Heavy quark spin symmetric molecular states from ${\bar D}^{(*)}\Sigma_c^{(*)}$ and other coupled channels in the light of the recent LHCb pentaquarks

TL;DR

This paper models pentaquark states as molecular states of heavy mesons and baryons, successfully reproducing observed LHCb pentaquarks and predicting additional states consistent with experimental hints.

Contribution

It introduces a heavy quark spin symmetric approach using an extended local hidden gauge model, fitting only one parameter to reproduce and predict pentaquark states.

Findings

Reproduces three LHCb pentaquarks with quantum numbers and structures.

Predicts additional states around 4374 MeV and 4520 MeV consistent with experimental hints.

Identifies a state at 5/2^- that does not couple to $J/\psi p$ in S-wave.

Abstract

We consider the ${\bar D}^{(*)}\Sigma_c^{(*)}$ states, together with $J/\psi N$ and other coupled channels, and take an interaction consistent with heavy quark spin symmetry, with the dynamical input obtained from an extension of the local hidden gauge approach. By fitting only one parameter to the recent three pentaquark states reported by the LHCb collaboration, we can reproduce the three of them in base to the mass and the width, providing for them the quantum numbers and approximate molecular structure as $1/2^-$ $\bar{D} \Sigma_c$, $1/2^-$ $\bar{D}^* \Sigma_c$, and $3/2^-$ $\bar{D}^* \Sigma_c$, and isospin $I=1/2$. We find another state around 4374 MeV, of $3/2^-$ $\bar{D} \Sigma_c^*$ structure, for which indications appear in the experimental spectrum. Two other near degenerate states of $1/2^-$ $\bar{D}^* \Sigma_c^*$ and $3/2^-$ $\bar{D}^* \Sigma_c^*$ nature are also found around 4520 MeV, which although less clear, are not incompatible with the observed spectrum. In addition, a $5/2^-$ $\bar D^* \Sigma_c^*$ state at the same energy appears, which however does not couple to $J/\psi p$ in $S-$wave, and hence it is not expected to show up in the LHCb experiment.

Tables

Table 1. Table 1: Dimensionless coupling constants of the ( I = 1 / 2 , J P = 1 / 2 − ) formulae-sequence 𝐼 1 2 superscript 𝐽 𝑃 1 superscript 2 (I=1/2,J^{P}=1/2^{-}) poles found in this work to the different channels.

$\left(4306.38+i7.62\right)$ MeV
	${\eta }_{c}N$	$J/\psi N$	$\overline{D}{\Lambda }_c$	$\overline{D}{\Sigma }_c$	${\overline{D}}^{\ast }{\Lambda }_c$	${\overline{D}}^{\ast }{\Sigma }_c$	${\overline{D}}^{\ast }{\Sigma }_c^{\ast }$
$g_i$	$0.67+i0.01$	$0.46-i0.03$	$0.01-i0.01$	$2.07-\mathrm{𝐢𝟎}.28$	$0.03+i0.25$	$0.06-i0.31$	$0.04-i0.15$
$\left|g_i\right|$	$0.67$	$0.46$	$0.01$	$2.09$	$0.25$	$0.31$	$0.16$
($4452.96+i11.72$) MeV
	${\eta }_{c}N$	$J/\psi N$	$\overline{D}{\Lambda }_c$	$\overline{D}{\Sigma }_c$	${\overline{D}}^{\ast }{\Lambda }_c$	${\overline{D}}^{\ast }{\Sigma }_c$	${\overline{D}}^{\ast }{\Sigma }_c^{\ast }$
$g_i$	$0.24+i0.03$	$0.88-0.11$	$0.09-i0.06$	$0.12-i0.02$	$0.11-i0.09$	$1.97-\mathrm{𝐢𝟎}.52$	$0.02+i0.19$
$\left|g_i\right|$	$0.25$	$0.89$	$0.11$	$0.13$	$0.14$	$2.03$	$0.19$
$\left(4520.45+i11.12\right)$ MeV
	${\eta }_{c}N$	$J/\psi N$	$\overline{D}{\Lambda }_c$	$\overline{D}{\Sigma }_c$	${\overline{D}}^{\ast }{\Lambda }_c$	${\overline{D}}^{\ast }{\Sigma }_c$	${\overline{D}}^{\ast }{\Sigma }_c^{\ast }$
$g_i$	$0.72-i0.10$	$0.45-i0.04$	$0.11-i0.06$	$0.06-i0.02$	$0.06-i0.05$	$0.07-i0.02$	$1.84-\mathrm{𝐢𝟎}.56$
$\left|g_i\right|$	$0.73$	$0.45$	$0.13$	$0.06$	$0.08$	$0.08$	$1.92$

