Model independent determination of the spins of the $P_{c}$(4440) and $P_{c}$(4457) from the spectroscopy of the triply charmed dibaryons

TL;DR

This paper proposes a model-independent method to determine the spins of the $P_c(4440)$ and $P_c(4457)$ pentaquarks using heavy-quark symmetry relations with triply charmed dibaryons, linking experimental data to fundamental QCD predictions.

Contribution

It introduces a novel, model-independent relation between pentaquark spin splittings and triply charmed dibaryon mass differences based on heavy antiquark-diquark symmetry.

Findings

Relates pentaquark spin splittings to dibaryon mass differences.

Suggests lattice QCD can test the pentaquark spin assignments.

Predicts existence of ten triply charmed dibaryons as molecular states.

Abstract

The LHCb collaboration has recently observed three narrow pentaquark states --- the $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$ ---that are located close to the $\bar{D} \Sigma_c$ and $\bar{D}^* \Sigma_c$ thresholds. Among the so-far proposed theoretical interpretations for these pentaquarks, the molecular hypothesis seems to be the preferred one. Nevertheless, in the molecular picture the spins of the $P_c(4440)$ and $P_c(4457)$ have not been unambiguously determined yet. In this letter we point out that heavy antiquark-diquark symmetry induces a model-independent relation between the spin-splitting in the masses of the $P_c(4440)$ and $P_c(4457)$ $\bar{D}^* \Sigma_c$ pentaquarks and the corresponding splitting for the $0^+$ and $1^+$ $\Xi_{cc} \Sigma_c$ triply charmed dibaryons. This is particularly relevant owing to a recent lattice QCD prediction of the $1^+$ triply charmed dibaryon, which suggests that a calculation of the mass of its $0^+$ partner might be within reach. This in turn would reveal the spins of the $P_c(4440)$ and $P_c(4457)$ pentaquarks, providing a highly nontrivial test of heavy-quark symmetry and the molecular nature of the pentaquarks. Furthermore, the molecular interpretation of the hidden-charm pentaquarks implies the existence of a total of ten triply charmed dibaryons as $\Xi_{cc}^{(*)} \Sigma_c^{(*)}$ molecules, which, if confirmed in the lattice, will largely expand our understanding of the non-perturbative strong interaction in the heavy-quark sector.

Tables

Table 1. Table 1: The lowest-order contact range potentials for the heavy antimeson - heavy baryon and doubly heavy baryon - heavy baryon systems, which depend on two unknown coupling constants C a subscript 𝐶 𝑎 C_{a} and C b subscript 𝐶 𝑏 C_{b} .

state	$J^P$	V	state	$J^P$	V
$\overline{D}{\Sigma }_c$	$1/2^{-}$	$C_a$	${\Xi }_{cc}{\Sigma }_c$	$0^{+}$	$C_a+\frac{2}{3}{C}_b$
				$1^{+}$	$C_a-\frac{2}{9}{C}_b$
$\overline{D}{\Sigma }_c^{\ast }$	$3/2^{-}$	$C_a$	${\Xi }_{cc}{\Sigma }_c^{\ast }$	$1^{+}$	$C_a+\frac{5}{9}{C}_b$
				$2^{+}$	$C_a-\frac{1}{3}{C}_b$
${\overline{D}}^{\ast }{\Sigma }_c$	$1/2^{-}$	$C_a-\frac{4}{3}{C}_b$	${\Xi }_{cc}^{\ast }{\Sigma }_c$	$1^{+}$	$C_a-\frac{10}{9}{C}_b$
	$3/2^{-}$	$C_a+\frac{2}{3}{C}_b$		$2^{+}$	$C_a+\frac{2}{3}{C}_b$
${\overline{D}}^{\ast }{\Sigma }_c^{\ast }$	$1/2^{-}$	$C_a-\frac{5}{3}{C}_b$	${\Xi }_{cc}^{\ast }{\Sigma }_c^{\ast }$	$0^{+}$	$C_a-\frac{5}{3}{C}_b$
	$3/2^{-}$	$C_a-\frac{2}{3}{C}_b$		$1^{+}$	$C_a-\frac{11}{9}{C}_b$
				$2^{+}$	$C_a-\frac{1}{3}{C}_b$
	$5/2^{-}$	$C_a+C_b$		$3^{+}$	$C_a+C_b$

