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

We study the $\pi^+ p$ elastic scattering process using an effective Lagrangian approach that incorporates the $s$-, $u$-, and $t$-channel amplitudes, including $\Delta^{++}(1232)$, $\Delta^{0}(1232)$, neutron, and $\rho^0$ 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 $\theta$. We find that entropy exhibits nontrivial angular and energy dependences, reflecting the interplay among resonance contributions and background amplitudes. In particular, the $\Delta^{++}$ 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.