Table 2. Table 2: Same as Table  1 for J P = 3 / 2 − superscript 𝐽 𝑃 3 superscript 2 J^{P}=3/2^{-} .

$\left(4374.33+i6.87\right)$ MeV	$J/\psi N$	${\overline{D}}^{\ast }{\Lambda }_c$	${\overline{D}}^{\ast }{\Sigma }_c$	$\overline{D}{\Sigma }_c^{\ast }$	${\overline{D}}^{\ast }{\Sigma }_c^{\ast }$
$g_i$	$0.73-i0.06$	$0.11-i0.13$	$0.02-i0.19$	$1.91-\mathrm{𝐢𝟎}.31$	$0.03-i0.30$
$\left|g_i\right|$	$0.73$	$0.18$	$0.19$	$1.94$	$0.30$
$\left(4452.48+i1.49\right)$ MeV	$J/\psi N$	${\overline{D}}^{\ast }{\Lambda }_c$	${\overline{D}}^{\ast }{\Sigma }_c$	$\overline{D}{\Sigma }_c^{\ast }$	${\overline{D}}^{\ast }{\Sigma }_c^{\ast }$
$g_i$	$0.30-i0.01$	$0.05-i0.04$	$1.82-\mathrm{𝐢𝟎}.08$	$0.08-i0.02$	$0.01-i0.19$
$\left|g_i\right|$	$0.30$	$0.07$	$1.82$	$0.08$	$0.19$
$\left(4519.01+i6.86\right)$ MeV	$J/\psi N$	${\overline{D}}^{\ast }{\Lambda }_c$	${\overline{D}}^{\ast }{\Sigma }_c$	$\overline{D}{\Sigma }_c^{\ast }$	${\overline{D}}^{\ast }{\Sigma }_c^{\ast }$
$g_i$	$0.66-i0.01$	$0.11-i0.07$	$0.10-i0.3$	$0.13-i0.02$	$1.79-\mathrm{𝐢𝟎}.36$
$\left|g_i\right|$	$0.66$	$0.13$	$0.10$	$0.13$	$1.82$

Table 3. Table 3: Identification of some of the I = 1 / 2 𝐼 1 2 I=1/2 resonances found in this work with experimental states.

Mass [MeV]	Width [MeV]	Main channel	$J^P$	Experimental state
4306.4	15.2	$\overline{D}{\Sigma }_c$	$1/2^{-}$	$P_c\left(4312\right)$
4453.0	23.4	${\overline{D}}^{\ast }{\Sigma }_c$	$1/2^{-}$	$P_c\left(4440\right)$
4452.5	3.0	${\overline{D}}^{\ast }{\Sigma }_c$	$3/2^{-}$	$P_c\left(4457\right)$

