Revisiting $$\Xi _{Q}-\Xi _{Q}^{\prime }$$ mixing in QCD sum rules

Abstract

Abstract In this work, we perform a QCD sum rules analysis on $$\Xi _{Q}-\Xi _{Q}^{\prime }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>Ξ</mml:mi> <mml:mi>Q</mml:mi> </mml:msub> <mml:mo>-</mml:mo> <mml:msubsup> <mml:mi>Ξ</mml:mi> <mml:mrow> <mml:mi>Q</mml:mi> </mml:mrow> <mml:mo>′</mml:mo> </mml:msubsup> </mml:mrow> </mml:math> mixing. Contributions from up to dimension-6 four-quark operators are considered. However, it turns out that only dimension-4 and dimension-5 operators contribute, which reveals the non-perturbative nature of mixing. In particular, we observe that only the diagrams with the two light quarks participating in gluon exchange contribute to the mixing. Our results indicate that the mixing angle $$\theta _{c}=(1.2\sim 2.8)^{\circ }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>θ</mml:mi> <mml:mi>c</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1.2</mml:mn> <mml:mo>∼</mml:mo> <mml:mn>2.8</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∘</mml:mo> </mml:msup> </mml:mrow> </mml:math> for the $$Q=c$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>=</mml:mo> <mml:mi>c</mml:mi> </mml:mrow> </mml:math> case and $$\theta _{b}=(0.28\sim 0.34)^{\circ }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>θ</mml:mi> <mml:mi>b</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0.28</mml:mn> <mml:mo>∼</mml:mo> <mml:mn>0.34</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>∘</mml:mo> </mml:msup> </mml:mrow> </mml:math> for the $$Q=b$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:mo>=</mml:mo> <mml:mi>b</mml:mi> </mml:mrow> </mml:math> case. Our prediction of $$\theta _{c}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>θ</mml:mi> <mml:mi>c</mml:mi> </mml:msub> </mml:math> is consistent with the most recent Lattice QCD result within error. Such a small mixing angle seems unlikely to resolve the tension between the recent experimental measurement from Belle and Lattice QCD calculation for the semileptonic decay $$\Xi _{c}^{0}\rightarrow \Xi ^{-} e^{+}\nu _{e}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mi>Ξ</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msubsup> <mml:mo>→</mml:mo> <mml:msup> <mml:mi>Ξ</mml:mi> <mml:mo>-</mml:mo> </mml:msup> <mml:msup> <mml:mi>e</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>e</mml:mi> </mml:msub> </mml:mrow> </mml:math> .