Coupled-channel effects of the ${\mathrm{\Sigma }}_c^{(*)}{\overline{D}}^{(*)}-{\mathrm{\Lambda }}_c(2595)\overline{D}$ system and molecular nature of the $P_c$ pentaquark states from one-boson exchange model

The effects of the ${\mathrm{\Sigma }}_c{\overline{D}}^{*}-{\mathrm{\Lambda }}_c(2595)\overline{D}$ coupled-channel dynamics and various one-boson-exchange (OBE) forces for the LHCb pentaquark states, $P_c(4440)$ and $P_c(4457)$, are reinvestigated. Both the pion and $\rho$-meson exchanges are considered for the ${\mathrm{\Sigma }}_c{\overline{D}}^{*}\text{−}{\mathrm{\Lambda }}_c(2595)\overline{D}$ coupled-channel dynamics. It is found that the role of the ${\mathrm{\Lambda }}_c(2595)\overline{D}$ channel in the descriptions of the $P_c(4440)$ and $P_c(4457)$ states is not significant with the OBE parameters constrained by other experimental sources. The naive OBE models with the short-distance $\delta (\overrightarrow{r})$ term of the one-pion exchange (OPE) keep failing to reproduce the $P_c(4440)$ and $P_c(4457)$ states simultaneously. The OPE potential with the full $\delta (\overrightarrow{r})$ term results in a too large mass splitting for the $J^P=1/2^{-}$ and $3/2^{-}{\mathrm{\Sigma }}_c{\overline{D}}^{*}$ bound states with total isospin $I=1/2$. The OBE model with only the OPE $\delta (\overrightarrow{r})$ term dropped may fit the splitting much better but somewhat underestimates the splitting. Since the $\delta (\overrightarrow{r})$ potential is from short-distance physics, which also contains contributions from the exchange of mesons heavier than those considered explicitly, we vary the strength of the $\delta (\overrightarrow{r})$ potential and find that the masses of the $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$ can be reproduced simultaneously with the $\delta (\overrightarrow{r})$ term in the OBE model reduced by about 80%. While two different spin assignments are possible to produce their masses, in the preferred description, the spin parities of the $P_c(4440)$ and $P_c(4457)$ are $3/2^{-}$ and $1/2^{-}$, respectively.

Figure 1

Central potential with different $a$ values.

Figure 2

Coupled-channel potentials for the $J^P=1/2^{+}$ and $J^P=3/2^{-}$ isodoublet systems with $\mathrm{\Lambda }=1\mathrm{GeV}$ and without the $\delta (\overrightarrow{r})$ term ($a=1$). (a) ${\mathrm{\Lambda }}_{c1}\overline{D}({}^2{S}_{1/2})\to {\mathrm{\Sigma }}_c\overline{D}({}^2{P}_{1/2})$ (b) ${\mathrm{\Lambda }}_{c1}\overline{D}({}^2{P}_{3/2})\to {\mathrm{\Sigma }}_c{\overline{D}}^{*}({}^4{S}_{3/2})$ (c) ${\mathrm{\Lambda }}_{c1}\overline{D}({}^2{S}_{1/2})\to {\mathrm{\Sigma }}_c^{*}{\overline{D}}^{*}({}^4{P}_{1/2})$ (d) ${\mathrm{\Lambda }}_{c1}\overline{D}({}^2{P}_{3/2})\to {\mathrm{\Sigma }}_c^{*}{\overline{D}}^{*}({}^4{S}_{3/2})$ (e) ${\mathrm{\Sigma }}_c{\overline{D}}^{*}({}^2{P}_{1/2})\to {\mathrm{\Sigma }}_c{\overline{D}}^{*}({}^2{P}_{1/2})$ (f) ${\mathrm{\Sigma }}_c{\overline{D}}^{*}({}^4{S}_{3/2})\to {\mathrm{\Sigma }}_c{\overline{D}}^{*}({}^4{S}_{3/2})$ (g) ${\mathrm{\Sigma }}_c{\overline{D}}^{*}({}^2{P}_{1/2})\to {\mathrm{\Sigma }}_c{\overline{D}}^{*}{}^2{P}_{1/2})$ (h) ${\mathrm{\Sigma }}_c{\overline{D}}^{*}({}^4{S}_{3/2})\to {\mathrm{\Sigma }}_c{\overline{D}}^{*}({}^4{S}_{3/2})$ (i) ${\mathrm{\Sigma }}_c^{*}{\overline{D}}^{*}({}^2{P}_{1/2})\to {\mathrm{\Sigma }}_c^{*}{\overline{D}}^{*}({}^2{P}_{1/2})$ (j) ${\mathrm{\Sigma }}_c^{*}{\overline{D}}^{*}({}^4{S}_{3/2})\to {\mathrm{\Sigma }}_c^{*}{\overline{D}}^{*}({}^4{S}_{3/2})$.

