Role of the $N^*(1535)$ in the $\Lambda^+_c \to \bar{K}^0 \eta p$ decay

TL;DR

This paper investigates the role of the $N^*(1535)$ resonance in the decay $oldsymbol{ ext{Lambda}_c^+ o ar{K}^0 ext{eta} p}$ using two theoretical models, aiming to understand its properties through invariant mass distributions.

Contribution

It compares the $N^*(1535)$ resonance behavior in $ ext{Lambda}_c^+$ decay using chiral unitary and effective Lagrangian approaches, highlighting differences and potential experimental tests.

Findings

Different resonance behaviors in the two models.

Invariant mass distribution sensitive to $N^*(1535)$ properties.

Potential for experimental validation at BEPC II and SuperKEKB.

Abstract

The nonleptonic weak decay of $\Lambda^+_c \to \bar{K}^0 \eta p$ is analyzed from the viewpoint of probing the $N^*(1535)$ resonance, which has a big decay branching ratio to $\eta N$. Up to an arbitrary normalization, the invariant mass distribution of $\eta p$ is calculated with both the chiral unitary approach and an effective Lagrangian model. Within the chiral unitary approach, the $N^*(1535)$ resonance is dynamically generated from the final state interaction of mesons and baryons in the strangeness zero sector. For the effective Lagrangian model, we take a Breit-Wigner formula to describe the distribution of the $N^*(1535)$ resonance. It is found that the behavior of the $N^*(1535)$ resonance in the $\Lambda^+_c \to \bar{K}^0 N^*(1535) \to \bar{K}^0 \eta p$ decay within the two approaches is different. The proposed $\Lambda^+_c$ decay mechanism can provide valuable information on the properties of the $N^*(1535)$ and can in principle be tested by facilities such as BEPC II and SuperKEKB.

Tables

Table 1. Table 1: Masses and spin-parities of the particles studied in the present work.

State	Mass (MeV)	Spin-parity ($J^P$)
${\Lambda }_c^{+}$	$2286.46$	${\frac{1}{2}}^{+}$
${\overline{K}}^0$	$497.61$	$0^{-}$
$\eta$	$547.86$	$0^{-}$
$p$	$938.27$	${\frac{1}{2}}^{+}$

Table 2. Table 2: Mass ( M R subscript 𝑀 𝑅 M_{R} ) and width ( Γ R subscript Γ 𝑅 \Gamma_{R} ) for N ∗ ​ ( 1535 ) superscript 𝑁 1535 N^{*}(1535) and N ∗ ​ ( 1650 ) superscript 𝑁 1650 N^{*}(1650) found in Refs.  Nieves:2001wt ; Bruns:2010sv ; Nieves:2011gb ; Khemchandani:2013nma ; Garzon:2014ida . The masses and widths from Ref.  Olive:2016xmw are deduced from a Breit-Wigner fit. The values of masses and widths are given in MeV.

Reference	$N^{\ast }\left(1535\right)$		$N^{\ast }\left(1650\right)$
	$M_R$	${\Gamma }_R$	$M_R$	${\Gamma }_R$
Nieves:2001wt	$1496.5\pm 0.4$	$83.3\pm 0.7$	$1684.3\pm 0.7$	$194.3\pm 0.8$
Bruns:2010sv	$1506$	$280$	$1692$	$92$
Nieves:2011gb ; Gamermann:2011mq	$1556$	$94$	$1639$	$76$
Khemchandani:2013nma	$1504$	$110$	$1668$	$56$
			$1673$	$134$ 111A twin pole structure for $N^{\ast }\left(1650\right)$ is obtained in Ref. Arndt:1995bj .
Garzon:2014ida	$1508.1$	$90.3$	$1672.3$	$158.2$
Olive:2016xmw	$1535\pm 10$	$150\pm 25$	$1655\pm 15$	$140\pm 30$
$\mathrm{Br}\left(\to \pi N\right)$	$35\sim 55\%$		$50\sim 70\%$
$\mathrm{Br}\left(\to \eta N\right)$	$32\sim 52\%$		$14\sim 22\%$ 222This value is quoted in PDG Olive:2016xmw , but it is originally taken from Ref. Anisovich:2011fc which is derived from a multichannel partial wave analysis of pion and photo-induced reactions off protons. On the other hand, from the coupled-channel analysis of $\eta$ meson production including all recent photo-production data on the proton, the value of $1\pm 2\%$ is obtained in Ref. Shklyar:2012js and of $1.4\%$ in Ref. Shklyar:2004ba .

