Quasi-two-body decays $B^+\to D_s^+ (R\to) K^+K^-$ in the perturbative QCD approach
A search for the decay $B^+\to D_s^+ K^+K^-$ has been reported by the LHCb Collaboration using $pp$ collision data corresponding to an integrated luminosity of $4.8\,\mathrm{fb}^{-1}$, collected at center-of-mass energies of 7, 8, and 13~TeV, in which no amplitude analysis of the $K^+K^-$ subsystem was performed. In this work, we study the resonant contributions to the decay $B^+\to D_s^+ K^+K^-$ within the perturbative QCD (PQCD) factorization framework. Contributions from the $S$-wave resonances $f_0(980)$, $f_0(1370)$, and $f_0(1500)$, the $P$-wave resonance $ϕ(1020)$, and the $D$-wave resonances $f_2(1270)$ and $f_2(1525)$ are taken into account. By introducing the corresponding two-meson distribution amplitudes for the $K^+K^-$ system, we perform a complete perturbative analysis of the quasi-two-body decays $B^+\to D_s^+(R\to)K^+K^-$, where $R$ denotes an intermediate resonance, and present the first PQCD predictions for the associated branching fractions. Using the narrow-width approximation, we further extract the branching fractions of the corresponding two-body decays $B^+\to D_s^+R$. Our results are consistent with the available experimental measurements and previous theoretical studies. Finally, we find that direct CP asymmetries vanish for these quasi-two-body decays within the Standard Model, so that any experimentally observed nonzero CP asymmetry would constitute a clear signal of physics beyond the Standard Model.