Bc meson decays into S-wave charmonium plus light meson pairs in the perturbative QCD approach

In this work, we explore the P -wave resonance contributions to the quasi-two-body charmonium decays of Bc→ Ψ (V→ ) P₁P₂ B c → Ψ ( V → ) P 1 P 2 using the perturbative QCD formalism at leading order, where Ψ Ψ denotes a S -wave charmonium state, such as η c(1S,2S),J/ψ η c ( 1 S , 2 S ) , J / ψ , and ψ (2S) ψ ( 2 S ) . Here, P₁P₂ P 1 P 2 represents a collinear π π π π ( Kπ K π ) pair in the final state, which was primarily produced through the vector resonance ρ (770) ρ ( 770 ) ( K^*(892) K ∗ ( 892 ) ). The contributions from the excited ρ (1450) ρ ′ ( 1450 ) , ρ (1700) ρ ″ ( 1700 ) and ρ (2254) ρ ″ ′