Equations

$$\frac{1}{\sqrt{2}}|u(ud-du)⟩.$$

$$|MB⟩=\frac{\sqrt{3}}{3}|\eta p⟩+\frac{\sqrt{2}}{2}|{\pi }^{0}p⟩+|{\pi }^{+}n⟩-\frac{\sqrt{6}}{3}|K^{+}\Lambda ⟩,$$

$$T^{MB}$$

$$\mathcal{M}=i\bar{u}(q)(A+B{\gamma }_5)u(p),$$

$${\mathcal{L}}_{N^{\ast }N\eta }=-i{g}_{N^{\ast }N\eta }\bar{N}\eta N^{\ast }+h.c.,$$

$$T^{N^{\ast }}=i{g}_{N^{\ast }N\eta }\bar{u}(p_3,s_p)G_{N^{\ast }}(q)(A+B{\gamma }_5)u(p,s_{{\Lambda }_c^{+}}),$$

$$\displaystyle G_{N^{*}}(q)=i\frac{\ooalign{\hfil/\hfil\crcr$q$}+M_{N^{*}}}{q^{2}-M^{2}_{N^{*}}+iM_{N^{*}}\Gamma_{N^{*}}(q^{2})},$$

$${\Gamma }_{N^{\ast }}(q^2)={\Gamma }_{N^{\ast }\to \pi N}(q^2)+{\Gamma }_{N^{\ast }\to \eta N}(q^2)+{\Gamma }_0,$$

$${\Gamma }_{N^{\ast }\to \pi N}(q^2)$$

$${\Gamma }_{N^{\ast }\to \eta N}(q^2)$$

$$|{\vec{p}}_{N\pi }|$$

$$|{\vec{p}}_{N\eta }|$$

$$\sum _{s_p}\bar{u}(p_3,s_p)u(p_3,s_p)$$

$$\sum _{s_{{\Lambda }_c^{+}}}\bar{u}(p,s_{{\Lambda }_c^{+}})u(p,s_{{\Lambda }_c^{+}})$$

$$\frac{1}{2}\overline{\sum _{s_{{\Lambda }_c^{+}}}}\sum _{s_p}{|T^{N^{\ast }}|}^2$$

$$D$$

$$a$$

$$b$$

$$c$$

$$p\cdot q$$

$$p_3\cdot q$$

$$p_3\cdot p$$

$$\frac{d\Gamma }{d{M}_{\eta p}}$$

$$p_{{\bar{K}}^0}$$

$$p_{\eta }^{\ast }$$

$$M_{\eta p}^{\max }$$

$$M_{\eta p}^{\min }$$

$$\frac{d\Gamma }{d{M}_{\eta p}}$$

$$R=\frac{f_2{B}^2}{f_1{A}^2}=\frac{f_2}{f_1}.$$

Full text

I Introduction

Understanding the nature of the $N^{*}(1535)$ with spin parity $J^{P}=1/2^{-}$ has always been one of the most challenging topics in hadron physics Klempt:2007cp ; Crede:2013sze . In classical constituent quark models, the $N^{*}(1535)$ is mainly composed of three valence quarks, and its mass should be lower than the radial excitation, the $N^{*}(1440)$, with $J^{P}=1/2^{+}$ Capstick:2000qj ; Olive:2016xmw . This is the long-standing mass reverse problem for the lowest spatial excited nucleon states. Another peculiar property of the $N^{*}(1535)$ is that it couples strongly to the channels with strangeness, such as $\eta N$ and $K\Lambda$, which is also difficult to understand in the naive constituent quark models.

