Establishing the first hidden-charm pentaquark with strangeness

Abstract

We study the $P_{cs}(4459)^0$ recently observed by LHCb using the method of QCD sum rules. Our results support its interpretation as the ${{\bar{D}}}^* \varXi _c$ hadronic molecular state of either $J^P=1/2^-$ or $3/2^-$. Within the hadronic molecular picture, the three LHCb experiments observing $P_c$ and $P_{cs}$ states (Aaij et al., Phys Rev Lett 115:072001, 2015; Aaij et al., Phys Rev Lett 122:222001, 2019; Aaij et al., arXiv:2012.10380 [hep-ex], 2012) can be well understood as a whole. This strongly supports the existence of hadronic molecules, whose studies can significantly improve our understanding on the construction of the subatomic world. To verify this picture, we propose to further investigate the $P_{cs}(4459)^0$ to examine whether it can be separated into two states, and to search for the ${{\bar{D}}} \varXi _c$ molecular state of $J^P=1/2^{-}$.

1 Introduction

Atomic nuclei are made of protons and neutrons, which are themselves composed of quarks and gluons. In the past century a huge number of subatomic particles, called hadrons, were discovered in particle experiments, whose properties are similar to the proton and neutron [4]. One naturally raises an interesting question: are there subatomic particles corresponding to the nucleus? Nowadays we call them “hadronic molecules”, whose studies can significantly improve our understanding on the construction of the subatomic world, as illustrated in Fig. 1.

It is not so easy to answer the above question. Most of the experimentally observed hadrons can be described as $q{{\bar{q}}}$ mesons or qqq baryons in the conventional quark model, while there can also exist (compact) $q{{\bar{q}}}q{{\bar{q}}}$ tetraquarks and $qqqq{{\bar{q}}}$ pentaquarks [5, 6], etc. With the experimental progress on this issue over the past decade, dozens of XYZ charmonium-like states were reported, providing us good opportunities to identify exotic hidden-charm tetraquarks [4]. However, these states may be explained as hadronic molecules, while they may also be explained as compact tetraquarks that are still hadrons.

Fortunately, in the LHCb experiments performed in 2015 and 2019 [1, 2], the famous hidden-charm pentaquark states, $P_c(4312)$, $P_c(4380)$, $P_c(4440)$, and $P_c(4457)$, were discovered. These four $P_c$ states contain at least five quarks ${{\bar{c}}} c uud$, so they are perfect candidates of hidden-charm pentaquark states. Among them, the three narrow states $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$ are just below the ${{\bar{D}}} \varSigma _c$ and ${{\bar{D}}}^{*} \varSigma _c$ thresholds, so their natural interpretations are the ${{\bar{D}}} \varSigma _c$ and ${{\bar{D}}}^{*} \varSigma _c$ hadronic molecular states [7,8,9,10,11], whose existence had been predicted in Refs. [12,13,14,15,16] before the LHCb experiment performed in 2015 [1]. However, there still exist other possible explanations [17,18,19,20,21,22], and we refer to the reviews [23,24,25,26,27,28] for detailed discussions.

It is natural to conjecture whether the hidden-charm pentaquark state with strangeness exists or not. Such a state is usually denoted as “$P_{cs}$”, whose quark content is ${{\bar{c}}} c s q q$ ($q=u/d$). There have been some but not many theoretical studies on it [29,30,31,32,33,34,35,36,37]. Especially, we proposed in Ref. [38] to search for the $P_{cs}$ in the $J/\psi \varLambda$ invariant mass spectrum of the $\varXi _b^- \rightarrow J/\psi K^- \varLambda$ decays. Besides, in Ref. [39] the authors studied the $P_{cs}$ using the chiral effective field theory, and calculated the mass of the $J^P = 1/2^-$ ${{\bar{D}}}^{*} \varXi _c$ molecular state to be $4456.9^{+3.2}_{-3.3}$ MeV. Note that both of these two references are based on the hadronic molecular picture.

Fig. 1

A possible way to construct the subatomic world

Very recently, the LHCb Collaboration reported the evidence of a hidden-charm pentaquark state with strangeness, $P_{cs}(4459)^0$, in the $J/\psi \varLambda$ invariant mass spectrum of the $\varXi _b^- \rightarrow J/\psi K^- \varLambda$ decays [3]. Its mass and width were measured to be:

$$\begin{aligned}&P_{cs}(4459)^0: M= 4458.8 \pm 2.9^{+4.7}_{-1.1} \text{ MeV } \, , \nonumber \\&\varGamma = 17.3 \pm 6.5^{+8.0}_{-5.7} \text{ MeV } \, , \end{aligned}$$

(1)

while its spin-parity quantum number was not determined since the statistic is not enough. Note that the channel observing the $P_{cs}(4459)^0$ is just the one proposed by us in Ref. [38], and the above mass value is almost identical to the mass of the $J^P = 1/2^-$ ${{\bar{D}}}^{*} \varXi _c$ molecular state predicted in Ref. [39]. Therefore, the present LHCb experiment [3] strongly supports the hadronic molecular picture once more.

Actually, as indicated by LHCb, the $P_{cs}(4459)^0$ is about 19 MeV below the ${{\bar{D}}}^{*0} \varXi _c^0$ threshold [3], so it is natural to interpret it as the ${{\bar{D}}}^{*} \varXi _c$ molecular state, with the spin-parity quantum number $J^P = 1/2^-$ or $3/2^-$. Accordingly, in this letter we shall study the ${{\bar{D}}}^{*} \varXi _c$ molecular states of $J^P = 1/2^-$ and $3/2^-$ using the method of QCD sum rules, and at the same time we shall also study the ${{\bar{D}}} \varXi _c$ molecular state of $J^P = 1/2^-$. We calculate masses of ${{\bar{D}}}^* \varXi _c$ molecular states to be $4.46^{+0.16}_{-0.14}$ GeV for the $J^P = 1/2^-$ one and $4.47^{+0.19}_{-0.15}$ GeV for the $J^P = 3/2^-$ one. These two values are both consistent with the experimental mass of the $P_{cs}(4459)^0$, supporting its interpretation as the ${{\bar{D}}}^* \varXi _c$ molecular state of either $J^P=1/2^-$ or $3/2^-$. We also calculate the mass of the $J^P = 1/2^-$ ${{\bar{D}}} \varXi _c$ molecular state to be $4.29^{+0.13}_{-0.12}$ GeV.

