Entanglement entropy for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>π</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:mi>p</mml:mi> </mml:mrow> </mml:math> elastic scattering using a spin-density matrix

The <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:mrow> <a:msup> <a:mi>π</a:mi> <a:mo>+</a:mo> </a:msup> <a:mi>p</a:mi> </a:mrow> </a:math> elastic scattering process is studied, using an effective Lagrangian approach that incorporates the <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mi>s</b:mi> </b:math> -, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mi>u</c:mi> </c:math> -, and <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"> <d:mi>t</d:mi> </d:math> -channel amplitudes, including <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"> <e:mrow> <e:msup> <e:mi mathvariant="normal">Δ</e:mi> <e:mrow> <e:mo>+</e:mo> <e:mo>+</e:mo> </e:mrow> </e:msup> <e:mrow> <e:mo>(</e:mo> <e:mn>1232</e:mn> <e:mo>)</e:mo> </e:mrow> </e:mrow> <e:mo>,</e:mo> <e:mo> </e:mo> <e:mrow> <e:msup> <e:mi mathvariant="normal">Δ</e:mi> <e:mn>0</e:mn> </e:msup> <e:mrow> <e:mo>(</e:mo> <e:mn>1232</e:mn> <e:mo>)</e:mo> </e:mrow> </e:mrow> </e:math> , neutron, and <h:math xmlns:h="http://www.w3.org/1998/Math/MathML"> <h:msup> <h:mi>ρ</h:mi> <h:mn>0</h:mn> </h:msup> </h:math> contributions. By constructing the spin-density matrices from the scattering amplitudes, the von Neumann (entanglement) entropy is derived and associated with the spin degrees of freedom of the initial and final state particles. The entropy is computed by performing a partial trace over the spin subsystems, and its behavior is analyzed as a function of the center-of-mass energy <i:math xmlns:i="http://www.w3.org/1998/Math/MathML"> <i:mi>W</i:mi> </i:math> and scattering angle <j:math xmlns:j="http://www.w3.org/1998/Math/MathML"> <j:mi>θ</j:mi> </j:math> . It is found that entropy exhibits nontrivial angular and energy dependences, reflecting the interplay among resonance contributions and background amplitudes. In particular, the <k:math xmlns:k="http://www.w3.org/1998/Math/MathML"> <k:msup> <k:mi mathvariant="normal">Δ</k:mi> <k:mrow> <k:mo>+</k:mo> <k:mo>+</k:mo> </k:mrow> </k:msup> </k:math> 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.