Renouncing the picture of baryons as three-quark bound states, a different point of view consists in describing meson-baryon scattering reactions by taking mesons and baryons as the relevant degrees of freedom at low energies. Then, baryon excited states manifest themselves as poles of the meson-baryon scattering amplitude in a certain Riemann sheet in the complex energy plane. For example, the unitary extensions of chiral perturbation theory have brought new light to studies of baryon resonances from meson-baryon interactions Kaiser:1995cy ; Kaiser:1996js . In the chiral unitary coupled-channels approach it was found that the $N^{*}(1535)$ resonance is dynamically generated as a meson-baryon state with its mass, width, and branching ratios in fair agreement with experiments Nieves:2001wt ; Inoue:2001ip ; Bruns:2010sv ; Nieves:2011gb ; Gamermann:2011mq ; Khemchandani:2013nma ; Garzon:2014ida . The numerical results obtained in those studies differ to some extent, but it was found that the $N^{*}(1535)$ resonance couples strongly to the $\eta N$ channel. Furthermore, it couples more strongly to $K\Sigma$ and $K\Lambda$ than to $\pi N$ Inoue:2001ip ; Bruns:2010sv ; Nieves:2011gb ; Gamermann:2011mq ; Khemchandani:2013nma ; Garzon:2014ida .

In the phenomenological studies, besides the large coupling of the $N^{*}(1535)$ to $\eta N$, a large value of the coupling of the $N^{*}(1535)$ to $K\Lambda$ is deduced in Refs. Liu:2005pm ; Geng:2008cv ; Mart:2013ida by a simultaneous fit to the BES data on $J/\psi\to p\bar{p}\eta$, $pK^{-}\bar{\Lambda}+\bar{p}K^{+}\Lambda$, the COSY data on $pp\to pK^{+}\Lambda$, and the CLAS data on $\gamma p\to K^{+}\Lambda$ reaction. There is also evidence for a large coupling of the $N^{*}(1535)$ to $\eta^{\prime}N$ from the analysis of the $\gamma p\to p\eta^{\prime}$ reaction Dugger:2005my and $pp\to pp\eta^{\prime}$ reaction Cao:2008st , and a large coupling of the $N^{*}(1535)$ to $\phi N$ from the $\pi^{-}p\to n\phi$, $pp\to pp\phi$ and $pn\to d\phi$ reactions Xie:2007qt ; Doring:2008sv ; Cao:2009ea .

The above-mentioned strange decay properties of the $N^{*}(1535)$ resonance can be easily understood if it contains large five-quark components  Liu:2005pm ; Helminen:2000jb ; Zou:2007mk . Within the pentaquark picture, the $N^{*}(1535)$ resonance could be the lowest $L=1$ orbital excited $uud$ state with a large admixture of $[ud][us]\bar{s}$ pentaquark component having $[ud]$, $[us]$, and $\bar{s}$ in the ground state. This makes the $N^{*}(1535)$ heavier than the $N^{*}(1440)$ and also gives a natural explanation of its larger couplings to the channels with strangeness Zou:2010tc .

Recently, it has been shown that the nonleptonic weak decays of charmed hadrons provide a useful platform to study hadronic resonances, some of which are subjects of intense debate about their nature Crede:2008vw ; Chen:2016qju . For instance, the $\Lambda^{+}_{c}\to\pi^{+}MB$ weak decays are studied in Ref. Miyahara:2015cja from the viewpoint of probing the $\Lambda(1405)$ and $\Lambda(1670)$ resonances, where $M$ and $B$ stand for mesons and baryons. In Ref. Hyodo:2011js , the $\pi\Sigma$ mass distribution was studied in the $\Lambda^{+}_{c}\to\pi^{+}\pi\Sigma$ decays with the aim of extracting the $\pi\Sigma$ scattering lengths. In Ref. Xie:2016evi , the $a_{0}(980)$ and $\Lambda(1670)$ states are investigated in the $\Lambda^{+}_{c}\to\pi^{+}\eta\Lambda$ decay taking into account the $\pi^{+}\eta$ and $\eta\Lambda$ final state interactions. The pure $I=1$ nature of the $\pi^{+}\eta$ channel is particularly beneficial to the study of the $a_{0}(980)$ state. The role of the $\Sigma^{*}(1380)$ state with $J^{P}=1/2^{-}$ in the $\Lambda^{+}_{c}\to\eta\pi^{+}\Lambda$ decay is also studied in Ref. Xie:2017xwx where the color-suppressed $W$-exchange diagram is considered for the production of the $\Sigma^{*}(1385)$ with $J^{P}=3/2^{+}$. In Ref. Lu:2016ogy the role of the exclusive $\Lambda^{+}_{c}$ decays into a neutron in testing the flavor symmetry and final state interaction was investigated. It was shown that the three body nonleptonic decays are of great interest to explore the final state interactions in $\Lambda_{c}^{+}$ decays.

