Three-body final state interactions in $B^+\to D\bar{D}K^+$ decays

Abstract

This paper presents a detailed analysis of the three-body final state interactions in the $B^+\to D\bar{D}K^+$ process, whose phase space is sufficient small. To precisely extract the resonance parameters, for instance the $χ_{c0/2}(3930)$ in the $D\bar{D}$ invariant mass distributions, in this process, one has to take into account final state interaction, especially the three-body final state interaction. We employ the dispersive Khuri-Treiman formalism, combined with a parameterization of the $D\bar{D}$ interaction based on Heavy Quark Spin Symmetry. By performing a simultaneous fit to the experimental data from LHCb, BaBar, and Belle collaborations, the scheme with three-body interaction successfully describes the invariant mass distributions of the three two-body subsystems. We precisely extract the pole structures of $χ_{c0}(3930)$ and $ψ(3770)$ as $3.913-0.018i~\mathrm{GeV}$ and $3.764-0.002i~\mathrm{GeV}$ in $B^+$ decay. By performing the pole trajectory analysis on a uniformized complex plane, we find that both of them stem from the input bare state.

Figures (9)

• Figure 1: The $B^+\to D\Bar{D}K^+$ decay reaction. (a) the contribution of $D\Bar{D}$ rescattering with right-hand cut; (b) the contribution of the resonance in the $\Bar{D}K^+$ channel projecting to $D\Bar{D}$ channel; (c) the contribution of the resonance in the $\Bar{D}K^+$ channel.
• Figure 2: The dependence of $t_{\pm}$ on the variation of $s+i\epsilon$ on $t$-plane, with the arrow indicating the $s$ increasing direction. Here we assign a small positive imaginary part $\epsilon$ to $s$. The blue solid and red dashed curves are for $t_+$ and $t_-$, respectively. The red point is the singularity position of $a_{0,2}^{(t)}(t)$.
• Figure 3: The left and right pictures are the real part and imaginary part of $b_{l,2}^{(s)}$, respectively. The blue, red and green curves are for $l=0,1,2$. The orange vertical dashed lines represent the positions of $(m_B-m_K)^2$ and $(m_B+m_K)^2$.
• Figure 4: The n loop rescattering Feynman diagram for $D$ and $\bar{D}$. The black dots denote the potentials from both contact and bare state.
• Figure 5: The invariant mass spectrum of $D\bar{D}$, $\bar{D}K$, and $DK$. The first line are the fit result of $B^+\to D^0\bar{D}^0K^+$, where we plot the experiment result of $BABAR$ and Belle, which are represented in red and blue hollow point; The second line are the result of $B^+\to D^+D^-K^+$, the red hollow point come from LHCb experiment; The third line are the result of $B^+\to D_s^+D_s^-K^+$ channel, the red hollow point are experiment values come from LHCb experiment. And the black dashed and green solid step lines are the fit result of scheme II and scheme I, respectively.