To verify the hadronic molecular picture, we propose to further investigate the $P_{cs}(4459)^0$ state in future experiments to examine whether it can be separated into two states, and to search for the $J^P = 1/2^-$ ${{\bar{D}}} \varXi _c$ molecular state. If the hadronic molecular picture turns out to be correct, our understanding on the construction of the subatomic world would be significantly improved. Besides, these studies are helpful to improve our understanding on the non-perturbative behaviors of the strong interaction at the low energy region.

2 Hidden-charm pentaquark currents

We use the ${{\bar{c}}}$, c, s, u, and d quarks to construct hidden-charm pentaquark interpolating currents with strangeness. To study ${{\bar{D}}}^{(*)} \varXi _c$ molecular states, we consider the following type of currents:

$$\begin{aligned} \eta (x) = [{{\bar{c}}}_a(x) \varGamma _1 u_b(x)] ~ \Big [[d^T_c(x) {\mathbb {C}} \varGamma _2 s_d(x)] ~ \varGamma _3 c_e(x) \Big ] \, , \end{aligned}$$

(2)

where $a \cdots e$ are color indices, $\varGamma _{1/2/3}$ are Dirac matrices, and ${\mathbb {C}} = i\gamma _2 \gamma _0$ is the charge-conjugation operator. The other type of currents:

$$\begin{aligned} \eta ^\prime (x) = [{{\bar{c}}}_a(x) \varGamma _1 d_b(x)] ~ \Big [[u^T_c(x) {\mathbb {C}} \varGamma _2 s_d(x)] ~ \varGamma _3 c_e(x) \Big ] \, , \end{aligned}$$

(3)

can be similarly studied, but they just lead to the same QCD sum rule results as the $\eta (x)$ currents. Hence, the present study can not distinguish the isospin of $P_{cs}$ states.

The $\eta (x)$ currents can be constructed by combining charmed meson operators and charmed baryon fields. We need the charmed meson operators $J_{{\mathcal {D}}}$, which couple to the ground-state charmed mesons ${\mathcal {D}} = {{\bar{D}}}^0/{{\bar{D}}}^{*0}$:

$$\begin{aligned} J_{{{\bar{D}}}^0}= & {} {{\bar{c}}}_d \gamma _5 u_d \, ,\nonumber \\ J_{{{\bar{D}}}^{*0}}= & {} {{\bar{c}}}_d \gamma _\mu u_d \, . \end{aligned}$$

(4)

We also need the charmed baryon field $J_{{\mathcal {B}}}$, which couples to the ground-state charmed baryon ${\mathcal {B}} = \varXi _c^0$:

$$\begin{aligned} J_{\varXi _c^0}= & {} \epsilon ^{abc} [d_a^T {\mathbb {C}} \gamma _{5} s_b] c_c \, . \end{aligned}$$

(5)

There can be altogether three ${{\bar{D}}}^{(*)} \varXi _c$ hadronic molecular states, that are ${{\bar{D}}} \varXi _c$ of $J^P = {1/2}^-$, ${{\bar{D}}}^* \varXi _c$ of $J^P = {1/2}^-$, and ${{\bar{D}}}^* \varXi _c$ of $J^P = {3/2}^-$. Their relevant interpolating currents are:

$$\begin{aligned} \eta _1= & {} [\delta ^{ab} {{\bar{c}}}_a \gamma _5 u_b] ~ [\epsilon ^{cde} d_c^T {\mathbb {C}} \gamma _5 s_d c_e] \nonumber \\= & {} {{\bar{D}}}^0 ~ \varXi _c^{0} \, , \end{aligned}$$

(6)

$$\begin{aligned} \eta _2= & {} [\delta ^{ab} {{\bar{c}}}_a \gamma _\nu u_b] ~ \gamma ^\nu \gamma _5 ~ [\epsilon ^{cde} d_c^T {\mathbb {C}} \gamma _5 s_d c_e]\nonumber \\= & {} {{\bar{D}}}^{*0}_\nu ~ \gamma ^\nu \gamma _5 ~ \varXi _c^{0} \, , \end{aligned}$$

(7)

$$\begin{aligned} \eta _3^\alpha= & {} P_{3/2}^{\alpha \nu } ~ [\delta ^{ab} {{\bar{c}}}_a \gamma _\nu u_b] ~ [\epsilon ^{cde} d_c^T {\mathbb {C}} \gamma _5 s_d c_e] \nonumber \\= & {} P_{3/2}^{\alpha \nu } ~ {{\bar{D}}}^{*0}_\nu ~ \varXi _c^{0} \, . \end{aligned}$$

(8)

In the above expressions, we have used ${\mathcal {D}}$ and ${\mathcal {B}}$ to denote the charmed meson operators $J_{{\mathcal {D}}}$ and the charmed baryon field $J_{{\mathcal {B}}}$; $P_{3/2}^{\mu \nu }$ is the spin-3/2 projection operator

$$\begin{aligned} P_{3/2}^{\mu \nu } = g^{\mu \nu } - {1 \over 4} \gamma ^\mu \gamma ^\nu \, . \end{aligned}$$

(9)