Table 2. Table 2: Binding energies for the pentaquarks composed of a charmed baryon and a charmed antimeson in the contact-range EFT defined in Eqs. ( 1 ) and ( 2 ). The two coupling constants, C a subscript 𝐶 𝑎 C_{a} and C b subscript 𝐶 𝑏 C_{b} , are determined by reproducing the P c ​ ( 4440 ) subscript 𝑃 𝑐 4440 P_{c}(4440) and P c ​ ( 4457 ) subscript 𝑃 𝑐 4457 P_{c}(4457) as 1 2 − superscript 1 2 \tfrac{1}{2}^{-} and 3 2 − superscript 3 2 \tfrac{3}{2}^{-} D ¯ ∗ ​ Σ c superscript ¯ 𝐷 subscript Σ 𝑐 \bar{D}^{*}\Sigma_{c} molecules (scenario A) or 3 2 − superscript 3 2 \tfrac{3}{2}^{-} and 1 2 − superscript 1 2 \tfrac{1}{2}^{-} D ¯ ∗ ​ Σ c superscript ¯ 𝐷 subscript Σ 𝑐 \bar{D}^{*}\Sigma_{c} molecules (scenario B), respectively. The results are obtained with a cutoff of Λ = 0.5 ​ ( 1.0 ) ​ GeV Λ 0.5 1.0 GeV \Lambda=0.5(1.0)\,{\rm GeV} .

State	$J^P$	Threshold	$\Lambda$(GeV)	Scenario A	Scenario B
$P_c\left(4312\right)$($\overline{D}{\Sigma }_c$)	${\frac{1}{2}}^{-}$	4320.8	0.5(1)	${8.8}_{-4.2}^{+5.0}$ $\left({7.6}_{-6.0}^{+9.2}\right)$	${14.4}_{-5.8}^{+6.5}$ $\left({12.8}_{-8.6}^{+11.8}\right)$
$P_c\left(?\right)$($\overline{D}{\Sigma }_c^{\ast }$)	${\frac{3}{2}}^{-}$	4385.3	0.5(1)	${9.1}_{-4.3}^{+5.0}$ $\left({8.1}_{-6.3}^{+9.4}\right)$	${14.7}_{-5.9}^{+6.6}$ $\left({13.5}_{-8.9}^{+12.0}\right)$
$P_c\left(4440\right)$(${\overline{D}}^{\ast }{\Sigma }_c$)	?	4462.2	0.5(1)	Input(${\frac{1}{2}}^{-}$)	Input(${\frac{3}{2}}^{-}$)
$P_c\left(4457\right)$(${\overline{D}}^{\ast }{\Sigma }_c$)	?	4462.2	0.5(1)	Input(${\frac{3}{2}}^{-}$)	Input(${\frac{1}{2}}^{-}$)
$P_c\left(?\right)$(${\overline{D}}^{\ast }{\Sigma }_c^{\ast }$)	${\frac{1}{2}}^{-}$	4526.7	0.5(1)	${25.6}_{-8.4}^{+9.0}$ $\left({26.3}_{-13.7}^{+16.6}\right)$	${3.0}_{-2.0}^{+2.8}$ $\left({3.4}_{-3.2}^{+6.3}\right)$
$P_c\left(?\right)$(${\overline{D}}^{\ast }{\Sigma }_c^{\ast }$)	${\frac{3}{2}}^{-}$	4526.7	0.5(1)	${15.8}_{-6.1}^{+6.7}$ $\left({15.9}_{-9.8}^{+12.8}\right)$	${10.0}_{-4.5}^{+5.2}$ $\left({10.0}_{-7.2}^{+10.2}\right)$
$P_c\left(?\right)$(${\overline{D}}^{\ast }{\Sigma }_c^{\ast }$)	${\frac{5}{2}}^{-}$	4526.7	0.5 (1)	${3.0}_{-2.0}^{+2.8}$ $\left({3.4}_{-3.2}^{+6.3}\right)$	${25.6}_{-8.4}^{+9.0}$ $\left({26.3}_{-13.7}^{+16.6}\right)$