Equations

$$M_{P_{c1}}=(4311.9\pm 0.7_{-0.6}^{+6.8})\mathrm{MeV},{\Gamma }_{P_{c1}}=(9.8\pm 2.7_{-4.5}^{+3.7})\mathrm{MeV},$$

$$M_{P_{c2}}=(4440.3\pm 1.3_{-4.7}^{+4.1})\mathrm{MeV},{\Gamma }_{P_{c2}}=(20.6\pm 4.9_{-10.1}^{+8.7})\mathrm{MeV},$$

$$M_{P_{c3}}=(4457.3\pm 0.6_{-1.7}^{+4.1})\mathrm{MeV},{\Gamma }_{P_{c3}}=(6.4\pm 2.0_{-1.9}^{+5.7})\mathrm{MeV}.$$

$$T=[1-VG{]}^{-1}V,$$

$$\left.\phantom{(}\begin{array}[]{ccccccc}\phantom{\sqrt{\frac{2}{3}}\text{$\mu_{13}$}}&\phantom{\frac{\sqrt{2}\text{$\mu_{13}$}}{3}}&\phantom{\sqrt{\frac{2}{3}}\text{$\mu_{23}$}}&\phantom{\frac{1}{3}\sqrt{\frac{2}{3}}(\text{$\mu_{3}$}-\text{$\lambda_{2}$})}&\phantom{\frac{\sqrt{2}\text{$\mu_{23}$}}{3}}&\phantom{\frac{1}{9}\sqrt{2}(\text{$\mu_{3}$}-\text{$\lambda_{2}$})}&\phantom{\frac{1}{9}(\text{$\lambda_{2}$}+8\text{$\mu_{3}$})}\\ \eta_{c}N&J/\psi N&\bar{D}\Lambda_{c}&\bar{D}\Sigma_{c}&\bar{D}^{*}\Lambda_{c}&\bar{D}^{*}\Sigma_{c}&\bar{D}^{*}\Sigma^{*}_{c}\end{array}\right.\phantom{)_{I=1/2}}$$

$$\left(\begin{array}[]{ccccccc}\text{$\mu_{1}$}&0&\frac{\text{$\mu_{12}$}}{2}&\frac{\text{$\mu_{13}$}}{2}&\frac{\sqrt{3}\text{$\mu_{12}$}}{2}&-\frac{\text{$\mu_{13}$}}{2\sqrt{3}}&\sqrt{\frac{2}{3}}\text{$\mu_{13}$}\\ \\ 0&\text{$\mu_{1}$}&\frac{\sqrt{3}\text{$\mu_{12}$}}{2}&-\frac{\text{$\mu_{13}$}}{2\sqrt{3}}&-\frac{\text{$\mu_{12}$}}{2}&\frac{5\text{$\mu_{13}$}}{6}&\frac{\sqrt{2}\text{$\mu_{13}$}}{3}\\ \\ \frac{\text{$\mu_{12}$}}{2}&\frac{\sqrt{3}\text{$\mu_{12}$}}{2}&\text{$\mu_{2}$}&0&0&\frac{\text{$\mu_{23}$}}{\sqrt{3}}&\sqrt{\frac{2}{3}}\text{$\mu_{23}$}\\ \\ \frac{\text{$\mu_{13}$}}{2}&-\frac{\text{$\mu_{13}$}}{2\sqrt{3}}&0&\frac{1}{3}(2\text{$\lambda_{2}$}+\text{$\mu_{3}$})&\frac{\text{$\mu_{23}$}}{\sqrt{3}}&\frac{2(\text{$\lambda_{2}$}-\text{$\mu_{3}$})}{3\sqrt{3}}&\frac{1}{3}\sqrt{\frac{2}{3}}(\text{$\mu_{3}$}-\text{$\lambda_{2}$})\\ \\ \frac{\sqrt{3}\text{$\mu_{12}$}}{2}&-\frac{\text{$\mu_{12}$}}{2}&0&\frac{\text{$\mu_{23}$}}{\sqrt{3}}&\text{$\mu_{2}$}&-\frac{2\text{$\mu_{23}$}}{3}&\frac{\sqrt{2}\text{$\mu_{23}$}}{3}\\ \\ -\frac{\text{$\mu_{13}$}}{2\sqrt{3}}&\frac{5\text{$\mu_{13}$}}{6}&\frac{\text{$\mu_{23}$}}{\sqrt{3}}&\frac{2(\text{$\lambda_{2}$}-\text{$\mu_{3}$})}{3\sqrt{3}}&-\frac{2\text{$\mu_{23}$}}{3}&\frac{1}{9}(2\text{$\lambda_{2}$}+7\text{$\mu_{3}$})&\frac{1}{9}\sqrt{2}(\text{$\mu_{3}$}-\text{$\lambda_{2}$})\\ \\ \sqrt{\frac{2}{3}}\text{$\mu_{13}$ }&\frac{\sqrt{2}\text{$\mu_{13}$}}{3}&\sqrt{\frac{2}{3}}\text{$\mu_{23}$ }&\frac{1}{3}\sqrt{\frac{2}{3}}(\text{$\mu_{3}$}-\text{$\lambda_{2}$})&\frac{\sqrt{2}\text{$\mu_{23}$}}{3}&\frac{1}{9}\sqrt{2}(\text{$\mu_{3}$}-\text{$\lambda_{2}$})&\frac{1}{9}(\text{$\lambda_{2}$}+8\text{$\mu_{3}$})\end{array}\right)_{I=1/2}$$