2.1 QCD sum rule studies

We use the method of QCD sum rules [40, 41] to study ${{\bar{D}}}^{(*)} \varXi _c$ molecular states through the currents $\eta _{1,2}$ of $J^P = 1/2^-$ and $\eta ^\alpha _{3}$ of $J^P = 3/2^-$. Taking the current $\eta _2$ as an example, we assume it couples to the ${{\bar{D}}}^* \varXi _c$ molecular state of $J^P = 1/2^-$ through

$$\begin{aligned} \langle 0 | \eta _2 | X; 1/2^- \rangle= & {} f_X u (p) \, , \end{aligned}$$

where u(p) is the Dirac spinor of this state, denoted as X for simplicity. The two-point correlation function extracted from $\eta _2$ can be written as:

$$\begin{aligned} \varPi \left( q^2\right)= & {} i \int d^4x e^{iq\cdot x} \langle 0 | T\left[ \eta _{2}(x) {{\bar{\eta }}}_{2}(0)\right] | 0 \rangle \nonumber \\= & {} (q\!\!\!/~ + M_{X}) ~ \varPi _0\left( q^2\right) \, . \end{aligned}$$

(10)

In QCD sum rule studies we calculate the two-point correlation function $\varPi _0\left( q^2\right)$ at both hadron and quark-gluon levels. At the hadron level, we use the dispersion relation to write it as

$$\begin{aligned} \varPi _0(q^2)={\frac{1}{\pi }}\int ^\infty _{s_<}\frac{\mathrm{Im} \varPi _0(s)}{s-q^2-i\varepsilon }ds \, , \end{aligned}$$

(11)

where $s_<$ is the physical threshold. We further define the imaginary part of the correlation function as the spectral density $\rho (s)$, which is usually evaluated at the hadron level by inserting intermediate hadron states $\sum _n|n\rangle \langle n|$:

$$\begin{aligned} \rho _{\mathrm{phen}}(s)\equiv & {} \mathrm{Im}\varPi _0(s)/\pi \nonumber \\= & {} \sum _n\delta (s-M^2_n)\langle 0|\eta |n\rangle \langle n|{\eta ^\dagger }|0\rangle \nonumber \\= & {} f_X^2\delta (s-m_X^2)+ \text{ continuum }. \end{aligned}$$

(12)

In the last step we have adopted the usual parametrization of one-pole dominance for the ground state X and a continuum contribution.

At the quark-gluon level we calculate $\varPi _0\left( q^2\right)$ using the method of operator product expansion (OPE), and derive its corresponding spectral density $\rho _{\mathrm{OPE}}(s)$. After performing the Borel transformation to Eq. (11) at both hadron and quark-gluon levels, we can approximate the continuum using the spectral density above a threshold value $s_0$ (quark-hadron duality), and obtain the sum rule equation

$$\begin{aligned} \varPi _0(s_0, M_B^2) \equiv f^2_X e^{-M_X^2/M_B^2} = \int ^{s_0}_{s_<} e^{-s/M_B^2}\rho _{\mathrm{OPE}}(s)ds \, . \end{aligned}$$

(13)

It can be used to further calculate $M_X$ through

$$\begin{aligned} M^2_X(s_0, M_B)= & {} \frac{\int ^{s_0}_{s_<} e^{-s/M_B^2}s\rho _{\mathrm{OPE}}(s)ds}{\int ^{s_0}_{s_<} e^{-s/M_B^2}\rho _{\mathrm{OPE}}(s)ds} \, . \end{aligned}$$

(14)

In the present study we calculate OPEs at the leading order of $\alpha _s$ and up to the $D(\mathrm{imension}) = 10$ terms, including the perturbative term, the charm quark mass, the quark condensates $\langle {{\bar{q}}} q \rangle /\langle {{\bar{s}}} s \rangle$, the gluon condensate $\langle g_s^2 GG \rangle$, the quark-gluon mixed condensates $\langle g_s {{\bar{q}}} \sigma G q \rangle /\langle g_s {{\bar{s}}} \sigma G s \rangle$, and their combinations. We summarized the obtained spectral densities $\rho _{1\cdots 3}(s)$ in “Appendix A”, which are extracted from the currents $\eta _{1\cdots 3}$, respectively. In the calculations we ignore the chirally suppressed terms with the up and down quark masses, and adopt the factorization assumption of vacuum saturation for higher dimensional condensates. We find that the $D=3$ quark condensates $\langle {{\bar{q}}} q \rangle /\langle {{\bar{s}}} s \rangle$ and the $D=5$ mixed condensates $\langle g_s {{\bar{q}}} \sigma G q \rangle /\langle g_s {{\bar{s}}} \sigma G s \rangle$ are multiplied by the charm quark mass, which are thus important power corrections.

2.2 Numerical analyses

We still use the current $\eta _2$ as an example to perform numerical analyses, where we use the following values for various QCD sum rule parameters [4, 42,43,44,45,46,47,48,49,50]:

$$\begin{aligned} m_s= & {} 96 ^{+8}_{-4} \text{ MeV } \, , \nonumber \\ m_c= & {} 1.275 ^{+0.025}_{-0.035} \text{ GeV } \, , \nonumber \\ \langle {{\bar{q}}}q \rangle= & {} -(0.240 \pm 0.010)^3 \text{ GeV}^3 \, ,\nonumber \\ \langle {{\bar{s}}}s \rangle= & {} (0.8\pm 0.1)\times \langle {{\bar{q}}}q \rangle \, , \nonumber \\ \langle g_s^2GG\rangle= & {} (0.48\pm 0.14) \text{ GeV}^4 \, , \nonumber \\ \langle g_s{{\bar{q}}}\sigma G q\rangle= & {} - M_0^2\times \langle {{\bar{q}}}q\rangle \, , \nonumber \\ \langle g_s{{\bar{s}}}\sigma G s\rangle= & {} - M_0^2\times \langle {{\bar{s}}}s\rangle \, , \nonumber \\ M_0^2= & {} (0.8 \pm 0.2) \text{ GeV}^2 \, . \end{aligned}$$

(15)

Here the running mass in the ${\overline{MS}}$ scheme is used for the charm quark.