Along this line, in the present work, we study the role of the $N^{*}(1535)$ resonance in the $\Lambda^{+}_{c}\to\bar{K}^{0}\eta p$ decay by taking the advantage of the strong coupling of the $N^{*}(1535)$ to the $\eta N$ channel and its large $uuds\bar{s}$ component. We calculate the invariant $\eta p$ mass distribution within the chiral unitary approach and an effective Lagrangian model. First, we follow the same approach used in Ref. Miyahara:2015cja to study the $\Lambda^{+}_{c}\to\pi^{+}MB$ decays, but with the hadronization of the $uud$ rather than the $sud$ cluster to get the final $\eta p$ and from the $s\bar{d}$ pair to get the $\bar{K}^{0}$. In this respect, the $N^{*}(1535)$ resonance is dynamically generated from the final state interaction of mesons and baryons in the $I=1/2$ sector where we have assumed that the $ud$ di-quark with $I=0$ in the $\Lambda^{+}_{c}$ is a spectator. Second, we study the $\Lambda^{+}_{c}\to\bar{K}^{0}N^{*}(1535)\to\bar{K}^{0}\eta p$ decay at the hadron level by taking a Breit-Wigner formula to describe the distribution of the $N^{*}(1535)$ resonance within the effective Lagrangian model. The contributions from other low-lying $N^{*}$ and $\Sigma^{*}$ resonances are discussed. Fortunately, it is found that these contributions may not affect much the results obtained here.

This article is organized as follows. In Sec. II, we present the theoretical formalism of the decay of $\Lambda^{+}_{c}\to\bar{K}^{0}\eta p$, explaining in detail the hadronization and final state interactions of the $\eta p$ pair. Numerical results and discussions are presented in Sec. III, followed by a short summary in the last section.

II Formalism

As shown in Refs. Miyahara:2015cja ; Xie:2016evi ; Miyahara:2016yyh , a Cabibbo allowed mechanism for the $\Lambda^{+}_{c}$ decay is that the charmed quark in $\Lambda^{+}_{c}$ turns into a strange quark with a $u\bar{d}$ pair by the weak interaction as shown in Fig. 1.

In addition to the $c$ quark decay process described above, in principle one can also have contributions from internal $W$-exchange ($c+d\to s+u$) diagrams. As discussed in Refs. Miyahara:2015cja ; Xie:2014tma ; Xie:2016evi ; Miyahara:2016yyh , these contributions are smaller than the $c$ decay process. Furthermore, including such contributions, the decay amplitudes would become more complex due to additional parameters from the weak interaction, and we can not determine or constrain these parameters at present. Hence, we will leave these contributions to future studies when more experimental data become available.

II.1 The $N^{*}(1535)$ as a dynamically generated state from meson-baryon scattering

We first discuss the decay of $\Lambda^{+}_{c}$ to produce the $\bar{K}^{0}$ from the $s\bar{d}$ pair and the insertion of a new $\bar{q}q$ pair with the quantum numbers of the vacuum, $\bar{u}u+\bar{d}d+\bar{s}s$, to construct the intermediate meson-baryon state $MB$ from the $uud$ cluster with the assumption that the $u$ and $d$ quarks in the $\Lambda^{+}_{c}$ are spectators in the weak decay corresponding to the mechanism of Fig. 2. Thus, after the hadronization these $u$ and $d$ quarks in the $\Lambda^{+}_{c}$ are part of the baryon, and the $u$ quark originated from the weak decay forms the meson. Furthermore, the $uud$ cluster with strangeness zero is combined into a pure $I=1/2$ state

[TABLE]

Following the procedure of Refs. Miyahara:2015cja ; Xie:2016evi ; Miyahara:2016yyh ; Roca:2015tea ; Feijoo:2015cca , one can straightforwardly obtain the meson-baryon states after the $\bar{q}q$ pair production as

[TABLE]

where we have omitted the $\eta^{\prime}p$ term because of its large mass threshold compared to other channels that we considered.

