Entanglement entropy for $π^+ p$ elastic scattering using spin-density matrix

TL;DR

This work analyzes π^+ p elastic scattering within an effective Lagrangian framework that includes $s$-, $t$-, and $u$-channel contributions from $Δ^{++}$, $Δ^{0}$, neutron, and $ρ^0$ intermediates. By constructing spin-density matrices in the canonical spin basis and performing partial traces, it computes the von Neumann entropies $S_{ ext{SDM}}^{i}$ and $S_{ ext{SDM}}^{f}$ to quantify spin entanglement with other degrees of freedom. The results show a nontrivial energy and angular dependence of $S_{ ext{SDM}}$, with relatively low entropy in the $ ext{Δ}^{++}$-dominated region and increased entanglement when background amplitudes interfere, highlighting the role of coherence and decoherence in hadronic amplitudes. This quantum-information perspective complements traditional cross-section analyses and points to potential experimental reconstruction of SDMs from polarization observables, offering a new diagnostic for resonance structure and nonperturbative QCD dynamics.

Abstract

We study the $π^+ p$ elastic scattering process using an effective Lagrangian approach that incorporates the $s$-, $u$-, and $t$-channel amplitudes, including $Δ^{++}(1232)$, $Δ^{0}(1232)$, neutron, and $ρ^0$ contributions. By constructing the spin-density matrices from the scattering amplitudes, we derive the von Neumann (entanglement) entropy associated with the spin degrees of freedom of the initial and final state particles. We compute the entropy by performing a partial trace over the spin subsystems, and its behavior is analyzed as a function of the center-of-mass energy $W$ and scattering angle $θ$. We find that entropy exhibits nontrivial angular and energy dependences, reflecting the interplay among resonance contributions and background amplitudes. In particular, the $Δ^{++}$ region shows relatively low entanglement, suggesting spin coherence dominated by the specific resonant contribution, while the background contributions increase entropy due to mixed spin configurations. These results indicate that entanglement entropy can serve as a novel probe into the quantum structure of hadronic scattering amplitudes, providing complementary insights beyond conventional cross-section analyses.

Figures (5)

• Figure 1: Relevant Feynman diagrams for $\pi^+p$ elastic scattering: (a) $\Delta^{++}$-pole diagram in the $s$ channel, (b) $\rho^0$-exchange in the $t$ channel, and (c) $n$ and $\Delta^0$ exchanges in the $u$ channel. The four momenta for the involved particles are also defined by $k_{1,2,3,4}$.
• Figure 2: (Color online) (a) Total cross-section for the $\pi^+p$ elastic scattering process as a function of c.m. energy $W$ [MeV]. We also show separate contributions and experimental data from PDG and SINR PDGplotPedroni:1978it. (b) Angular-dependent differential cross-section as a function of the scattering angle $\theta$ of the outgoing $\pi^+p$ in the c.m. frame at $W\approx1.285$ GeV with the experimental data from Ref. Bussey:1973gz. (c) Differential cross-section as a function of $W$ and $\cos\theta$.
• Figure 3: Real and imaginary matrix elements of $\rho_\mathrm{SDM}$ in separate panels as functions of $W=(W_\mathrm{th}-2.0)$ GeV and $|\cos\theta|\leq1$ for the horizontal and vertical axes, respectively. $W_\mathrm{th}=M_{\pi^+}+M_p$ denotes the c.m. threshold energy.
• Figure 4: (Color online) von Neumann entropy for the SDM, i.e., $S_\mathrm{SDM}$ as a function of $W$ [MeV] and $\cos\theta$, including all the contributions, $\Delta^{++}$, $\Delta^0$, $n$, and $\rho^0$.
• Figure 5: (Color online) Similar to Fig. \ref{['FIG4']}; $S_\mathrm{SDM}$ as a function of $W$ [MeV] and $\cos\theta$ for the (a) $\Delta^{++}+\Delta^{0}$, (b) $\Delta^{++}+n$, (c) $\Delta^{++}+\rho^0$, (d) $\Delta^0+n$, (e) $\Delta^0+\rho^0$, and (f) $n+\rho^0$ contributions.