Comprehending isospin breaking effects of $X(3872)$ in a Friedrichs-model-like scheme

Recently, we have shown that the $X(3872)$ state can be naturally generated as a bound state by incorporating the hadron interactions into the Godfrey-Isgur quark model using a Friedrichs-like model combined with the quark pair creation model, in which the wave function for the $X(3872)$ as a combination of the bare $c\overline{c}$ state and the continuum states can also be obtained. Under this scheme, we now investigate the isospin-breaking effect of $X(3872)$ in its decays to $J/\psi {\pi }^{+}{\pi }^{-}$ and $J/\psi {\pi }^{+}{\pi }^{-}{\pi }^0$. By coupling its dominant continuum parts to $J/\psi \rho$ and $J/\psi \omega$ through the quark rearrangement process, one could obtain the reasonable ratio of $\mathcal{B}(X(3872)\to J/\psi {\pi }^{+}{\pi }^{-}{\pi }^0)/\mathcal{B}(X(3872)\to J/\psi {\pi }^{+}{\pi }^{-})\simeq (0.58–0.92)$. It is also shown that the $\overline{D}{D}^{*}$ invariant mass distributions in the $B\to \overline{D}{D}^{*}K$ decays could be understood qualitatively at the same time. This scheme may provide more insight into the enigmatic nature of the $X(3872)$ state.

Figure 1

Momentum definitions in the quark-quark transition amplitude.

Figure 2

The four-quark rearrangement diagrams of $AB\to CD$ meson-meson scatterings. The arrows represent the quark line directions.

Figure 3

Comparison of the integrand of Eq. (28) for charged and neutral channels without or with the denominators ($N$ or $N/D$) respectively. The solid line represents the neutral $\overline{D}{D}^{*}$ channel and the dashed one the charged channel. The left panel shows the contributions of the numerator only and the right panel shows the whole integrand with the denominators.

Figure 4

The experimental ${\overline{D}}^0{D}^{0*}$ mass distributions in $B\to X({\overline{D}}^0{D}^{0*})K$ decays measured by BABAR (left) [68] and Belle (middle and right) [67] compared with $r|T({\overline{D}}^0{D}^{0*}\to {\overline{D}}^0{D}^{0*}){|}^2·\text{phase space}$, where $r$ is a rescaling factor chosen by hand. The dotted, solid, and dashed lines represent the curves with the $X(3872)$ mass at 3.8710, 3.8714, and 3.8717 GeV, respectively.