There are two free parameters in Eqs. (14): the Borel mass $M_B$ and the threshold value $s_0$. We use two criteria to constrain the Borel mass $M_B$ for a fixed $s_0$. The first criterion is used to insure the convergence of the OPE series. It is done by requiring the $D=10$ terms $\big (m_c \langle {{\bar{q}}} q \rangle ^3$ and $\langle g_s {{\bar{q}}} \sigma G q \rangle ^2\big )$ to be less than 20%, which can determine the lower limit $M_B^{min}$:

$$\begin{aligned} \text{ Convergence } \text{(CVG) } \equiv \left| \frac{ \varPi ^{D=10}(\infty , M_B) }{ \varPi (\infty , M_B) }\right| \le 20\% \, . \end{aligned}$$

(16)

This criterion leads to $\left( M_B^{min}\right) ^2 = 2.93$ $\hbox {GeV}^2$, when setting $s_0 = 25.8$ $\hbox {GeV}^2$.

The second criterion is used to insure the validity of one-pole parametrization. It is done by requiring the pole contribution to be larger than 50%, which can determine the upper limit $M_B^{max}$:

$$\begin{aligned} \text{ Pole } \text{ Contribution } \text{(PC) } \equiv \frac{ \varPi (s_0, M_B) }{ \varPi (\infty , M_B) } \ge 50\% \, . \end{aligned}$$

(17)

This criterion leads to $\left( M_B^{max}\right) ^2 = 3.07$ $\hbox {GeV}^2$, when setting $s_0 = 25.8$ $\hbox {GeV}^2$.

Fig. 2

The variation of $M_X$ with respect to the Borel mass $M_B$, calculated using the current $\eta _2$. The short-dashed, solid, and long-dashed curves are obtained by setting $s_0 = 24.8$, 25.8, and 26.8 $\hbox {GeV}^2$, respectively

Altogether we extract the working region of Borel mass to be 2.93 $\hbox {GeV}^2< M_B^2 < 3.07$ $\hbox {GeV}^2$ for the current $\eta _2$ with the threshold value $s_0 = 25.8$ $\hbox {GeV}^2$. We show the variation of $M_{X}$ with respect to the Borel mass $M_B$ in Fig. 2 in a broader region 2.7 $\hbox {GeV}^2\le M_B^2 \le 3.2$ $\hbox {GeV}^2$, and find it more stable inside the above Borel window.

Redoing the same procedures by changing $s_0$, we find that there are non-vanishing Borel windows as long as $s_0 \ge s_0^{min} = 24.8$ $\hbox {GeV}^2$. Accordingly, we choose the threshold value $s_0$ to be about 1.0 GeV larger with the uncertainty $\pm 1.0$ GeV, i.e., 24.8 $\hbox {GeV}^2\le s_0\le 26.8$ $\hbox {GeV}^2$. Altogether, our working regions for the current $\eta _2$ are determined to be 24.8 $\hbox {GeV}^2\le s_0\le 26.8$ $\hbox {GeV}^2$ and 2.93 $\hbox {GeV}^2< M_B^2 < 3.07$ $\hbox {GeV}^2$, where the mass is extracted to be:

$$\begin{aligned} M_{{{\bar{D}}}^* \varXi _c; 1/2^-} = 4.46^{+0.16}_{-0.14} \text{ GeV } \, . \end{aligned}$$

(18)

Here the central value corresponds to $M_B^2=3.00$ $\hbox {GeV}^2$ and $s_0 = 25.8$ $\hbox {GeV}^2$. Its uncertainty comes from the Borel mass $M_B$, the threshold value $s_0$, the charm quark mass $m_c$, and various QCD sum rule parameters listed in Eqs. (15). This mass value is consistent with the experimental mass of the $P_{cs}(4459)^0$, supporting its interpretation as the ${{\bar{D}}}^* \varXi _c$ molecular state of $J^P=1/2^-$.

Similarly, we use the currents $\eta _{1}$ and $\eta ^\alpha _{3}$ to perform numerical analyses, and extract masses of the $J^P=1/2^-$ ${{\bar{D}}} \varXi _c$ molecular state and the $J^P=3/2^-$ ${{\bar{D}}}^* \varXi _c$ molecular state to be:

$$\begin{aligned} M_{{{\bar{D}}} \varXi _c; 1/2^-}= & {} 4.29^{+0.13}_{-0.12} \text{ GeV } \, , \end{aligned}$$

(19)

$$\begin{aligned} M_{{{\bar{D}}}^* \varXi _c; 3/2^-}= & {} 4.47^{+0.19}_{-0.15} \text{ GeV } \, . \end{aligned}$$

(20)

Hence, our results also support the interpretation of the $P_{cs}(4459)^0$ as the ${{\bar{D}}}^* \varXi _c$ molecular state of $J^P=3/2^-$. We summarize all the above results in Table 1.

Generally speaking, understanding the nature of exotic hadrons is a complicated topic, since different structures with the same quantum numbers can contribute to the same state, and different structures may have similar masses. However, these different structures may lead to different decay processes. Therefore, to determine whether the $P_{cs}(4459)^0$ is the ${{\bar{D}}}^* \varXi _c$ molecular state of $J^P=1/2^-$ or the one of $J^P=3/2^-$ in our framework, we shall further calculate its width in our future study, to be compared with its experimental value $\varGamma _{P_{cs}(4459)^0} = 17.3 \pm 6.5^{+8.0}_{-5.7}$ MeV [3].

Table 1 Masses of the ${{\bar{D}}}^{(*)} \varXi _c$ hadronic molecular states, extracted from the currents $\eta ^{(\alpha )}_{1\cdots 3}$

3 Summary and discussions

Very recently, the LHCb Collaboration reported the evidence of a hidden-charm pentaquark state with strangeness, $P_{cs}(4459)^0$, in the $J/\psi \varLambda$ invariant mass spectrum of the $\varXi _b^- \rightarrow J/\psi K^- \varLambda$ decays [3]. This state contains at least five quarks ${{\bar{c}}} c s qq$ ($q=u/d$), with one of them the strange quark. This LHCb experiment indicates that there probably exist many more exotic hadrons with strangeness to be discovered in the near future, so a new hadron spectrum is waiting to be constructed.