Table 3. Table 3: Same as Table 2 but for the dibaryons composed of a doubly charmed baryon ( Ξ c ​ c subscript Ξ 𝑐 𝑐 \Xi_{cc} , Ξ c ​ c ∗ superscript subscript Ξ 𝑐 𝑐 \Xi_{cc}^{*} ) and a singly charmed baryon ( Σ c subscript Σ 𝑐 \Sigma_{c} , Σ c ∗ superscript subscript Σ 𝑐 \Sigma_{c}^{*} ).

State	$J^P$	Threshold	$\Lambda$(GeV)	Scenario(A)	Scenario(B)
${\Xi }_{cc}{\Sigma }_c$	$0^{+}$	6074.9	0.5(1)	${10.0}_{-6.4}^{+7.9}$ $\left({17.9}_{-14.4}^{+20.8}\right)$	${30.4}_{-14.2}^{+15.5}$ $\left({43.2}_{-27.2}^{+33.1}\right)$
${\Xi }_{cc}{\Sigma }_c$	$1^{+}$	6074.9	0.5(1)	${18.4}_{-9.9}^{+11.3}$ $\left({28.3}_{-20.1}^{+26.3}\right)$	${20.7}_{-10.8}^{+12.1}$ $\left({31.2}_{-21.6}^{+27.7}\right)$
${\Xi }_{cc}{\Sigma }_c^{\ast }$	$1^{+}$	6139.5	0.5(1)	${11.3}_{-7.0}^{+8.4}$ $\left({20.0}_{-17.2}^{+25.4}\right)$	${29.6}_{-13.8}^{+15.1}$ $\left({42.8}_{-26.9}^{+32.7}\right)$
${\Xi }_{cc}{\Sigma }_c^{\ast }$	$2^{+}$	6139.5	0.5(1)	${20.0}_{-10.4}^{+11.8}$ $\left({30.7}_{-21.2}^{+27.2}\right)$	${20.0}_{-10.4}^{+11.8}$ $\left({30.8}_{-21.3}^{+27.3}\right)$
${\Xi }_{cc}^{\ast }{\Sigma }_c$	$1^{+}$	6180.9	0.5(1)	${28.2}_{-13.4}^{+14.7}$ $\left({41.0}_{-26.0}^{+32.0}\right)$	${12.2}_{-7.4}^{+8.8}$ $\left({21.0}_{-16.2}^{+22.4}\right)$
${\Xi }_{cc}^{\ast }{\Sigma }_c$	$2^{+}$	6180.9	0.5(1)	${10.2}_{-6.5}^{+8.0}$ $\left({18.5}_{-14.8}^{+21.0}\right)$	${30.7}_{-14.2}^{+15.5}$ $\left({44.1}_{-27.5}^{+33.3}\right)$
${\Xi }_{cc}^{\ast }{\Sigma }_c^{\ast }$	$0^{+}$	6245.5	0.5(1)	${35.0}_{-15.6}^{+16.8}$ $\left({50.2}_{-29.9}^{+35.6}\right)$	${7.6}_{-5.3}^{+6.7}$ $\left({15.8}_{-13.0}^{+19.2}\right)$
${\Xi }_{cc}^{\ast }{\Sigma }_c^{\ast }$	$1^{+}$	6245.5	0.5(1)	${29.9}_{-13.9}^{+15.2}$ $\left({43.7}_{-27.2}^{+32.9}\right)$	${11.5}_{-7.0}^{+8.5}$ $\left({20.7}_{-15.9}^{+22.0}\right)$
${\Xi }_{cc}^{\ast }{\Sigma }_c^{\ast }$	$2^{+}$	6245.5	0.5(1)	${20.3}_{-10.5}^{+11.8}$ $\left({31.6}_{-21.6}^{+27.5}\right)$	${20.3}_{-10.5}^{+11.8}$ $\left({31.6}_{-21.6}^{+27.5}\right)$
${\Xi }_{cc}^{\ast }{\Sigma }_c^{\ast }$	$3^{+}$	6245.5	0.5(1)	${7.6}_{-5.3}^{+6.7}$ $\left({15.8}_{-13.0}^{+19.2}\right)$	${35.0}_{-15.6}^{+16.8}$ $\left({50.2}_{-30.0}^{+35.6}\right)$