After the production of a meson-baryon pair, the final-state interaction between them takes place, which can be parameterized by the re-scattering shown in Fig. 3 at the hadronic level for the $\Lambda^{+}_{c}\to\bar{K}^{0}\eta p$ decay. The final-state interaction of $MB$, in $I=1/2$, leads to the dynamical generation of the $N^{*}(1535)$ resonance Inoue:2001ip ; Doring:2005bx . In Fig. 3, we also show the tree level diagram for the $\Lambda^{+}_{c}\to\bar{K}^{0}\eta p$ decay.

According to Eq. (2), we can write down the $\Lambda^{+}_{c}\to\bar{K}^{0}\eta p$ decay amplitude of Fig. 3 as Oller:1997yg ,

[TABLE]

where $V_{P}$ expresses the weak and hadronization strength, which is assumed to be a constant and independent of the final state interaction. In the above equation, $G_{MB}$ denotes the one-meson-one-baryon loop function, which depends on the invariant mass of the final $\eta p$ system, $M_{\eta p}$. The meson-baryon scattering amplitudes $t_{MB\to\eta p}$ are those obtained in the chiral unitary approach, which depend also on $M_{\eta p}$. Details can be found in Refs. Inoue:2001ip ; Doring:2005bx .

II.2 Effective Lagrangian approach and the $N^{*}(1535)$ resonance as a Breit-Wigner resonance

On the other hand, because the $N^{*}(1535)$ has a large $uuds\bar{s}$ component, it can also be produced via the process shown in Fig. 4 (a), similar to the $P^{+}_{c}$ states produced in the $\Lambda^{0}_{b}\to K^{-}P^{+}_{c}$ decay Aaij:2015tga . After the $N^{*}(1535)$ is formed with $uuds\bar{s}$, it decays into $\eta p$, which is the dominant decay channel of the $N^{*}(1535)$ resonance. We show the hadron level diagram for the decay of $\Lambda^{+}_{c}\to\bar{K}^{0}N^{*}(1535)\to\bar{K}^{0}\eta p$ in Fig. 4 (b).

Before going further, we emphasize that the strangeness component of $N^{*}(1535)$ can not be guaranteed from the decay process shown in Fig. 4. Indeed, the $N^{*}(1535)$ can also be produced from the process shown in Fig. 2, where the $s\bar{d}$ forms the $\bar{K}^{0}$, while the $N^{*}(1535)$ is constructed from the $uud$ cluster and then it decays into $\eta p$ because of its large coupling to this channel.

The general decay amplitudes for $\Lambda^{+}_{c}\to\bar{K}^{0}N^{*}(1535)$ can be decomposed into two different structures with the corresponding coefficients $A$ and $B$,

[TABLE]

where $q$ and $p$ are the momentum of the $N^{*}(1535)$ and $\Lambda^{+}_{c}$, respectively. The coefficients $A$ and $B$ for charmed baryons decaying into ground meson and baryon states, in general, can be calculated in the framework of the pole model Cheng:1991sn or within the perturbative $QCD$ approach Lu:2009cm . In the present case, because the $N^{*}(1535)$ resonance is not well understood in the classical quark model, the values of $A$ and $B$ in Eq. (4) are very difficult to be pined down, and we have to determine them with future experimental data. In this work, we take $A=B$ and we come back to this issue later.

To get the whole decay amplitude of $\Lambda^{+}_{c}\to\bar{K}^{0}N^{*}(1535)\to\bar{K}^{0}\eta p$ as shown in Fig. 4 (b), we use the effective Lagrangian density of Refs. Bai:2001ua ; Liu:2005pm ; Xie:2007qt for the $N^{*}(1535)N\eta$ vertex,

[TABLE]

where $N$, $\eta$, and $N^{*}$ represent the fields of the proton, the $\eta$ meson, and the $N^{*}(1535)$ resonance, respectively.

