Based on the perturbative quantum chromodynamics (pQCD) approach and the quasi-two-body approximation, we have studied the three-body decays $$B^{0}_{s}\rightarrow \psi (3686,3770)K \pi $$ , which include the contributions of the intermediate resonances $$\overline{K }^{*}_{0}(1430)^{0}$$ , $$\overline{K }^{*}(892)^{0}$$ , $$\overline{K }^{*}(1410)^{0}$$ , $$\overline{K }^{*}(1680)^{0}$$ , and $$\overline{K }^{*}_{2}(1430)^{0}$$ . The time-like form factors corresponding to the distribution amplitudes of the S-, P-, and D-wave of the kaon–pion pair were adopted in parameterized form, and describe the interactions between $$K $$ and $$\pi $$ in the resonance region. First, the decays $$B^{0}_{s}\rightarrow \psi (2S,1D)K ^{-}\pi ^{+}$$ were calculated, followed by the calculation of the branching ratios of the decays $$B^{0}_{s}\rightarrow \psi (3686,3770)K ^{-}\pi ^{+}$$ using the 2S–1D mixing scheme. In addition, the pQCD predictions for the decays $$B^{0}_{s}\rightarrow \psi (2S,1D)K \pi $$ and $$B^{0}_{s}\rightarrow \psi (3686,3770)K \pi $$ were obtained using the narrow-width approximation relation given by the Clebsch–Gordan coefficients. Our work shows that the $$\overline{K }^{*}(892)^{0}$$ resonance is the main contributor to the total decay, and the branching ratio and the longitudinal polarization fraction of the $$\psi (2S)\overline{K }^{*}(892)^{0}$$ decay mode agree well with the currently available data within errors. Furthermore, the theoretical predictions of the $$\psi (2S)$$ and $$\psi (3686)$$ decay modes are very close, indicating that they can be regarded as the same meson state. Finally, the pQCD predictions for branching ratios of decays $$B^{0}_{s}\rightarrow \psi (3686,3770)K \pi $$ are of the order of $$10^{-5}$$ and $$10^{-6}$$ , respectively, which can be verified using the ongoing LHCb and Belle II experiments.
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"title": "Contributions of S-, P-, and D-wave resonances to the quasi-two-body decays $$B^{0}_{s} \\rightarrow \\psi (3686,3770)K \\pi $$ <math> <mrow> <msubsup> <mi>B</mi> <mi>s</mi> <mn>0</mn> </msubsup> <mo>\u2192</mo> <mi>\u03c8</mi> <mrow> <mo>(</mo> <mn>3686</mn> <mo>,</mo> <mn>3770</mn> <mo>)</mo> </mrow> <mi>K</mi> <mi>\u03c0</mi> </mrow> </math> in the perturbative QCD approach"
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"value": "Based on the perturbative quantum chromodynamics (pQCD) approach and the quasi-two-body approximation, we have studied the three-body decays $$B^{0}_{s}\\rightarrow \\psi (3686,3770)K \\pi $$ <math> <mrow> <msubsup> <mi>B</mi> <mi>s</mi> <mn>0</mn> </msubsup> <mo>\u2192</mo> <mi>\u03c8</mi> <mrow> <mo>(</mo> <mn>3686</mn> <mo>,</mo> <mn>3770</mn> <mo>)</mo> </mrow> <mi>K</mi> <mi>\u03c0</mi> </mrow> </math> , which include the contributions of the intermediate resonances $$\\overline{K }^{*}_{0}(1430)^{0}$$ <math> <mrow> <msubsup> <mover> <mi>K</mi> <mo>\u00af</mo> </mover> <mn>0</mn> <mrow> <mrow></mrow> <mo>\u2217</mo> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mn>1430</mn> <mo>)</mo> </mrow> <mn>0</mn> </msup> </mrow> </math> , $$\\overline{K }^{*}(892)^{0}$$ <math> <mrow> <msup> <mover> <mi>K</mi> <mo>\u00af</mo> </mover> <mrow> <mrow></mrow> <mo>\u2217</mo> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mn>892</mn> <mo>)</mo> </mrow> <mn>0</mn> </msup> </mrow> </math> , $$\\overline{K }^{*}(1410)^{0}$$ <math> <mrow> <msup> <mover> <mi>K</mi> <mo>\u00af</mo> </mover> <mrow> <mrow></mrow> <mo>\u2217</mo> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mn>1410</mn> <mo>)</mo> </mrow> <mn>0</mn> </msup> </mrow> </math> , $$\\overline{K }^{*}(1680)^{0}$$ <math> <mrow> <msup> <mover> <mi>K</mi> <mo>\u00af</mo> </mover> <mrow> <mrow></mrow> <mo>\u2217</mo> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mn>1680</mn> <mo>)</mo> </mrow> <mn>0</mn> </msup> </mrow> </math> , and $$\\overline{K }^{*}_{2}(1430)^{0}$$ <math> <mrow> <msubsup> <mover> <mi>K</mi> <mo>\u00af</mo> </mover> <mn>2</mn> <mrow> <mrow></mrow> <mo>\u2217</mo> </mrow> </msubsup> <msup> <mrow> <mo>(</mo> <mn>1430</mn> <mo>)</mo> </mrow> <mn>0</mn> </msup> </mrow> </math> . The time-like form factors corresponding to the distribution amplitudes of the S-, P-, and D-wave of the kaon\u2013pion pair were adopted in parameterized form, and describe the interactions between $$K $$ <math> <mi>K</mi> </math> and $$\\pi $$ <math> <mi>\u03c0</mi> </math> in the resonance region. First, the decays $$B^{0}_{s}\\rightarrow \\psi (2S,1D)K ^{-}\\pi ^{+}$$ <math> <mrow> <msubsup> <mi>B</mi> <mi>s</mi> <mn>0</mn> </msubsup> <mo>\u2192</mo> <mi>\u03c8</mi> <mrow> <mo>(</mo> <mn>2</mn> <mi>S</mi> <mo>,</mo> <mn>1</mn> <mi>D</mi> <mo>)</mo> </mrow> <msup> <mi>K</mi> <mo>-</mo> </msup> <msup> <mi>\u03c0</mi> <mo>+</mo> </msup> </mrow> </math> were calculated, followed by the calculation of the branching ratios of the decays $$B^{0}_{s}\\rightarrow \\psi (3686,3770)K ^{-}\\pi ^{+}$$ <math> <mrow> <msubsup> <mi>B</mi> <mi>s</mi> <mn>0</mn> </msubsup> <mo>\u2192</mo> <mi>\u03c8</mi> <mrow> <mo>(</mo> <mn>3686</mn> <mo>,</mo> <mn>3770</mn> <mo>)</mo> </mrow> <msup> <mi>K</mi> <mo>-</mo> </msup> <msup> <mi>\u03c0</mi> <mo>+</mo> </msup> </mrow> </math> using the 2S\u20131D mixing scheme. In addition, the pQCD predictions for the decays $$B^{0}_{s}\\rightarrow \\psi (2S,1D)K \\pi $$ <math> <mrow> <msubsup> <mi>B</mi> <mi>s</mi> <mn>0</mn> </msubsup> <mo>\u2192</mo> <mi>\u03c8</mi> <mrow> <mo>(</mo> <mn>2</mn> <mi>S</mi> <mo>,</mo> <mn>1</mn> <mi>D</mi> <mo>)</mo> </mrow> <mi>K</mi> <mi>\u03c0</mi> </mrow> </math> and $$B^{0}_{s}\\rightarrow \\psi (3686,3770)K \\pi $$ <math> <mrow> <msubsup> <mi>B</mi> <mi>s</mi> <mn>0</mn> </msubsup> <mo>\u2192</mo> <mi>\u03c8</mi> <mrow> <mo>(</mo> <mn>3686</mn> <mo>,</mo> <mn>3770</mn> <mo>)</mo> </mrow> <mi>K</mi> <mi>\u03c0</mi> </mrow> </math> were obtained using the narrow-width approximation relation given by the Clebsch\u2013Gordan coefficients. Our work shows that the $$\\overline{K }^{*}(892)^{0}$$ <math> <mrow> <msup> <mover> <mi>K</mi> <mo>\u00af</mo> </mover> <mrow> <mrow></mrow> <mo>\u2217</mo> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mn>892</mn> <mo>)</mo> </mrow> <mn>0</mn> </msup> </mrow> </math> resonance is the main contributor to the total decay, and the branching ratio and the longitudinal polarization fraction of the $$\\psi (2S)\\overline{K }^{*}(892)^{0}$$ <math> <mrow> <mi>\u03c8</mi> <mrow> <mo>(</mo> <mn>2</mn> <mi>S</mi> <mo>)</mo> </mrow> <msup> <mover> <mi>K</mi> <mo>\u00af</mo> </mover> <mrow> <mrow></mrow> <mo>\u2217</mo> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mn>892</mn> <mo>)</mo> </mrow> <mn>0</mn> </msup> </mrow> </math> decay mode agree well with the currently available data within errors. Furthermore, the theoretical predictions of the $$\\psi (2S)$$ <math> <mrow> <mi>\u03c8</mi> <mo>(</mo> <mn>2</mn> <mi>S</mi> <mo>)</mo> </mrow> </math> and $$\\psi (3686)$$ <math> <mrow> <mi>\u03c8</mi> <mo>(</mo> <mn>3686</mn> <mo>)</mo> </mrow> </math> decay modes are very close, indicating that they can be regarded as the same meson state. Finally, the pQCD predictions for branching ratios of decays $$B^{0}_{s}\\rightarrow \\psi (3686,3770)K \\pi $$ <math> <mrow> <msubsup> <mi>B</mi> <mi>s</mi> <mn>0</mn> </msubsup> <mo>\u2192</mo> <mi>\u03c8</mi> <mrow> <mo>(</mo> <mn>3686</mn> <mo>,</mo> <mn>3770</mn> <mo>)</mo> </mrow> <mi>K</mi> <mi>\u03c0</mi> </mrow> </math> are of the order of $$10^{-5}$$ <math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>5</mn> </mrow> </msup> </math> and $$10^{-6}$$ <math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>6</mn> </mrow> </msup> </math> , respectively, which can be verified using the ongoing LHCb and Belle II experiments."
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