Equations

$${\mathcal{L}}_{\mathrm{contact}}$$

$$V_C=C_a+C_b{\vec{\sigma }}_L\cdot {\vec{S}}_L.$$

$$⟨\bar{D}|{\vec{\sigma }}_L|\bar{D}⟩$$

$$⟨{\Xi }_{cc}|{\vec{\sigma }}_L|{\Xi }_{cc}⟩$$

$$⟨{\Sigma }_c|{\vec{S}}_L|{\Sigma }_c⟩$$

$$\varphi (k)+\int \frac{d^{3}p}{(2\pi {)}^3}⟨k|V|p⟩\frac{\varphi (p)}{B+\frac{p^2}{2\mu }}=0,$$

$$⟨p|V_{\Lambda }|p^{\prime }⟩=C(\Lambda )f(\frac{p}{\Lambda })f(\frac{p^{\prime }}{\Lambda }),$$

$$V({\bar{D}}^{\ast }{\Sigma }_c,J=\frac{1}{2})$$

$$V({\bar{D}}^{\ast }{\Sigma }_c,J=\frac{3}{2})$$

$$M(1^{+})-M(0^{+})$$

Full text

Recently the LHCb collaboration has reported the observation of three pentaquark states Aaij et al. (2019) — the $P_{c}(4312)$, $P_{c}(4440)$ and $P_{c}(4457)$ — very close to two meson-baryon thresholds, the $\bar{D}\Sigma_{c}$ in the case of the $P_{c}(4312)$ and the $\bar{D}^{\ast}\Sigma_{c}$ in the case of the $P_{c}(4440)$ and $P_{c}(4457)$. This interesting coincidence has been promptly interpreted by a series of theoretical works Chen et al. (2019a, b); Liu et al. (2019a); Guo et al. (2019); Xiao et al. (2019a); Shimizu et al. (2019); Guo and Oller (2019) as evidence for their molecular nature, a hypothesis which is further fueled by the predictions a few years ago of the existence of these pentaquarks Wu et al. (2010, 2011); Xiao et al. (2013); Karliner and Rosner (2015); Wang et al. (2011); Yang et al. (2012). In addition to the molecular picture, there are other explanations, e.g., hadro-charmonium Eides et al. (2019) or compact pentaquark states Ali and Parkhomenko (2019); Wang (2019); Cheng and Liu (2019). It is also worth noticing that the $P_{c}(4440)$ and $P_{c}(4457)$ were previously identified as a single state, the $P_{c}(4450)$ Aaij et al. (2015).

However the molecular interpretation cannot unambiguously determine the spins of the $P_{c}(4440)$ and $P_{c}(4457)$, which can be either $1/2$ or $3/2$. Initial studies showed a preference for the $P_{c}(4440)$ and $P_{c}(4457)$ being identified with $J=1/2$ and $3/2$ molecules respectively Chen et al. (2019b); Liu et al. (2019a); Xiao et al. (2019a), while recent theoretical works prefer the opposite identification Yamaguchi et al. (2019); Pavon Valderrama (2019a); Liu et al. (2019b).