The invariant decay amplitude of the $\Lambda^{+}_{c}\to\bar{K}^{0}N^{*}(1535)\to\bar{K}^{0}\eta p$ decay is

[TABLE]

where $p_{3}$ is the momentum of the final proton. The $s_{p}$ and $s_{\Lambda^{+}_{c}}$ are the spin polarization variables for the proton and $\Lambda^{+}_{c}$ baryon, respectively. The $G_{N^{*}}(q)$ is the propagator of the $N^{*}(1535)$, which is given by a Breit-Wigner (BW) form as,

[TABLE]

where $M_{N^{*}}$ and $\Gamma_{N^{*}}(q^{2})$ are the mass and total decay width of the $N^{*}(1535)$, respectively. We take $M_{N^{*}}=1535$ MeV as in the PDG Olive:2016xmw . For $\Gamma_{N^{*}}(q^{2})$, since the dominant decay channels for the $N^{*}(1535)$ resonance are $\pi N$ and $\eta N$ Olive:2016xmw , we take the following form as used in Refs. Wu:2009nw ; Xie:2013wfa

[TABLE]

with

[TABLE]

Here

[TABLE]

where $\lambda$ is the Källen function with $\lambda(x,y,z)=(x-y-z)^{2}-4yz$. In the present work, we take $g^{2}_{N^{*}N\pi}/4\pi=0.037$ and $g^{2}_{N^{*}N\eta}/4\pi=0.28$ as used in Ref. Lu:2014yba . With these values we can get $\Gamma_{N^{*}\to N\pi}=67.5$ MeV and $\Gamma_{N^{*}\to N\eta}=63$ MeV if we take $\sqrt{q^{2}}=1535$ MeV. In this work, we choose $\Gamma_{0}=19.5$ MeV for $\Gamma_{N^{*}}(\sqrt{q^{2}}=1535~{}{\rm MeV})=150$ MeV.

In the effective Lagrangian approach, the sum over polarizations and the Dirac spinors can be easily done thanks to

[TABLE]

Finally, we obtain

[TABLE]

with

[TABLE]

and

[TABLE]

with $M^{2}_{\eta p}=q^{2}$.

II.3 Invariant mass distributions of the $\Lambda^{+}_{c}\to\bar{K}^{0}\eta p$ decay

With all the ingredients obtained in the previous subsection, one can write down the invariant $\eta p$ mass distribution of the $\Lambda^{+}_{c}\to\bar{K}^{0}\eta p$ decay as:

[TABLE]

where $T$ is the total decay amplitude. The $p_{\bar{K}^{0}}$ and $p^{*}_{\eta}$ are the three-momenta of the outgoing $\bar{K}^{0}$ meson in the $\Lambda^{+}_{c}$ rest frame and the outgoing $\eta$ meson in the center of mass frame of the final $\eta p$ system, respectively. They are given by

[TABLE]

The range of $M_{\eta p}$ is

[TABLE]

III Numerical results and discussion

In this section, we first show the numerical results for the $d\Gamma/dM_{\eta p}$ with three models: Model I takes $T=T^{MB}$; Model II takes $T=T^{N^{*}}$ and $\Gamma_{N^{*}}$ is energy dependent as in Eq. (10); Model III takes $T=T^{N^{*}}$ and $\Gamma_{N^{*}}=150$ MeV as a constant. Next, we will discuss the impact of the contributions from other $N^{*}$ and $\Sigma^{*}$ states.

III.1 Invariant $\eta p$ mass distributions

In Fig. 5, we show the $\eta p$ invariant mass distribution obtained with the mass values shown in Table 1, where the solid, dashed and dotted curves represent the numerical results obtained with Model I, II, and III, respectively. The results of Model I are obtained with $V_{P}=1$ MeV*-1*. The results of Model II with $A=B=45.2$ and Model III with $A=B=47.4$ are normalized to the peak of Model I.

For Model I, a peak around $1524$ MeV corresponding to the $N^{*}(1535)$ resonance can be clearly seen as in Refs. Inoue:2001ip ; Doring:2005bx . The peaks of Model II and III move to $1532$ and $1553$ MeV, respectively. The peak position of Model II is very close to the central value, 1535 MeV, estimated in the PDG Olive:2016xmw for the $N^{*}(1535)$. The peak position of Model I is also close to the value 1535 MeV, but with a narrow width. For Model III where a constant decay width of the $N^{*}(1535)$ is used, the peak position moves $20$ MeV away from the Breit-Wigner mass $1535$ MeV. Besides, the resonant shapes of Model II and III are broader than the result of Model I.

Because the mass of the $N^{*}(1535)$ is close to the $\eta N$ threshold and has a large coupling to this channel, the approximation of a BW form with a constant width is not very realistic Liu:2005pm . We should take the coupled channel BW formula as in Eq. (10), which will reduce the BW mass of the $N^{*}(1535)$ Liu:2005pm .