$$\left.\phantom{(}\begin{array}[]{ccccc}\phantom{\frac{\sqrt{5}\text{$\mu_{13}$}}{3}}&\phantom{\frac{\sqrt{5}\text{$\mu_{23}$}}{3}}&\phantom{\frac{1}{9}\sqrt{5}(\text{$\mu_{3}$}-\text{$\lambda_{2}$})}&\phantom{\frac{1}{3}\sqrt{\frac{5}{3}}(\text{$\lambda_{2}$}-\text{$\mu_{3}$})}&\phantom{\frac{1}{9}(4\text{$\lambda_{2}$}+5\text{$\mu_{3}$})}\\ J/\psi N&\bar{D}^{*}\Lambda_{c}&\bar{D}^{*}\Sigma_{c}&\bar{D}\Sigma^{*}_{c}&\bar{D}^{*}\Sigma^{*}_{c}\end{array}\right.\phantom{)_{I=1/2}}$$

$$\left(\begin{array}[]{ccccc}\text{$\mu_{1}$}&\text{$\mu_{12}$}&\frac{\text{$\mu_{13}$}}{3}&-\frac{\text{$\mu_{13}$}}{\sqrt{3}}&\frac{\sqrt{5}\text{$\mu_{13}$}}{3}\\ \\ \text{$\mu_{12}$}&\text{$\mu_{2}$}&\frac{\text{$\mu_{23}$}}{3}&-\frac{\text{$\mu_{23}$}}{\sqrt{3}}&\frac{\sqrt{5}\text{$\mu_{23}$}}{3}\\ \\ \frac{\text{$\mu_{13}$}}{3}&\frac{\text{$\mu_{23}$}}{3}&\frac{1}{9}(8\text{$\lambda_{2}$}+\text{$\mu_{3}$})&\frac{\text{$\lambda_{2}$}-\text{$\mu_{3}$}}{3\sqrt{3}}&\frac{1}{9}\sqrt{5}(\text{$\mu_{3}$}-\text{$\lambda_{2}$})\\ \\ -\frac{\text{$\mu_{13}$}}{\sqrt{3}}&-\frac{\text{$\mu_{23}$}}{\sqrt{3}}&\frac{\text{$\lambda_{2}$}-\text{$\mu_{3}$}}{3\sqrt{3}}&\frac{1}{3}(2\text{$\lambda_{2}$}+\text{$\mu_{3}$})&\frac{1}{3}\sqrt{\frac{5}{3}}(\text{$\lambda_{2}$}-\text{$\mu_{3}$})\\ \\ \frac{\sqrt{5}\text{$\mu_{13}$}}{3}&\frac{\sqrt{5}\text{$\mu_{23}$}}{3}&\frac{1}{9}\sqrt{5}(\text{$\mu_{3}$}-\text{$\lambda_{2}$})&\frac{1}{3}\sqrt{\frac{5}{3}}(\text{$\lambda_{2}$}-\text{$\mu_{3}$})&\frac{1}{9}(4\text{$\lambda_{2}$}+5\text{$\mu_{3}$})\\ \end{array}\right)_{I=1/2}$$