The $P_{cs}(4459)^0$ is about 19 MeV below the ${{\bar{D}}}^{*0} \varXi _c^0$ threshold, so it is natural to interpret it as the ${{\bar{D}}}^{*} \varXi _c$ molecular state. Accordingly, in this letter we use the method of QCD sum rules to study the ${{\bar{D}}}^{*} \varXi _c$ molecular states of $J^P=1/2^-$ and $3/2^-$, and at the same time we also study the ${{\bar{D}}} \varXi _c$ molecular state of $J^P=1/2^-$. We evaluate their masses to be

$$\begin{aligned} M_{{{\bar{D}}} \varXi _c; 1/2^-}= & {} 4.29^{+0.13}_{-0.12} \text{ GeV } \, ,\\ M_{{{\bar{D}}}^* \varXi _c; 1/2^-}= & {} 4.46^{+0.16}_{-0.14} \text{ GeV } \, ,\\ M_{{{\bar{D}}}^* \varXi _c; 3/2^-}= & {} 4.47^{+0.19}_{-0.15} \text{ GeV } \, . \end{aligned}$$

Hence, our QCD sum rule results support the interpretation of the $P_{cs}(4459)^0$ as the ${{\bar{D}}}^* \varXi _c$ molecular state of either $J^P=1/2^-$ or $3/2^-$.

Fig. 3

Possible production mechanisms of the $P_c^+$ and $P_{cs}^0$ states in $\varLambda _b^0/\varXi _b^-$ decays

Within the hadronic molecular picture, the three LHCb experiments observing $P_c$ and $P_{cs}$ states [1,2,3] can be well understood as a whole:

• The transition of $\varXi _b^-\rightarrow J/\psi K^-\varLambda$ is dominated by the Cabibbo-favored weak decay of $b\rightarrow c + {\bar{c}}s$ via the V-A current. This leads to an intuitive expectation that the s and d pair in $\varXi _b^-$ may exist as a spectator, so that their total spin $S=0$ is conserved. As a consequence, if this sd pair is to be combined with a c quark to form a charmed baryon, it will favor the $\varXi _c$ instead of the $\varXi _c^\prime$ and $\varXi _c^*$, of which the total spin of the sd pair is $S = 1$. Accordingly, the $P_{cs}(4459)^0$ possibly as the ${{\bar{D}}}^* \varXi _c$ molecular state was observed in this channel by LHCb [3], other than those possibly existing ${{\bar{D}}}^{(*)} \varXi _c^\prime$ and ${{\bar{D}}}^{(*)} \varXi _c^*$ molecular states. We illustrate this process in Fig. 3(a).

• The transition of $\varLambda _b^0\rightarrow J/\psi K^- p$ can be similarly analysed, which may favor the $\varLambda _c$ instead of the $\varSigma _c$ and $\varSigma _c^*$. However, in this case the $\varLambda _c$ is probably not bounded with the charmed mesons due to the lack of $\pi$ exchanges. It is only because of the significantly larger data sample (680k for the $\varLambda _b^0\rightarrow J/\psi K^- p$ decays and only 4k for the $\varXi _b^-\rightarrow J/\psi K^-\varLambda$ decays [3]), that the $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$ possibly as the ${{\bar{D}}} \varSigma _c$ and ${{\bar{D}}}^{*} \varSigma _c$ molecular states were observed in this channel by LHCb [1, 2]. We illustrate the relevant two processes in Fig. 3b, c.

Therefore, the three LHCb experiments observing $P_c$ and $P_{cs}$ states [1,2,3] strongly support the hadronic molecular picture and the existence of “hadronic molecules”.

To further verify the above hadronic molecular picture, we propose to investigate the $P_{cs}(4459)^0$ in future experiments to examine whether it can be separated into two states. We also propose to search for the ${{\bar{D}}} \varXi _c$ molecular state of $J^P=1/2^-$, whose mass is predicted to be $4.29^{+0.13}_{-0.12}$ GeV. If the hadronic molecular picture turns out to be correct, our understanding on the construction of the subatomic world be significantly improved, and our understanding on the non-perturbative behaviors of the strong interaction at the low energy region would also be significantly improved.

Appendix: Spectral densities

In this appendix we list the spectral densities $\rho _{1\cdots 3}(s)$ extracted for the currents $\eta _{1\cdots 3}$. In the following expressions, ${\mathcal {F}}(s) = \left[ (\alpha +\beta )m_c^2-\alpha \beta s\right]$, ${\mathcal {H}}(s) = \left[ m_c^2-\alpha (1-\alpha ) s\right]$, and the integration limits are $\alpha _{min}=\frac{1-\sqrt{1-4m_c^2/s}}{2}$, $\alpha _{max}=\frac{1+\sqrt{1-4m_c^2/s}}{2}$, $\beta _{min}=\frac{\alpha m_c^2}{\alpha s-m_c^2}$, and $\beta _{max}=1-\alpha$. In the calculations we take into account both the $m_s$ and $m_s^2$ terms, but here we list only the $m_s$ terms.

The spectral density $\rho _{1}(s)$ extracted for the current $\eta _{1}$ is

$$\begin{aligned} \rho _{1}(s)= & {} \rho ^{pert}_{1}(s) + \rho ^{\langle {{\bar{q}}}q\rangle }_{1}(s) + \rho ^{\langle GG\rangle }_{1}(s)+ \rho ^{\langle {{\bar{q}}}Gq\rangle }_{1}(s) + \rho ^{\langle {{\bar{q}}}q\rangle ^2}_{1}(s) \nonumber \\&+\, \rho ^{\langle {{\bar{q}}}q\rangle \langle {{\bar{q}}}Gq\rangle }_{1}(s)+ \rho ^{\langle {{\bar{q}}}Gq\rangle ^2}_{1}(s) + \rho ^{\langle {{\bar{q}}}q\rangle ^3}_{1}(s) \, , \end{aligned}$$

