Three-body decay of ${\mathrm{\Lambda }}_c^{*}(2765)$ and determination of its spin-parity

We study three-body decays of ${\mathrm{\Lambda }}_c^{*}(2765)\to {\mathrm{\Lambda }}_c^{+}{\pi }^{+}{\pi }^{-}$ by using effective Lagrangians in a nonrelativistic framework. We consider the sequential decays through ${\mathrm{\Sigma }}_c(2455)\pi$ and ${\mathrm{\Sigma }}_c^{*}(2520)\pi$ in intermediate states which are dominant contributions. The coupling constants in the effective Lagrangians are computed in the quark model. We demonstrate that the ratio $R=\mathrm{\Gamma }({\mathrm{\Lambda }}_c^{*}\to {\mathrm{\Sigma }}_c^{*}(2520)\pi )/\mathrm{\Gamma }({\mathrm{\Lambda }}_c^{*}\to {\mathrm{\Sigma }}_c(2455)\pi )$ and angular correlations are sensitive to the spin and parity of ${\mathrm{\Lambda }}_c^{*}(2765)$. Thus, the measurement of these observables in experimental facilities such as Belle and LHCb can provide useful constraints to determine the spin and parity of ${\mathrm{\Lambda }}_c^{*}(2765)$.

Figure 1

Three-body decay of ${\mathrm{\Lambda }}_c^{*}(2765)$ into ${\mathrm{\Lambda }}_c^{+}{\pi }^{+}{\pi }^{-}$. Blue arrows represent sequential processes, and a red arrow corresponds to the nonsequential process.

Figure 2

Feynman diagrams of three-body decays of ${\mathrm{\Lambda }}_c^{*+}$ into ${\mathrm{\Lambda }}_c^{+}{\pi }^{+}{\pi }^{-}$. The first two diagrams represent sequential processes going through ${\mathrm{\Sigma }}_c^{(*)0}$ and ${\mathrm{\Sigma }}_c^{(*)++}$. The last diagram corresponds to the direct process.

Figure 3

Relative angle of two pions defined in the ${\mathrm{\Sigma }}_c^{*}$ resonance rest frame.

Figure 4

Resonance band (vertical solid line) on which angular correlations are studied for different combinations of invariant masses.

Figure 5

The Dalitz plots of ${\mathrm{\Lambda }}_c^{*}(2765)$ with mass 2715 (lower), 2765 (middle), and 2815 MeV (upper). The bands formed by intermediate states ${\mathrm{\Sigma }}_c^{(*)}$ are indicated in the figure as eye’s guides.

Figure 6

The Dalitz plots in the ($m_{23}^2,m_{12}^2)$ plane and the invariant mass plots of ${\mathrm{\Lambda }}_c^{+}{\pi }^{-}$. The spins and parities of ${\mathrm{\Lambda }}_c^{*}(2765)$ as $J^P=1/2^{\pm }$, $3/2^{\pm }$, and $5/2^{\pm }$, along with the corresponding brown muck spin $j$, are indicated in each panel. The Dalitz plots are made by fixing the initial mass at 2765 MeV.

Figure 7

The same as in Fig. 6 for the spins and parities of ${\mathrm{\Lambda }}_c^{*}(2765)$ as $1/2^{+}$ and $3/2^{+}$ with different configurations, and $7/2^{+}$.

Figure 8

The typical angular correlations along ${\mathrm{\Sigma }}_c^{*}$ resonance band where we consider only one of the ${\mathrm{\Sigma }}_c^{*}$ resonances appearing in the left diagram of Fig. 2 and neglect any interference terms. The spin and parity $J^P$ of ${\mathrm{\Lambda }}_c^{*}(2765)$ along with the brown muck spin $j$ are indicated in each figure.

Figure 9

The convoluted Dalitz and invariant mass plots for ${\mathrm{\Lambda }}_c^{*}(2765)$ with $J(j{)}^P=1/2(0{)}^{+}$.

Figure 10

The convoluted square Dalitz plots for ${\mathrm{\Lambda }}_c^{*}(2765)$ with $J(j{)}^P=1/2(0{)}^{+}$ which consider (a) only ${\mathrm{\Sigma }}_c^{(*)0}$, (b) ${\mathrm{\Sigma }}_c^{(*)0}$ and ${\mathrm{\Sigma }}_c^{(*)++}$, and (c) total amplitudes including interference terms. Note that we do not include any interference terms for (a) and (b). Their corresponding angular correlations along ${\mathrm{\Sigma }}_c^{*0}$ with a mass cut $M_{{\mathrm{\Sigma }}_c^{*0}}\pm {\mathrm{\Gamma }}_{{\mathrm{\Sigma }}_c^{*0}}$ are given in the bottom panel.