$${\bar{D}}^{\ast }{\Sigma }_c^{\ast }:({\lambda }_2{)}_{I=1/2}$$

$$\begin{array}{rl}{\mu }_1& =0,{\mu }_{23}=0,{\lambda }_2={\mu }_3,{\mu }_{13}=-{\mu }_{12},\\ {\mu }_2& =\frac{1}{4f^2}(k^0+k^{\prime 0}),{\mu }_3=-\frac{1}{4f^2}(k^0+k^{\prime 0}),\\ {\mu }_{12}& =-\sqrt{6}\frac{m_{\rho }^2}{p_{D^{\ast }}^2-m_{D^{\ast }}^2}\frac{1}{4f^2}(k^0+k^{\prime 0}),\end{array}$$

Full text

The discovery of some pentaquarks signals by the LHCb collaboration in 2015 Aaij:2015tga ; Aaij:2015fea generated a wave of enthusiasm in the hadron physics community. Two states were reported, one at 4380 MeV and width $\Gamma\sim 205\textrm{ MeV}$ and another one at 4450 MeV and width 40 MeV. Actually there had been several predictions for hidden charm molecular states in this region prior to the experimental discovery Wu:2010jy ; Wang:2011rga ; Yang:2011wz ; Yuan:2012wz ; Wu:2012md ; Xiao:2013yca ; Uchino:2015uha ; Karliner:2015ina . The hidden charm molecular states would have some resemblance with the $N^{*}(1535)$ resonance, which in the chiral unitary approach has large $K\Lambda$, $K\Sigma$ components Kaiser:1995cy ; Inoue:2001ip ; Nieves:2001wt ; Hyodo:2008xr ; Gamermann:2011mq . Large $s\bar{s}$ components in that resonance have also been claimed in Xie:2007qt from the study of the $pp\to pp\phi$ and $\pi^{-}p\to n\phi$ reactions.

A wave of theoretical papers with very different approaches, stimulated by the LHCb findings, were produced trying to match the masses and spin parity quantum numbers suggested in the experimental work, $(3/2^{-},5/2^{+})$, $(3/2^{+},5/2^{-})$, $(5/2^{+},3/2^{-})$ for the two states, and other less likely combinations. In the meanwhile it has become apparent that the hadron community took too seriously these suggestions since the LHCb collaboration no longer sticks to any preference for these quantum numbers tomasz . We refer to review papers for references to all these works Chen:2016qju ; Lebed:2016hpi ; Esposito:2016noz ; Guo:2017jvc ; Ali:2017jda ; Olsen:2017bmm ; Karliner:2017qhf ; Cerri:2018ypt ; Liu:2019zoy .

With the advent of Run-2 data, the LHCb collaboration updated the results of Aaij:2015tga ; Aaij:2015fea reporting the observation of three clear narrow structures exp3 , branded as

[TABLE]

As one can see, the old peak at 4450 MeV is now split into two states at 4440 MeV and 4457 MeV, the last one very narrow, and a fluctuation observed in the old spectrum has given rise to a neat peak around 4312 MeV.

The new experimental findings have already had a reply from the theoretical community. In Chen:2019bip sum rules are used that provide several scenarios to explain these states, the most favored ones being of $\Sigma_{c}^{(*)}D^{(*)}$ molecular nature. In Liu:2019tjn heavy quark spin symmetry (HQSS) is used with $\Sigma_{c}\bar{D}$, $\Sigma_{c}\bar{D}^{*}$, $\Sigma_{c}^{*}\bar{D}$, $\Sigma_{c}^{*}\bar{D}^{*}$ as single channels and seven bound states are found, three of which can be associated with the experimental states. One should mention that in that line there is previous work, including other coupled channels, and which also predicts seven states with isospin $I=1/2$, and the widths of the states Xiao:2013yca .

