}_h^{(0)}(R)+c_2\mathfrak{t}_h^{(2)}(R)$. A soft-wall AdS/QCD appendix shows that the pole-subtracted integrated trace--energy correlator closes on this same $\{A^S,A\}$ basis and supplies a model-dependent benchmark ratio for $c_0/c_2$; for lattice comparison the coefficients are left free and extracted from data. For representative nucleon dipole inputs, the pure endpoints predict distinct single-extremum scales, $R_{\rm EE}^{(0)}\sim0.84~\mathrm{fm}$ and $R_{\rm EE}^{(2)}\sim0.43~\mathrm{fm}$, enabling discrimination among scalar control, spin-2 control, and genuine mixing through the turning-point location, the sign change of the slope across it, and the fitted ratio of template weights.

Figures (12)

• Figure 1: Cut-and-glue definition of the reduced density matrix kernel. At the Euclidean time cut $\tau=0$, the complement $\mathcal{B}$ is glued by imposing $\Phi(0^+,\mathcal{B})=\Phi(0^-,\mathcal{B})$ and integrating over $\Phi_{\mathcal{B}}$, while the region $\mathcal{A}$ is left open so that independent boundary data $\Phi_{\mathcal{A},+}$ and $\Phi_{\mathcal{A},-}$ live on the two sides of the cut. The resulting path integral defines $\rho_{\mathcal{A}}[\Phi_{\mathcal{A},+},\Phi_{\mathcal{A},-}]$ and is the basic building block for the replica construction of $\mathop{\mathrm{Tr}}\nolimits(\rho_{\mathcal{A}}^n)$ illustrated in Fig. \ref{['fig:replica-gluing-renyi']}.
• Figure 2: Spherical region $\mathcal{A}=B_R$ bounded by the entangling surface $\Sigma=\partial\mathcal{A}$ (inner circle is a 1D illustration of the 2D surface).
• Figure 3: Replica gluing for Rényi entropies (shown for $n=2$). Start from the cut-and-glue kernel on each replica sheet (Fig. \ref{['fig:cut-glue']}): fields are glued across $\tau=0$ on $\mathcal{B}$ within each sheet, while the open cut on $\mathcal{A}$ carries independent boundary data $\Phi_{\mathcal{A},\pm}^{(j)}$. To form $\mathop{\mathrm{Tr}}\nolimits(\rho_{\mathcal{A}}^{\,n})$, glue the $\mathcal{A}$ edges cyclically across sheets, $\Phi_{\mathcal{A},-}^{(j)}=\Phi_{\mathcal{A},+}^{(j+1)}$ (with $j=n$ glued back to $j=1$), producing the $n$-sheeted manifold $\mathcal{M}_n$ and yielding $\mathop{\mathrm{Tr}}\nolimits(\rho_{\mathcal{A}}^{\,n})=Z_n/(Z_1)^n$ and $S_n=(1-n)^{-1}\ln\mathop{\mathrm{Tr}}\nolimits(\rho_{\mathcal{A}}^{\,n})$; the von Neumann entropy follows by analytic continuation $n\to1$.
• Figure 4: Spin-0 and spin-2 endpoint templates and bulk proxies. Top: the spin-0/trace endpoint $\mathfrak t_h^{(0)}(R)=R^3\rho_{S}(R)$ and the spin-2 endpoint $\mathfrak t_h^{(2)}(R)=R^3\rho_{A}(R)$, together with the interior proxies from \ref{['eq:cbulk-def']}. Bottom: their derivatives. The surface-evaluated endpoints have a single pronounced extremum, whereas the interior proxies remain smooth and monotone for positive dipole densities. These endpoint curves (shown here for the representative dipole inputs used in this work) form the basis of the mixed family $\mathfrak t_h^{\rm mix}(R;c_0,c_2)$.
• Figure 5: Surface-plus-bulk distortions of the endpoint templates. Top: the spin-0 and spin-2 surface-evaluated endpoints together with corresponding surface-plus-bulk proxies. Bottom: their derivatives. Adding a smooth bulk component can shift the first extremum, but it does not generate new oscillatory structure. This is why a low-curvature remainder is not expected to fake a stable turning point associated with the wrong GFF channel. These plots are illustrative for the dipole inputs adopted here and are not intended as general theorems.