$P_c(4457)\to P_c(4312)\pi /\gamma$ in the molecular picture

The three pentaquark states, $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$, discovered by the LHCb Collaboration in 2019, can be nicely arranged into a multiplet of ${\overline{D}}^{(*)}{\mathrm{\Sigma }}_c^{(*)}$ of seven molecules dictated by heavy quark spin symmetry. In this work we employ the effective Lagrangian approach to investigate the two decay modes of $P_c(4457)$, $P_c(4457)\to P_c(4312)\pi$, and $P_c(4457)\to P_c(4312)\gamma$, via the triangle mechanism, assuming that $P_c(4457)$ and $P_c(4312)$ are ${\overline{D}}^{*}{\mathrm{\Sigma }}_c$ and $\overline{D}{\mathrm{\Sigma }}_c$ bound states but the spin of $P_c(4457)$ can be either $1/2$ or $3/2$. Our results show that the spin of $P_c(4457)$ can not be discriminated through these two decay modes. The decay widths of $P_c(4457)\to P_c(4312)\pi$ and $P_c(4457)\to P_c(4312)\gamma$ are estimated to be of order of 100 keV and 1 keV, respectively. The ratio of the partial decay widths of $P_c(4457)\to P_c(4312)\pi$ to $P_c(4457)\to P_c(4312)\gamma$ is similar to the ratio of $D^{*}\to D\pi$ to $D^{*}\to D\gamma$, which could be used to check the molecular nature of $P_c(4457)$ and $P_c(4312)$ if they can be observed in the future.

Figure 1

Mass splittings between $P_c(4457)$, $P_c(4440)$, and $P_c(4312)$.

Figure 2

Triangle diagram of pionic and radiative decay of $P_c(4457)$ to $P_c(4312)$ with the spin of $P_c(4457)$ being either $1/2$ or $3/2$.

Figure 3

Mass operators of $P_c(4312)$ and $P_c(4457)$ as $\overline{D}{\mathrm{\Sigma }}_c$ and ${\overline{D}}^{*}{\mathrm{\Sigma }}_c$ bound states.

Figure 4

Contact diagram of the radiative decay of $P_c(4457)$ to $P_c(4312)$ with the spin of $P_c(4457)$ being either $1/2$ or $3/2$.

Figure 5

Decay widths of $P_c(4457{)}^{1/2}\to P_c(4312)\pi$ (a) and $P_c(4457{)}^{3/2}\to P_c(4312)\pi$ (b) as a function of the cutoff.

Figure 6

Ratio of the partial decay widths of $P_c(4457{)}^{3/2}\to P_c(4312)\pi$ to $P_c(4457{)}^{1/2}\to P_c(4312)\pi$ as a function of the cutoff.

Figure 7

Decay width of $P_c(4457{)}^{1/2}\to P_c(4312)\gamma$ (a) and $P_c(4457{)}^{3/2}\to P_c(4312)\gamma$ (b) as a function of the cutoff.

Figure 8

Ratio of the decay width of $P_c(4457{)}^{3/2}\to P_c(4312)\gamma$ to $P_c(4457{)}^{1/2}\to P_c(4312)\gamma$ as a function of the cutoff.

Figure 9

Ratio of the radiative decay width and the pionic decay width of $P_c(4457)$ with spin 1/2(a) and 3/2(b) as a function of the cutoff.