Another work He:2019ify considers again the $\Sigma_{c}^{(*)}D^{(*)}$ coupled channels and, using meson exchange for the dynamics, generates three states that are associated to the new experimental resonances. There is also an interesting suggestion to look into the isospin suppressed $\Lambda_{b}\to J/\psi\Delta K^{-}$ reaction, showing that the ratio of rates for $J/\psi\Delta$ to $J/\psi p$ production is largely enhanced due to the molecular nature of the states Guo:2019fdo .

The blind predictions for the molecular hidden charm states have necessarily uncertainties, which are tied to the cutoff or subtraction constants needed to regularize the loops involved in the calculations. The differences in the results found among different approaches are mostly due to this point (see Refs. Wu:2010jy and Wu:2012md for instance). In this sense, differences of masses between the $3/2^{-}$ and $1/2^{-}$ states are more reliable. Thus, in Wu:2010jy one finds that this difference is 149 MeV and in Wu:2012md it is 141 MeV. Actually these numbers are very close to the differences between the masses of the $P_{c}(4457)$ and $P_{c}(4312)$, which is 145 MeV. In Xiao:2013yca this difference is 155 MeV.

In the works of Refs. Wu:2010jy ; Wu:2012md $\bar{D}\Sigma_{c}$ and $\bar{D}^{*}\Sigma_{c}$, among other coupled channels, were used, but not $\bar{D}\Sigma_{c}^{*}$, $\bar{D}^{*}\Sigma_{c}^{*}$. HQSS IW89 ; hssq2 ; MW00 relates the strength of the interaction of these channels and they were considered in Xiao:2013yca . The advent of the LHCb data offers an opportunity to tune the regulator of the loops to adjust to some experimental data. This is the purpose of the present work. It is similar to the study of Ref. Liu:2019tjn , but includes more channels than the $\Sigma_{c}^{(*)}D^{(*)}$ used in Liu:2019tjn , and in addition we work with coupled channels rather than using single channels, which allows us to obtain also the widths.

In Xiao:2013yca the Bethe-Salpeter equation is used with the coupled channels in $I=1/2$, $\eta_{c}N$, $J/\psi N$, $\bar{D}\Lambda_{c}$, $\bar{D}\Sigma_{c}$, $\bar{D}^{*}\Lambda_{c}$, $\bar{D}^{*}\Sigma_{c}$, $\bar{D}^{*}\Sigma_{c}^{*}$ for spin parity $J^{P}=1/2^{-}$ and $J/\psi N$, $\bar{D}^{*}\Lambda_{c}$, $\bar{D}^{*}\Sigma_{c}$, $\bar{D}\Sigma_{c}^{*}$, $\bar{D}^{*}\Sigma_{c}^{*}$ for $J^{P}=3/2^{-}$. In addition a single channel for $\bar{D}^{*}\Sigma_{c}^{*}$ in the $J^{P}=5/2^{-}$ sector is also studied. The Bethe-Salpeter equation in matrix form for the scattering matrix reads

[TABLE]

where $G$ is the loop function of the meson-baryon intermediate states and the potential $V$, respecting leading order (LO) HQSS constraints, is given in Eqs. (3)–(5) (taken from Ref. Xiao:2013yca ).

• •

$J=1/2$, $I=1/2$

[TABLE]

[TABLE]

• •

$J=3/2$, $I=1/2$

[TABLE]

[TABLE]

• •

$J=5/2$, $I=1/2$

[TABLE]

LO HQSS interactions for $I=3/2$ can be also found in Ref. Xiao:2013yca .

