Radiative and dilepton decays of the hadronic molecule $Z_c^{+}(3900)$

The newly observed hidden-charm meson $Z_c^{+}(3900)$ with quantum numbers $J^P=1^{+}$ is considered as a hadronic molecule composed of $\overline{D}{D}^{*}$. We give detailed predictions for the decay modes $Z_c^{+}\to J/\psi {\pi }^{+}\gamma$ and $Z_c^{+}\to J/\psi {\pi }^{+}{\ell }^{+}{\ell }^{-}(\ell =e,\mu )$ in a phenomenological Lagrangian approach.

Figure 1

Mass operator for the $Z_c$.

Figure 2

Photon attached to the nonlocal vertex.

Figure 3

Notation for the kinematics of the process $Z_c^{+}\to J/\psi {\pi }^{+}\gamma$.

Figure 4

Notation for the kinematics of the process $Z_c^{+}\to J/\psi {\pi }^{+}\gamma [\to e^{+}{e}^{-}]$.

Figure 5

${\overline{D}}^{*0}{D}^{+}$ meson loop diagrams contributing to the three-body decay $Z_c^{+}\to J/\psi {\pi }^{+}\gamma$.

Figure 6

${\overline{D}}^0{D}^{*+}$ meson loop diagrams contributing to the three-body decay $Z_c^{+}\to J/\psi {\pi }^{+}\gamma$.

Figure 7

Choice of kinematical variables in the four-particle decay $Z_c^{+}\to J/\psi {\pi }^{+}{\ell }^{+}{\ell }^{-}$.

Figure 8

Double differential decay rate $d^2{\mathrm{\Gamma }}_3/d{s}_{12}d{s}_{31}$ in ${\mathrm{GeV}}^{-3}$ for the three-body decay $Z_c^{+}\to J/\psi {\pi }^{+}\gamma$ ($\epsilon =3\mathrm{MeV}$ and $\mathrm{\Lambda }=650\mathrm{MeV}$).

Figure 9

Double differential decay rate $d^2{\mathrm{\Gamma }}_3/d{s}_{12}d{s}_{31}$ in ${\mathrm{GeV}}^{-3}$ for the three-body decay $Z_c^{+}\to J/\psi {\pi }^{+}\gamma$ ($\epsilon =3\mathrm{MeV}$ and $\mathrm{\Lambda }=650\mathrm{MeV}$). The contour is nearly the same as in Fig. 8 except for $s_{31}\to m_{\pi }^2$ and $s_{12}\to m_{Z_c}^2$ where the bremsstrahlung singularity is located. This area is now excluded with an energy cut at $\delta m_{\gamma }=150\mathrm{MeV}$.

Figure 10

Partial decay rate ${\mathrm{\Gamma }}_3$ in ${\mathrm{MeV}}^{-1}$ as a function of $\delta m_{\gamma }$ for selected values of $\epsilon$ and $\mathrm{\Lambda }$ for the three-body decay $Z_c^{+}\to J/\psi {\pi }^{+}\gamma$.

Figure 11

Differential decay rate $d{\mathrm{\Gamma }}_3/d{s}_{31}$ in ${\mathrm{GeV}}^{-1}$ for selected values of $\epsilon$ and $\mathrm{\Lambda }$ for the three-body decay $Z_c^{+}\to J/\psi {\pi }^{+}\gamma$ with $\delta m_{\gamma }=150\mathrm{MeV}$.

Figure 12

Partial decay rate ${\mathrm{\Gamma }}_3$ in keV in dependence on $\epsilon$ and $\mathrm{\Lambda }$ for the three-body decay $Z_c^{+}\to J/\psi {\pi }^{+}\gamma$ with $\delta m_{\gamma }=150\mathrm{MeV}$.

Figure 13

Double differential decay rate $d^2{\mathrm{\Gamma }}_4/d{s}_{12}d{s}_{34}$ in ${\mathrm{GeV}}^{-3}$ for $\epsilon =3\mathrm{MeV}$ and $\mathrm{\Lambda }=650\mathrm{MeV}$ for the four-body decay $Z_c^{+}\to J/\psi {\pi }^{+}{e}^{+}{e}^{-}$.

Figure 14

Double differential decay rate $d^2{\mathrm{\Gamma }}_4/d{s}_{12}d{s}_{34}$ in ${\mathrm{GeV}}^{-3}$ for $\epsilon =3\mathrm{MeV}$ and $\mathrm{\Lambda }=650\mathrm{MeV}$ for the four-body decay $Z_c^{+}\to J/\psi {\pi }^{+}{\mu }^{+}{\mu }^{-}$.

Figure 15

Differential decay rate $d{\mathrm{\Gamma }}_4/d{s}_{34}$ in ${\mathrm{GeV}}^{-1}$ for selected values of $\epsilon$ and $\mathrm{\Lambda }$ for the four-body decay $Z_c^{+}\to J/\psi {\pi }^{+}{e}^{+}{e}^{-}$.

Figure 16

Differential decay rate $d{\mathrm{\Gamma }}_4/d{s}_{34}$ in ${\mathrm{GeV}}^{-1}$ for selected values of $\epsilon$ and $\mathrm{\Lambda }$ for the four-body decay $Z_c^{+}\to J/\psi {\pi }^{+}{\mu }^{+}{\mu }^{-}$.

Figure 17

Partial decay rate ${\mathrm{\Gamma }}_4$ in keV as a function of $\epsilon$ and $\mathrm{\Lambda }$ for the four-body decay $Z_c^{+}\to J/\psi {\pi }^{+}{e}^{+}{e}^{-}$.

Figure 18

Partial decay rate ${\mathrm{\Gamma }}_4$ in keV as a function of $\epsilon$ and $\mathrm{\Lambda }$ for the four-body decay $Z_c^{+}\to J/\psi {\pi }^{+}{\mu }^{+}{\mu }^{-}$.

Figure 19

Vertices contributing to the three-body decay $Z_c^{+}\to J/\psi {\pi }^{+}\gamma$.