Addressing the $S$-wave scalar $f_0(1500)$-resonance in quasi-four-body FCNC rare $$B_s \to f_0(1500) (\to π^+ π^- ) \ell^+ \ell^- / ν\bar ν$$ decays

Abstract: The $f_0(1500)$ is a quite special resonance in the light scalar spectrum. Compared with other nearby resonance, it has an unexpectedly narrow decay width. Since its mass region overlaps strongly with $f_0(1370)$ and $f_0(980)$, physicists have long debated its internal structure. To explore this issue, this work adopted the conventional quark-antiquark $q\bar{q}$ picture and investigated the behavior of the $f_0(1500)$-resonance in quasi-four-body decay channels. Based on this, we constructed a twist-2 light-cone distribution amplitude(LCDA) schemes based on the light-cone harmonic oscillator model, and presented their moments $$\langle ξⁿ <em>{2;f_0(1500)} \rangle |</em>μ$$ and Gegenbauer moments $a_{n;f_0(1500)}(μ)$ at $μ<em>0=1~\mathrm{GeV}$ and $μ_k= 3~\mathrm{GeV}$ for $n=1,3,5$. Meanwhile, the $B_s\to f_0(1500)$ transition form factors (TFFs) are calculated by using the QCD light-cone sum rule. Then, we obtained the three TFFs at large recoil point, {\it i.e.,} $$f</em> + ^{B_s f_0(1500)}(0)= 0.390_{-0.046}^{ + 0.047}$$, $$f_-^{B_s f_0(1500)}(0)= -0.460_{-0.055}^{ + 0.051}$$, and $$f_{\rm T}^{B_s f_0(1500)}(0)= 0.568^{ + 0.069}_{-0.065}$$. In addition, we extrapolated TFFs to the whole physical $q<sup>2$-region by using the simplified $z(q<sup>2)$-series expansion. Then we computed the branching fractions of the quasi-four-body rare decays $$B_s \to f_0(1500)(\to π^ + π^{-)\ell^} + \ell<sup>-$$ and $B_s \to f_0(1500)(\to π^ + π<sup>-)ν\barν$. For comparison, we also present the results of the corresponding three-body decays obtained under the narrow-width approximation. Finally, we show the distribution of the double-differential decay width $d^2Γ/ds dq<sup>2$ for the quasi-four-body decay $$B_s\to f_0(1500)(\toπ^ + π^{-)μ^} + μ<sup>-$$. We hope that our predictions can provide a useful theoretical reference for future experimental measurements and phenomenological research.