Note, that the single channel interactions used in Liu:2019tjn are recovered from Eqs. (3)–(5), identifying the terms $C_{a}$ and $C_{b}$ introduced in that reference to $(2\lambda_{2}/3+\mu_{3}/3)_{I=1/2}$ and $(\lambda_{2}/3-\mu_{3}/3)_{I=1/2}$, respectively.

There are seven parameters relying upon HQSS only, but when one imposes a particular dynamics, restrictions among them appear, as shown in Garcia-Recio:2013gaa . In the present work we shall consider the same constraints as in Xiao:2013yca , which stem from the use of an extension of the local hidden gauge approach, where the source of interaction is the exchange of vector mesons Bando:1984ej ; Bando:1987br ; Meissner:1987ge . Detailed discussions justifying this extension to the charm, or bottom sector, are given in Sakai:2017avl ; Debastiani:2017ewu . These constraints are for $I=1/2$

[TABLE]

with $f_{\pi}=93\textrm{ MeV}$, and $k^{0}$, $k^{\prime 0}$ the center of mass energies of the mesons in the $MB\to M^{\prime}B^{\prime}$ transition. In addition, $p^{2}_{D^{*}}$ applies to the $t-$channel exchanged $D^{*}$ in the tree level of some suppressed transitions ($\eta_{c}N\to\bar{D}\Lambda_{c}$ for instance).

The novelty with respect to Ref. Xiao:2013yca is a different choice of the subtraction constant to renormalize the meson-baryon loops ($G$) in dimensional regularization. A subtraction constant $a(\mu)=-2.3$ with $\mu=1$ GeV was used in Xiao:2013yca . This value was justified since it falls in the range of “natural values” discussed in Oller:2000fj and was also used in Wu:2010jy . The scheme produces seven states, three of which can be clearly associated to the recently found experimental resonances. The new information allows us to take a new value of $a(\mu=1\,{\rm GeV})=-2.09$, such that the sum of masses of the three theoretical states matches the experimental results. With this constraint we fix the only free parameter of the model of Ref. Xiao:2013yca . The results are reported in Tables 1 and 2 for $J^{P}=1/2^{-}$ and $3/2^{-}$, respectively. In addition we get a mass of 4519.23 MeV and a zero width for the single channel $\bar{D}^{*}\Sigma_{c}^{*}$ with $J=5/2^{-}$. This channel obviously does not couple to $J/\psi N$ so we should not see it in the $\Lambda_{b}\to J/\psi pK^{-}$ experiment. The states in Tables 1 and 2 all couple to $J/\psi N$ and in principle they could be seen in the experiment, although we cannot predict their strength in the spectrum. In Table. 3, we show the results for the three resonances that we identify with the experimental states. The main channel is taken from the largest coupling. We find the last two states nearly degenerate, yet, the widths of the states force us to identify the $3/2^{-}$ state with the $P_{c}(4457)$. Note that the masses divert only in a few MeV from the experimental ones, and the three widths obtained are compatible with the experiment. The results of Table 3 are similar to those of Liu:2019tjn , where the input has been adjusted to reproduce the $P_{c}(4440)$ and $P_{c}(4457)$ states. There is only a small difference since in Liu:2019tjn the $J^{P}$ assignments to the $P_{c}(4440)$ and $P_{c}(4457)$ are opposite to ours. Our approach, providing the width, gives us one additional reason to support our assignment. As to the molecular nature of the states, the single channel calculation of Liu:2019tjn gives the same state as those written in Table 3 as our main channel.

We should note that the reason why $\mu_{23}=0$ in Eq. (6) is the neglect of pion exchange which was found small, although not negligible in Xiao:2013yca . Its consideration would break the near degeneracy that we have in the two higher states of Table 3, as was found in Uchino:2015uha , where, however, the effect of pion exchange was found more important as a consequence of the choice of large cutoffs that made the binding much larger.