Given this situation 111Even experiments might need larger statistics to give a conclusive answer., one might turn to lattice QCD for a model independent answer. But meson-baryon systems are difficult to study directly in the lattice, more so if there are coupled channels. After the 2015 pentaquark peaks Aaij et al. (2015), there have been only two lattice QCD studies on $J/\psi N$ and $\eta_{c}N$ interactions Sugiura et al. (2019); Skerbis and Prelovsek (2019), none of which found pentaquarks. These studies, which were performed within a single-channel approximation, concluded that further calculations including coupled-channels were needed.

To circumvent the aforementioned difficulties, we propose to learn more about the $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ pentaquarks from the $\Xi_{cc}^{(*)}\Sigma_{c}^{(*)}$ dibaryons, which are connected by heavy antiquark-diquark symmetry (HADS).

Hadronic molecules containing heavy quarks ($c$, $b$) are constrained by heavy-quark symmetry (HQS) Isgur and Wise (1989, 1990), which can be exploited to predict unobserved states Nieves and Valderrama (2012); Guo et al. (2013a); Xiao et al. (2019b); Peng et al. (2019) or explain relations among experimentally known states Bondar et al. (2011); Mehen and Powell (2011); Guo et al. (2013b). Two manifestations of HQS are relevant for the present work: heavy-quark spin symmetry (HQSS) (the interaction between heavy hadrons is independent of the orientation of the heavy quark spin) and HADS Savage and Wise (1990) (a pair of heavy-quarks behaves like a heavy antiquark). Regarding the pentaquark states, HQSS predicts the existence of seven molecular states Xiao et al. (2013); Liu et al. (2019a); Xiao et al. (2019a); Sakai et al. (2019); Yamaguchi et al. (2019); Pavon Valderrama (2019a). Namely, from HQSS and the LHCb pentaquarks Aaij et al. (2019) one expects four more unobserved hidden-charm pentaquarks.

But the latest discovery of the LHCb pentaquarks leads to more consequences. From HADS, we expect the light-quark structures of the ${\bar{D}}^{(*)}$ and the ${\Xi}_{cc}^{(*)}$ to be almost identical: if the $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ system binds, the ${\Xi}_{cc}^{(*)}\Sigma_{c}^{(*)}$ system should also bind. Specifically we can update the triply charmed dibaryon spectrum, from the four dibaryons originally predicted in Ref. Liu et al. (2018a) to ten dibaryons in the present manuscript. As we will show, this could help determine the spins of the $P_{c}(4440)$ and $P_{c}(4457)$. Besides, it seems to be much easier for lattice QCD to study the $\Xi_{cc}^{(*)}\Sigma_{c}^{(*)}$ systems than the $\bar{D}^{(*)}\Sigma_{c}^{(*)}$ systems, which up to now lattice QCD studies have not been able to simulate. On the other hand, a recent lattice QCD study Junnarkar and Mathur (2019) has reported the likely existence of a triply charmed $\Xi_{cc}\Sigma_{c}$ dibaryon with spin-parity $J^{P}=1^{+}$, isospin $I=\tfrac{1}{2}$ and a binding energy of $B_{2}=(8\pm 17)\,{\rm MeV}$, where Ref. Junnarkar and Mathur (2019) indicates that owing to the large uncertainties its existence is not guaranteed. Here we predict this dibaryon to have a binding energy of $B_{2}\sim(15-30)\,{\rm MeV}$, which is compatible within its large error with the results of Ref. Junnarkar and Mathur (2019), but clearly leans towards the conclusion that the $1^{+}$ $\Xi_{cc}\Sigma_{c}$ dibaryon binds. More importantly, as we will show below, if lattice QCD is able to calculate the mass of the $J^{P}=0^{+}$ configuration or the mass splitting between the $0^{+}$ and $1^{+}$ $\Xi_{cc}\Sigma_{c}$ dibaryons, it will provide a model-independent determination of the spins of the $P_{c}(4440)$ and $P_{c}(4457)$.

