The BABAR Collaboration data of the $D^0\to K_S^0{K}^{+}{K}^{-}$ process are analyzed within a quasi-two-body factorization framework. Starting from the weak effective Hamiltonian, one has to evaluate matrix elements of $D^0$ transitions to two kaons for the tree amplitudes and the transitions between one kaon and two kaons for the annihilation ones ($W$-exchange). In earlier studies, assuming these transitions to proceed through the dominant intermediate resonances, we approximated them as being proportional to the kaon form factors. Here, to obtain a good fit, one has to multiply the scalar-kaon form factors, derived from unitary relativistic coupled-channel models or in a dispersion relation approach, by phenomenological energy-dependent functions. The final state kaon-kaon interactions in the $S$-, $P$-, and $D$- waves are taken into account. All $S$-wave channels are treated in a unitary way. In other respects, it is shown in a model-independent manner that the $K^{+}{K}^{-}$ and ${\overline{K}}^0{K}^{+}S$-wave effective mass squared distributions, corrected for phase space, are significantly different. At variance with the BABAR analysis, it means that the $f_0(980)$ resonance must be included in the phenomenological analysis of the $D^0\to K_S^0{K}^{+}{K}^{-}$ data. The best fit described in the main text has 19 free parameters and indicates (i) the dominance of annihilation amplitudes, (ii) a large dominance of the $f_0(980)$ meson in the near threshold $K^{+}{K}^{-}$ invariant mass distribution, and (iii) a sizable branching fraction to the $[\rho (770{)}^{+}+\rho (1450{)}^{+}+\rho (1700{)}^{+}]K_S^0$ final states. A first Appendix provides an update of the determination of the isoscalar-scalar meson-meson amplitudes based on an enlarged set of data embodying new precise low energy $\pi \pi$ data. A second Appendix proposes two alternative fits using the scalar-kaon form factors calculated from the Muskhelishvili-Omnès dispersion relation approach. These fits have ${\chi }^2$ quite close to that of the best fit but they show important contributions from both the $f_0$ and $a_0^0$ mesons and a weaker role of the ${\rho }^{+}$ mesons.

• Received 7 May 2021
• Accepted 24 May 2021

DOI:https://doi.org/10.1103/PhysRevD.103.114028

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Published by the American Physical Society