From the results of Model I and II shown in Fig. 5, we see that these two different descriptions of the $N^{*}(1535)$ resonance give different invariant $\eta p$ mass distributions. The findings here are similar to that obtained in Refs. Doring:2008sv ; Geng:2008cv . For the $N^{*}(1535)$, the amplitude square obtained with the chiral unitary approach does not behave like an usual BW resonance, even at the peak position (see Fig. 1 of Ref. Geng:2008cv ). It is expected that future experimental measurements may test our model predictions and clarify this issue.

One might be tempted to think that the discrepancy between Model I and II (or III) is due to the inclusion of the $p$-wave contribution for Model II and III shown in Eq. (4) with the $B$ term. We have explored such a possibility from the comparison of the contributions of the $A$ and $B$ terms. For doing this, we first rewrite $d\Gamma/dM_{\eta p}$ for Model II and III as,

[TABLE]

Then we define the ratio $R$ as

[TABLE]

In the last step, we have taken $A=B$.

In Fig. 6 we show the numerical results for $R$ as a function of $M_{\eta p}$. We see clearly that $R$ is less than $2.8$ percent for the whole possible $M_{\eta p}$ in the $\Lambda^{+}_{c}\to\bar{K}^{0}\eta p$ decay. This means that the contribution of the $p$-wave $B$ term is rather small in comparison with the contribution from the $s$-wave $A$ term and can be neglected safely. This study provides further support for the factorization scheme of the hard process (the weak decay and hadronization) for Model I where only the $s$-wave contribution is considered between any two particles of the final $\bar{K}^{0}\eta p$. Such a factorization scheme seems to work fairly well in the present case.

It should be noted that the $B$ term is very small compared with the $A$ term, which is tied to the fact that we take $A=B$. A model independent calculation of the values of $A$ and $B$ is most welcome and will ultimately test our model calculations.

III.2 Contributions from other processes

Up to now, we have considered only the contribution from $N^{*}(1535)$, while the contributions from other nucleon resonances, such as, $N^{*}(1650)\frac{1}{2}^{-}$, $N^{*}(1710)\frac{1}{2}^{+}$, and $N^{*}(1720)\frac{3}{2}^{+}$, are not taken into account. The $N^{*}(1710)$ and $N^{*}(1720)$ decay into $\eta p$ in $p$-wave and the decay of $\Lambda^{+}_{c}\to\bar{K}^{0}N^{*}(1710)$ and $\Lambda^{+}_{c}\to\bar{K}^{0}N^{*}(1720)$ have very limited phase space, hence, their contributions should be much suppressed.

It is interesting to note that both $N^{*}(1535)$ and $N^{*}(1650)$ are dynamically generated from the analysis of the $s$-wave $\pi N$ scattering Nieves:2001wt ; Bruns:2010sv . We list the results obtained in Refs. Nieves:2001wt ; Bruns:2010sv ; Nieves:2011gb ; Khemchandani:2013nma ; Garzon:2014ida for $N^{*}(1535)$ and $N^{*}(1650)$ in Table 2, where we show also the BW mass and width, branching ratios to $\pi N$ and $\eta N$ of $N^{*}(1535)$ and $N^{*}(1650)$ that are estimated by PDG Olive:2016xmw for comparison. We see that the $N^{*}(1650)$ and the $N^{*}(1535)$ are much separated in mass Nieves:2001wt ; Bruns:2010sv . Therefore, the contribution from $N^{*}(1650)$ will not overlap too much with that from $N^{*}(1535)$. Furthermore, the branching ratio of $N^{*}(1650)$ to $\eta N$ is very small compared with the one to $\pi N$. We thus conclude that the contribution from $N^{*}(1650)$ to the invariant $\eta p$ mass distribution is small or at least it will not change too much the numerical results shown in Fig. 5 even if it has a sizable contribution.

