Revisiting $$B_s \rightarrow PP$$ and PV decays with contributions from $${\phi }_{B2}$$ with the perturbative QCD approach

Abstract The $$B_s \rightarrow PP$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>s</mml:mi> </mml:msub> <mml:mo>→</mml:mo> <mml:mi>P</mml:mi> <mml:mi>P</mml:mi> </mml:mrow> </mml:math> and PV decay modes are revisited at leading order within the perturbative QCD (pQCD) approach, incorporating the $$B_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> mesonic wave function (WF) $${\phi }_{B2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mi>B</mml:mi> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> . Here, P represents the pseudoscalar mesons $${\pi }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>π</mml:mi> </mml:math> and K , while V denotes the ground-state vector mesons. The investigation incorporates two key refinements: the contribution of the subleading-twist WF $$\phi _{B2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mi>B</mml:mi> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> of the $$B_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> meson and the effects of higher-order terms in the distribution amplitudes (DAs) of the final-state mesons. We emphasize that $$\phi _{B2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mi>B</mml:mi> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> is not merely a numerical correction, but a necessary ingredient to preserve gauge invariance. Employing the minimum $$\chi ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>χ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> method, we optimize the shape parameter $$\omega _{B_s}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ω</mml:mi> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:msub> </mml:math> of the $$B_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> meson WF and systematically calculate the branching ratios and CP violation parameters for these decay modes. Our results demonstrate that the inclusion of $$\phi _{B2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mi>B</mml:mi> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> significantly impacts both the branching ratios and CP asymmetries, resulting in better agreement with existing experimental data for specific channels. This underscores the need to account for $$\phi _{B2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mrow> <mml:mi>B</mml:mi> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> </mml:math> in theoretical studies of $$B_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> weak decays. While the higher-order corrections in the final-state meson DAs yield comparatively smaller effects, they still enhance the theoretical predictions. These findings highlight the importance of refining both WF modeling and higher-order contributions in pQCD calculations. Future high-precision experimental measurements will further test these predictions, while continued theoretical efforts are essential for exploring additional interaction mechanisms and systematic uncertainties. The interplay between experimental advancements and theoretical improvements remains critical for a deeper understanding of $$B_s$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>s</mml:mi> </mml:msub> </mml:math> meson decay dynamics.