(21)

where

$$\begin{aligned} \rho ^{pert}_{1}(s)= & {} \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \int ^{\beta _{max}}_{\beta _{min}}d\beta \Bigg \{ {\mathcal {F}}(s)^5 \times \frac{(1 - \alpha - \beta )^3}{163840 \pi ^8 \alpha ^5 \beta ^4} \Bigg \} \, ,\\ \rho ^{\langle {{\bar{q}}}q\rangle }_{1}(s)= & {} \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \int ^{\beta _{max}}_{\beta _{min}}d\beta \Bigg \{ {\mathcal {F}}(s)^3 \\&\times \frac{(1 - \alpha - \beta ) ( (\alpha +\beta -1) \beta m_c \langle {{\bar{q}}}q\rangle + \beta m_s (\langle {{\bar{s}}}s\rangle - 2 \langle {{\bar{q}}}q\rangle ))}{2048 \pi ^6 \alpha ^3 \beta ^3} \Bigg \} \, ,\\ \rho ^{\langle GG\rangle }_{1}(s)= & {} {\langle g_s^2GG\rangle } \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \int ^{\beta _{max}}_{\beta _{min}}d\beta \Bigg \{ m_c^2 {\mathcal {F}}(s)^2 \\&\times \frac{(1 - \alpha - \beta )^3 \left( \alpha ^3+\beta ^3\right) }{196608 \pi ^8 \alpha ^5 \beta ^4}\\&+ {\mathcal {F}}(s)^3 \times \frac{(\alpha +\beta -1) \left( 3 \alpha ^3-\alpha ^2 (\beta +3)-2 \alpha (\beta -1) \beta -(\beta -1)^2 \beta \right) }{196608 \pi ^8 \alpha ^5 \beta ^3} \Bigg \} \, ,\\ \rho ^{\langle {{\bar{q}}}Gq\rangle }_{1}(s)= & {} \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \int ^{\beta _{max}}_{\beta _{min}}d\beta \Bigg \{ {\mathcal {F}}(s)^2 \\&\times \frac{-3 \left( \alpha ^2+\alpha (3 \beta -2)+2 \beta ^2-3 \beta +1\right) {m_c\langle g_s{{\bar{q}}}\sigma Gq\rangle } + 3 \beta ^2 {m_s\langle g_s{{\bar{q}}}\sigma Gq\rangle } }{4096 \pi ^6 \alpha ^2 \beta ^3} \Bigg \} \, ,\\ \rho ^{\langle {{\bar{q}}}q\rangle ^2}_{1}(s)= & {} \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \int ^{\beta _{max}}_{\beta _{min}}d\beta \Bigg \{\frac{- {\mathcal {F}}(s)^2 \times \langle {{\bar{q}}}q\rangle \langle {{\bar{s}}}s\rangle + m_c m_s {\mathcal {F}}(s) \times \alpha (2 \langle {{\bar{q}}}q\rangle ^2 - \langle {{\bar{q}}}q\rangle \langle {{\bar{s}}}s\rangle )}{256 \pi ^4 \alpha ^2 \beta } \Bigg \} \, ,\\ \rho ^{\langle {{\bar{q}}}q\rangle \langle {{\bar{q}}}Gq\rangle }_{1}(s)= & {} \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \Bigg \{\int ^{\beta _{max}}_{\beta _{min}}d\beta \Bigg \{ \frac{m_c m_s \langle g_s{{\bar{q}}}\sigma Gq\rangle (2 \langle {{\bar{q}}}q\rangle - \langle {{\bar{s}}}s\rangle )}{512 \pi ^4 \beta } \Bigg \}\\&+ {\mathcal {H}}(s) \times \frac{ \langle {{\bar{q}}}q\rangle \langle g_s{{\bar{s}}}\sigma Gs\rangle + \langle {{\bar{s}}}s\rangle \langle g_s{{\bar{q}}}\sigma Gq\rangle }{512 \pi ^4 \alpha } \\&-\frac{ m_c m_s \langle g_s{{\bar{q}}}\sigma Gq\rangle (4 \langle {{\bar{q}}}q\rangle - \langle {{\bar{s}}}s\rangle )}{1024 \pi ^4} \Bigg \} \, ,\\ \rho ^{\langle {{\bar{q}}}Gq\rangle ^2}_{1}(s)= & {} \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \Bigg \{\frac{(\alpha -1) \langle g_s{{\bar{q}}}\sigma Gq\rangle \langle g_s{{\bar{s}}}\sigma Gs\rangle }{1024 \pi ^4}\Bigg \}\\&+ \int ^{1}_{0}d\alpha \Bigg \{ \delta \left( s - {m_c^2 \over \alpha (1-\alpha )}\right) \\&\times \frac{\langle g_s{{\bar{q}}}\sigma Gq\rangle \left( m_c^3 m_s \langle g_s{{\bar{q}}}\sigma Gq\rangle /M_B^2 + (1-\alpha ) m_c^2 \langle g_s{{\bar{s}}}\sigma Gs\rangle - \alpha (\alpha +1) m_c m_s \langle g_s{{\bar{q}}}\sigma Gq\rangle \right) }{2048 \pi ^4 (\alpha -1) \alpha } \Bigg \} \, ,\\ \rho ^{\langle {{\bar{q}}}q\rangle ^3}_{1}(s)= & {} {m_c\langle {{\bar{q}}}q\rangle ^2 \langle {{\bar{s}}}s\rangle } \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \Bigg \{\frac{1}{96 \pi ^2} \Bigg \} \, . \end{aligned}$$

The spectral density $\rho _{2}(s)$ extracted for the current $\eta _{2}$ is

$$\begin{aligned} \rho _{2}(s)= & {} \rho ^{pert}_{2}(s) + \rho ^{\langle {{\bar{q}}}q\rangle }_{2}(s) + \rho ^{\langle GG\rangle }_{2}(s)+ \rho ^{\langle {{\bar{q}}}Gq\rangle }_{2}(s) + \rho ^{\langle {{\bar{q}}}q\rangle ^2}_{2}(s) \nonumber \\&+ \rho ^{\langle {{\bar{q}}}q\rangle \langle {{\bar{q}}}Gq\rangle }_{2}(s)+ \rho ^{\langle {{\bar{q}}}Gq\rangle ^2}_{2}(s) + \rho ^{\langle {{\bar{q}}}q\rangle ^3}_{2}(s) \, , \end{aligned}$$