On the other hand, there should be also contributions from $\Sigma^{*}$ resonances that have significant branching ratio to $\bar{K}^{0}p$. Those $\Sigma^{*}$ resonances are: $\Sigma^{*}(1660)\frac{1}{2}^{+}$, $\Sigma^{*}(1670)\frac{3}{2}^{-}$, and $\Sigma^{*}(1750)\frac{1}{2}^{-}$. We show the Dalitz plot for the $\Lambda^{+}_{c}\to\bar{K}^{0}\eta p$ decay in Fig. 7. In the $N^{*}(1535)$ energy region, the Dalitz plot overlaps with these $\Sigma^{*}$ resonances from $1600$ to $1800$ MeV in the $\bar{K}^{0}p$ channel, which may make the analysis of $N^{*}(1535)$ difficult. Fortunately, the $\Sigma^{*}(1660)\frac{1}{2}^{+}$ and $\Sigma^{*}(1670)\frac{3}{2}^{-}$ decay into $\bar{K}^{0}p$ in $p$-wave and $D$-wave, respectively. These contributions will be suppressed because of the higher partial waves involved. For the $\Sigma^{*}(1750)\frac{1}{2}^{-}$, it decays into $\bar{K}^{0}p$ in $s$-wave. However, it lies in the kinematic end-point region and therefore the decay of $\Lambda^{+}_{c}\to\eta\Sigma^{*}(1750)$ has a relative small phase space.

In summary, the contributions from other $N^{*}$ and $\Sigma^{*}$ resonances should be small compared with the contribution from the $N^{*}(1535)$, and we expect that their contributions will not change much the model predictions presented in the present work. On the other hand, if future experimental measurements provide enough data to disentangle the contributions from these resonances, one can also study them. It should be kept in mind that our study made some assumptions and hence it can be improved once more data become available.

IV Conclusions

In the present work we have studied the invariant $\eta p$ mass distribution in the $\Lambda^{+}_{c}\to\bar{K}^{0}\eta p$ decay to better understand the $N^{*}(1535)$ resonance. First, we employed the molecular picture where the $N^{*}(1535)$ is dynamically generated from the meson-baryon interaction. In such a scenario, the weak interaction part is dominated by the $c$ quark decay process: $c(ud)\to(s+u+\bar{d})(ud)$, while the hadronization part takes place by the $uud$ cluster picking up a $q\bar{q}$ pair from the vacuum and hadronizes into a meson-baryon pair, while the $s\bar{d}$ pair from the weak decay turns into a $\bar{K}^{0}$. The following final state interactions of the meson-baryon pairs are described in the chiral unitary model that dynamically generates the $N^{*}(1535)$ resonance in the $I=1/2$ sector. Second, we studied the $\Lambda^{+}_{c}\to\bar{K}^{0}N^{*}(1535)\to\bar{K}^{0}\eta p$ decay with a Breit-Wigner formula to describe the distribution of the $N^{*}(1535)$ in the effective Lagrangian model. The above two descriptions for the $N^{*}(1535)$ resonance give different invariant $\eta p$ mass distributions. Furthermore, we showed in a qualitative way that the contributions from other $N^{*}$ and $\Sigma^{*}$ resonance are relatively small and will not affect much the results obtained in the present study.

On the experimental side, the decay mode $\Lambda^{+}_{c}\to\bar{K}^{0}\eta p$ has been observed Olive:2016xmw ; Ammar:1995je and the branching ratio $\mathrm{Br}(\Lambda^{+}_{c}\to\bar{K}^{0}\eta p)$ is determined to be $(1.6\pm 0.4)\%$, which is one of the dominant decay modes of the $\Lambda^{+}_{c}$ state. For the decay of $\Lambda^{+}_{c}\to\bar{K}^{0}\eta p$, the final $\eta p$ is in pure isospin $I=1/2$. Hence, this decay can be an ideal process to study the $N^{*}(1535)$ resonance, which has a large branching ratio to $\eta N$ and decays into $\eta N$ in $s$-wave. Future experimental measurements of the invariant $\eta p$ mass distribution studied in the present work will be very helpful to test our model calculations and constrain the properties of the $N^{*}(1535)$ resonance. For example, a corresponding experimental measurement could in principle be done at BESIII Ablikim:2015flg and Belle.

V ACKNOWLEDGMENTS

This work is partly supported by the National Natural Science Foundation of China under Grant No.11475227, No. 11375024, No. 11522539, No. 11505158, No. 11475015, and No. 11647601. It is also supported by the Youth Innovation Promotion Association CAS (No. 2016367).