Figure 3

Diagonal $S$-wave potentials for the $J^P=1/2^{-}$ and $J^P=3/2^{-}$ isodoublet systems with $\mathrm{\Lambda }=1\mathrm{GeV}$. $a=1$ means without the $\delta (\overrightarrow{r})$ term. (a) $V_{11/22}$ potentials which are independent of spin and $a$. (b) $V_{44},J^P=1/2^{-},a=0$ (c) $V_{44},J^P=1/2^{-},a=1$ (d) $V_{44},J^P=3/2^{-},a=0$ (e) $V_{44},J^P=3/2^{-},a=1$ (f) $V_{55},J^P=1/2^{-},a=0$ (g) $V_{55},J^P=1/2^{-},a=1$ (h) $V_{55},J^P=3/2^{-},a=0$ (i) $V_{55},J^P=3/2^{-},a=1$.

Figure 4

Reduced wave functions $u(r)=rR(r)$ corresponding to the partial wave components considered in the single-channel analysis with $\mathrm{\Lambda }=1.6\mathrm{GeV}$ and $a=0.78$. The hadron pairs inside the parentheses are the corresponding channels. (a) $J^P=1/2^{-}({\mathrm{\Sigma }}_c\overline{D})$ (b) $J^P=3/2^{-}({\mathrm{\Sigma }}_c^{*}\overline{D})$ (c) $J^P=3/2^{-}({\mathrm{\Sigma }}_c{\overline{D}}^{*})$ (d) $J^P=1/2^{-}({\mathrm{\Sigma }}_c{\overline{D}}^{*})$ (e) $J^P=1/2^{-}({\mathrm{\Sigma }}_c^{*}{\overline{D}}^{*})$ (f) $J^P=3/2^{-}({\mathrm{\Sigma }}_c^{*}{\overline{D}}^{*})$ (g) $J^P=5/2^{-}({\mathrm{\Sigma }}_c^{*}{\overline{D}}^{*})$.

Figure 5

Coupled-channel results of isodoublet system ${\mathrm{\Sigma }}_c^{(*)}{\overline{D}}^{(*)}$ by varying the cutoff $\mathrm{\Lambda }$. Upper (lower) panel shows the mass (decay width through ${\mathrm{\Sigma }}_c\overline{D}$ and ${\mathrm{\Sigma }}_c^{*}\overline{D}$ channel) of corresponding states. Horizontal gray bands are representing the experimental uncertainties of $P_c$ masses, and vertical gray bands stand for the cutoff range where masses of $P_c$ states can be simultaneously fitted. Lines for the $J^P=3/2^{-}({\mathrm{\Sigma }}_c^{*}\overline{D})$ state are not shown, and its mass always lies in the experimental mass of $P_c(4380)$. (a) $a=0.55$ (b) $a=0.79$.

Figure 6

The $S$-wave potentials for the ${\mathrm{\Sigma }}_c{\overline{D}}^{*}$ channel with spin parities of $J^P=1/2^{-}$ and $J^P=3/2^{-}$, where the total potentials (summing up all light-meson-exchange potentials) are shown.