(22)

where

$$\begin{aligned} \rho ^{pert}_{2}(s)= & {} \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \int ^{\beta _{max}}_{\beta _{min}}d\beta \Bigg \{ {\mathcal {F}}(s)^5\times \frac{(1 - \alpha - \beta )^3}{81920 \pi ^8 \alpha ^5 \beta ^4} \Bigg \} \, ,\\ \rho ^{\langle {{\bar{q}}}q\rangle }_{2}(s)= & {} \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \int ^{\beta _{max}}_{\beta _{min}}d\beta \Bigg \{ {\mathcal {F}}(s)^3\\&\times \frac{ (1 - \alpha - \beta ) (2 (\alpha +\beta -1) m_c \langle {{\bar{q}}}q\rangle + \beta m_s (\langle {{\bar{s}}}s\rangle - 2 \langle {{\bar{q}}}q\rangle ))}{1024 \pi ^6 \alpha ^3 \beta ^3} \Bigg \} \, ,\\ \rho ^{\langle GG\rangle }_{2}(s)= & {} {\langle g_s^2GG\rangle } \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \int ^{\beta _{max}}_{\beta _{min}}d\beta \Bigg \{ m_c^2 {\mathcal {F}}(s)^2 \times \frac{ (1 - \alpha - \beta )^3 \left( \alpha ^3+\beta ^3\right) }{98304 \pi ^8 \alpha ^5 \beta ^4}\\&+ {\mathcal {F}}(s)^3 \times \frac{ (1 - \alpha - \beta ) \left( 3 \alpha ^3+\alpha ^2 (7 \beta -3)+2 \alpha (\beta -1) \beta +(\beta -1)^2 \beta \right) }{98304 \pi ^8 \alpha ^5 \beta ^3} \Bigg \} \, ,\\ \rho ^{\langle {{\bar{q}}}Gq\rangle }_{2}(s)= & {} \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \int ^{\beta _{max}}_{\beta _{min}}d\beta \Bigg \{ {\mathcal {F}}(s)^2 \times \frac{- 3 \langle g_s{{\bar{q}}}\sigma Gq\rangle (2 (\alpha +\beta -1) m_c - \beta m_s)}{2048 \pi ^6 \alpha ^2 \beta ^2} \Bigg \} \, ,\\ \rho ^{\langle {{\bar{q}}}q\rangle ^2}_{2}(s)= & {} \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \int ^{\beta _{max}}_{\beta _{min}}d\beta \Bigg \{\frac{ - {\mathcal {F}}(s)^2 \times \langle {{\bar{q}}}q\rangle \langle {{\bar{s}}}s\rangle - m_c m_s {\mathcal {F}}(s) \times 2 \alpha ( \langle {{\bar{s}}}s\rangle - 2 \langle {{\bar{q}}}q\rangle )}{128 \pi ^4 \alpha ^2 \beta } \Bigg \} \, ,\\ \rho ^{\langle {{\bar{q}}}q\rangle \langle {{\bar{q}}}Gq\rangle }_{2}(s)= & {} \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \Bigg \{ {\mathcal {H}}(s) \\&\times \frac{\langle {{\bar{q}}}q\rangle \langle g_s{{\bar{s}}}\sigma Gs\rangle + \langle {{\bar{s}}}s\rangle \langle g_s{{\bar{q}}}\sigma Gq\rangle }{256 \pi ^4 \alpha } -\frac{ m_c m_s \langle g_s{{\bar{q}}}\sigma Gq\rangle (4 \langle {{\bar{q}}}q\rangle - \langle {{\bar{s}}}s\rangle )}{256 \pi ^4} \Bigg \} \, ,\\ \rho ^{\langle {{\bar{q}}}Gq\rangle ^2}_{2}(s)= & {} \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \Bigg \{\frac{(\alpha -1) \langle g_s{{\bar{q}}}\sigma Gq\rangle \langle g_s{{\bar{s}}}\sigma Gs\rangle }{512 \pi ^4} \Bigg \} + \int ^{1}_{0}d\alpha \Bigg \{ \delta \left( s - {m_c^2 \over \alpha (1-\alpha )}\right) \\&\times \frac{ \langle g_s{{\bar{q}}}\sigma Gq\rangle \left( 2 m_c^3 m_s \langle g_s{{\bar{q}}}\sigma Gq\rangle / M_B^2 + ( 1-\alpha ) m_c^2 \langle g_s{{\bar{s}}}\sigma Gs\rangle - 2 (\alpha -1) \alpha m_s \langle g_s{{\bar{q}}}\sigma Gq\rangle \right) }{1024 \pi ^4 (\alpha -1) \alpha } \Bigg \} \, ,\\ \rho ^{\langle {{\bar{q}}}q\rangle ^3}_{2}(s)= & {} {m_c\langle {{\bar{q}}}q\rangle ^2 \langle {{\bar{s}}}s\rangle } \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \Bigg \{\frac{1}{24 \pi ^2} \Bigg \}. \end{aligned}$$

The spectral density $\rho _{3}(s)$ extracted for the current $\eta _{3}^\alpha$ is

$$\begin{aligned} \rho _{3}(s)= & {} \rho ^{pert}_{3}(s) + \rho ^{\langle {{\bar{q}}}q\rangle }_{3}(s) + \rho ^{\langle GG\rangle }_{3}(s)+ \rho ^{\langle {{\bar{q}}}Gq\rangle }_{3}(s) + \rho ^{\langle {{\bar{q}}}q\rangle ^2}_{3}(s) \nonumber \\&+ \rho ^{\langle {{\bar{q}}}q\rangle \langle {{\bar{q}}}Gq\rangle }_{3}(s)+ \rho ^{\langle {{\bar{q}}}Gq\rangle ^2}_{3}(s) + \rho ^{\langle {{\bar{q}}}q\rangle ^3}_{3}(s) \, , \end{aligned}$$

