Bypassing spin-analyzing power dependence for quantum entanglement at colliders: a case study of $$ \Lambda \overline{\Lambda} $$

A bstract We study, as a concrete case study using the $$ \Lambda \left(\to p{\pi}^{-}\right)\overline{\Lambda}\left(\to \overline{p}{\pi}^{+}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> <mml:mfenced> <mml:mrow> <mml:mo>→</mml:mo> <mml:mi>p</mml:mi> <mml:msup> <mml:mi>π</mml:mi> <mml:mo>−</mml:mo> </mml:msup> </mml:mrow> </mml:mfenced> <mml:mover> <mml:mi>Λ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mfenced> <mml:mrow> <mml:mo>→</mml:mo> <mml:mover> <mml:mi>p</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:msup> <mml:mi>π</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> </mml:mfenced> </mml:math> system, whether quantum entanglement in fermion pairs produced at colliders can be certified solely using angular information from final-state decays, while remaining independent of the parity-violating decay parameters α Λ and $$ {\alpha}_{\overline{\Lambda}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mover> <mml:mi>Λ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:msub> </mml:math> . Building on a general decomposition of any angular observable in terms of Wigner d-functions, we show that the expectation value must take the form $$ {\mathcal{O}}_0+{\mathcal{O}}_1{\alpha}_{\Lambda}+{\mathcal{O}}_2{\alpha}_{\overline{\Lambda}}+{\mathcal{O}}_3{\alpha}_{\Lambda}{\alpha}_{\overline{\Lambda}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>O</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>Λ</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>O</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:msub> <mml:mi>α</mml:mi> <mml:mover> <mml:mi>Λ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:msub> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>O</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>Λ</mml:mi> </mml:msub> <mml:msub> <mml:mi>α</mml:mi> <mml:mover> <mml:mi>Λ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:msub> </mml:math> , with coefficients 𝒪 i ( i = 0 , 1 , 2 , 3) linear in the spin-density matrix elements $$ {\alpha}_{k,j}{\alpha}_{m,n}^{\ast } $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:msubsup> <mml:mi>α</mml:mi> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:mo>∗</mml:mo> </mml:msubsup> </mml:math> . We obtain the value ranges of observables over the general and separable spaces of α k,j , and demonstrate a sufficient entanglement condition for pure states, extending it to mixed states by convexity. In constructing an α Λ - and $$ {\alpha}_{\overline{\Lambda}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mover> <mml:mi>Λ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:msub> </mml:math> -independent witness from angular observables alone, we find that there are obstacles to probe quantum entanglement via the inequality-type and ratio-type ways. In particular, for the ratio-type criterion 〈 A 〉/〈 B 〉, the presence of zeros of 〈 B 〉 in both the general and separable spaces of $$ {\alpha}_{k,j}\left(k,j=\pm \frac{1}{2}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>α</mml:mi> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> </mml:mrow> </mml:msub> <mml:mfenced> <mml:mi>k</mml:mi> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mo>±</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:mfenced> </mml:math> </jats:in