It looks strange that the widths obtained here are smaller than those reported in Xiao:2013yca in spite that the masses of the states are bigger and, hence, there is more phase space for decay. The answer has to be found in the fact that the couplings have also become smaller. This is not an accident but the consequence of one important property. Indeed it is well known that in the case of a one channel bound state, the coupling square, $g^{2}$, goes as the square root of the binding energy as a consequence of the most celebrated Weinberg’s compositeness condition Weinberg:1965zz ; Baru:2003qq . It is, however, less known that in the case of coupled channels, all couplings go to zero close to the threshold of one channel toki ; Gamermann:2009uq . In the present case the $P_{c}(4312)$ is close to the $\Sigma_{c}\bar{D}$ threshold and the $P_{c}(4440)$ and $P_{c}(4457)$ are very close to the $\Sigma_{c}\bar{D}^{*}$ threshold.

The association that we have done of the states found in this work with the experimental ones agrees with the one proposed in He:2019ify where, however, the widths are not evaluated. One should also note that in Liu:2019tjn and here we find seven states, while only three states are reported in He:2019ify . Actually, it is worth noting that in Liu:2019tjn a $3/2^{-}$ $\bar{D}\Sigma_{c}^{*}$ state is reported at 4371 MeV, while we find a state in Table 2, coupling mostly to $\bar{D}\Sigma_{c}^{*}$, at 4374 MeV with a width of about 14 MeV. It is interesting to call the attention to the fact that the $J/\psi p$ spectrum of Ref. exp3 shows a peak around 4370 MeV that could have well been identified with a new state. The strength of this peak is only about 1/2 of that of the $P_{c}(4312)$ and it is clearly distinguishable from other minor peaks that can be consistent with statistical fluctuations. We find two more states that can decay to $J/\psi p$ in Tables 1 and 2, a state of $1/2^{-}$ at 4520 MeV and a $3/2^{-}$ state at 4519 MeV, which couple mostly to $\bar{D}^{*}\Sigma_{c}^{*}$. The single channel results reported in Liu:2019tjn also find these two states at 4523 MeV and 4517 MeV, respectively, in their option A. With the risk of stretching too much the imagination there is indeed a peak in $J/\psi p$ spectrum of exp3 in that region that, however, it could as well be a statistical fluctuation. Note that we also obtain a near degenerate state with this nature for $5/2^{-}$. This state appears at 4500 MeV in option A and at 4523 in option B of Liu:2019tjn .

In summary, the molecular picture in the coupled channels to $J/\psi p$ in S-wave, using constraints of HQSS and dynamics from the extension of the local hidden gauge approach, basically an extension of the chiral unitary approach to the charm sector, renders six states that couple to $J/\psi p$. Three of these resonances can be identified with the three states reported in exp3 in base to their masses and widths. In addition, we provide a prediction of their $J^{P}$ quantum numbers and of the nature of these states as basically $1/2^{-}$ $\bar{D}\Sigma_{c}$, $1/2^{-}$ $\bar{D}^{*}\Sigma_{c}$ and $3/2^{-}$ $\bar{D}^{*}\Sigma_{c}$. We find a fourth state, which couples mostly to $\bar{D}\Sigma_{c}^{*}$ with $3/2^{-}$, for which there are indications in the $J/\psi p$ spectrum of exp3 . The other two states, of $\bar{D}^{*}\Sigma^{*}_{c}$ nature, are around 4520 MeV (close to the threshold of this meson-baryon pair), and although there are small peaks in that region in exp3 , one can only speculate at the present stage. They are also near degenerate with a $5/2^{-}$ state of the same nature, which however is not expected to show up in the LHCb experiment. This degeneracy is obvious from the diagonal $\bar{D}^{*}\Sigma_{c}^{*}$ interactions given in Eqs. (3)–(5), taking into account that the hidden gauge model used here leads to $\lambda_{2}=\mu_{3}$ for $I=1/2$.

Acknowledgments

This research has been supported by the Spanish Ministerio de Ciencia, Innovación y Universidades and European FEDER funds under Contracts FIS2017-84038-C2-1-P, FIS2017-84038-C2-2-P and SEV-2014-0398.