(23)

where

$$\begin{aligned} \rho ^{pert}_{3}(s)= & {} \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \int ^{\beta _{max}}_{\beta _{min}}d\beta \Bigg \{ {\mathcal {F}}(s)^5 \\&\times \frac{(1 - \alpha - \beta )^3 (\alpha +\beta +4)}{1310720 \pi ^8 \alpha ^5 \beta ^4} \Bigg \} \, ,\\ \rho ^{\langle {{\bar{q}}}q\rangle }_{3}(s)= & {} \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \int ^{\beta _{max}}_{\beta _{min}}d\beta \Bigg \{ {\mathcal {F}}(s)^3 \\&\times \frac{(\alpha +\beta -1) (6 (1 - \alpha - \beta ) m_c \langle {{\bar{q}}}q\rangle + \beta (2 \alpha +2 \beta +3) m_s (2 \langle {{\bar{q}}}q\rangle - \langle {{\bar{s}}}s\rangle ))}{16384 \pi ^6 \alpha ^3 \beta ^3} \Bigg \} \, ,\\ \rho ^{\langle GG\rangle }_{3}(s)= & {} {\langle g_s^2GG\rangle } \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \int ^{\beta _{max}}_{\beta _{min}}d\beta \Bigg \{ m_c^2 {\mathcal {F}}(s)^2 \\&\times \frac{(1 - \alpha - \beta )^3 (\alpha +\beta +4) \left( \alpha ^3+\beta ^3\right) }{1572864 \pi ^8 \alpha ^5 \beta ^4}\\&+ {\mathcal {F}}(s)^3 \times \Bigg (\frac{ 4 \alpha ^5-3 \alpha ^4 (3 \beta +11)-6 \alpha ^3 \left( 6 \beta ^2+13 \beta -9\right) -\alpha ^2 \left( 32 \beta ^3+51 \beta ^2-108 \beta +25\right) }{4718592 \pi ^8 \alpha ^5 \beta ^3}\\&\times \frac{-3 \alpha (\beta -1)^2 \beta (4 \beta +11)-3 (\beta -1)^3 \beta (\beta +4)}{4718592 \pi ^8 \alpha ^5 \beta ^3} \Bigg ) \Bigg \} \, ,\\ \rho ^{\langle {{\bar{q}}}Gq\rangle }_{3}(s)= & {} \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \int ^{\beta _{max}}_{\beta _{min}}d\beta \Bigg \{ {\mathcal {F}}(s)^2 \\&\times \frac{3 \langle g_s{{\bar{q}}}\sigma Gq\rangle (-6 (\alpha +\beta -1) m_c + \beta (4 \alpha +4 \beta +1) m_s)}{32768 \pi ^6 \alpha ^2 \beta ^2} \Bigg \} \, ,\\ \rho ^{\langle {{\bar{q}}}q\rangle ^2}_{3}(s)= & {} \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \int ^{\beta _{max}}_{\beta _{min}}d\beta \Bigg \{\frac{ - {\mathcal {F}}(s)^2 \times (4 \alpha +4 \beta +1) \langle {{\bar{q}}}q\rangle \langle {{\bar{s}}}s\rangle - m_c m_s {\mathcal {F}}(s) \times 6 \alpha (\langle {{\bar{s}}}s\rangle - 2 \langle {{\bar{q}}}q\rangle )}{2048 \pi ^4 \alpha ^2 \beta } \Bigg \} \, ,\\ \rho ^{\langle {{\bar{q}}}q\rangle \langle {{\bar{q}}}Gq\rangle }_{3}(s)= & {} \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \Bigg \{ \int ^{\beta _{max}}_{\beta _{min}}d\beta \Bigg \{{\mathcal {F}}(s) \times \frac{- \langle {{\bar{q}}}q\rangle \langle g_s{{\bar{s}}}\sigma Gs\rangle - \langle {{\bar{s}}}s\rangle \langle g_s{{\bar{q}}}\sigma Gq\rangle }{1024 \pi ^4 \alpha } \Bigg \}\\&+ {\mathcal {H}}(s) \times \frac{5\langle {{\bar{q}}}q\rangle \langle g_s{{\bar{s}}}\sigma Gs\rangle + 5 \langle {{\bar{s}}}s\rangle \langle g_s{{\bar{q}}}\sigma Gq\rangle }{4096 \pi ^4 \alpha } -\frac{3 m_c m_s \langle g_s{{\bar{q}}}\sigma Gq\rangle (4 \langle {{\bar{q}}}q\rangle - \langle {{\bar{s}}}s\rangle )}{4096 \pi ^4} \Bigg \} \, ,\\ \rho ^{\langle {{\bar{q}}}Gq\rangle ^2}_{3}(s)= & {} \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \Bigg \{\frac{3 (\alpha -1) \langle g_s{{\bar{q}}}\sigma Gq\rangle \langle g_s{{\bar{s}}}\sigma Gs\rangle }{8192 \pi ^4}\Bigg \} + \int ^{1}_{0}d\alpha \Bigg \{ \delta \left( s - {m_c^2 \over \alpha (1-\alpha )}\right) \\&\times \frac{\langle g_s{{\bar{q}}}\sigma Gq\rangle \left( 6 m_c^3 m_s \langle g_s{{\bar{q}}}\sigma Gq\rangle /M_B^2 - 5 (\alpha -1) m_c^2 \langle g_s{{\bar{s}}}\sigma Gs\rangle - 6 (\alpha -1) \alpha m_c m_s \langle g_s{{\bar{q}}}\sigma Gq\rangle \right) }{16384 \pi ^4 (\alpha -1) \alpha } \Bigg \} \, ,\\ \rho ^{\langle {{\bar{q}}}q\rangle ^3}_{3}(s)= & {} {m_c\langle {{\bar{q}}}q\rangle ^2 \langle {{\bar{s}}}s\rangle } \int ^{\alpha _{max}}_{\alpha _{min}}d\alpha \Bigg \{\frac{1}{128 \pi ^2}\Bigg \} \, . \end{aligned}$$