Now we explain how we predict the binding energies of the triply charmed dibaryons. For this we construct a contact-range effective field theory (EFT) that respects HQS, in the line of Refs. Liu et al. (2018b, 2019a). We will follow a streamlined approach, where the starting point is the “brown muck” within the heavy hadrons, i.e., the light-quark subfields. The idea is that the heavy quark is merely a spectator, therefore it is more economic to write the interaction between the heavy hadrons in terms of the light-quark degrees of freedom Pavon Valderrama (2019b). Thus, a heavy antimeson can be written as a non-relativistic light-quark subfield $q_{L}$. Owing to HADS the “brown muck” in a doubly heavy baryon is basically the same as that in a heavy antimeson, i.e., we can represent the doubly heavy baryon with the same light-quark subfield $q_{L}$ as the heavy antimeson. Finally, the (singly) heavy baryons can be represented with a $S=1$ light-diquark subfield $d_{L}$.

With these pieces, we can write the lowest-dimensional, contact-range Lagrangian involving a light-quark and light-diquark subfield as

[TABLE]

where $\vec{\sigma}_{L}$ are the Pauli matrices for the $q_{L}$ subfield and $\vec{S}_{L}$ the spin-1 matrices for the $d_{L}$ subfield. From this Lagrangian, we end up with the non-relativistic potential

[TABLE]

This is supplemented by a series of rules to translate the light-quark spin operators into the heavy hadron ones. For the heavy antimeson fields we have

[TABLE]

where $\vec{S}_{1}$ are the spin-1 matrices as applied to the ${\bar{D}}^{*}$ meson. For the doubly heavy baryons

[TABLE]

where $\vec{\sigma}_{1}$ and $\vec{\Sigma}_{1}$ are the Pauli and spin-$\tfrac{3}{2}$ matrices as applied to the $\Xi_{cc}$ and $\Xi_{cc}^{*}$ baryons. Finally for the singly heavy hadrons

[TABLE]

with $\vec{\sigma}_{2}$ and $\vec{\Sigma}_{2}$ as in Eq. (4), but applied to the $\Sigma_{c}$ and $\Sigma_{c}^{*}$ baryons. With these rules and the contact-range potentials of Eq. (2) we arrive at the potentials of Table 1. We only consider contact-range interactions, as previous studies of heavy hadron-hadron systems Valderrama (2012); Lu et al. (2019), indicate that pion exchanges are perturbative in the charm sector.

The potentials of Table 1 are subject to a certain degree of uncertainty. They are derived from HQS, which is only exact in the limit of infinite heavy-quark mass, $m_{Q}\to\infty$. For finite $m_{Q}$ we expect small violations, which for HQSS are of the order of $\Lambda_{\rm QCD}/m_{Q}$ with $\Lambda_{QCD}\sim(200-300)\,{\rm MeV}$ the QCD scale, i.e., we expect a $15\%$ error in the charm sector for the pentaquark potentials. For HADS, the uncertainty is of the order of $\Lambda_{QCD}/(m_{Q}v)$ Savage and Wise (1990), with $v$ the expected velocity for the heavy-quark pair. From the estimation of Ref. Hu and Mehen (2006), $m_{Q}v\sim 0.8\,{\rm GeV}$ for a charm quark pair, we arrive at a $25-40\%$ uncertainty for HADS. This figure has been conjectured to be larger, up to the point that HADS might not be applicable in the charm sector Cohen and Hohler (2006). But recent lattice QCD calculations Padmanath et al. (2015); Chen and Chiu (2017); Alexandrou and Kallidonis (2017); Mathur and Padmanath (2019) of the mass splitting between the $J=\frac{1}{2}$ and $J=\frac{3}{2}$ doubly charmed baryons are usually $20-25\%$ smaller than the HADS prediction, suggesting our lower estimation for the HADS uncertainty. From this we will settle on a $25\%$ error for the dibaryon potentials.

To obtain concrete predictions we have to solve a non-relativistic bound state equation with the contact-range potentials of Table 1. If we work in momentum space we simply solve the Lippmann-Schwinger equation for the bound state pole, which reads

[TABLE]

where $\phi(k)$ is the vertex function, $B$ the binding energy and $\mu$ the reduced mass. To solve this equation we first regularize the potential

[TABLE]

with $\Lambda$ the cutoff, $f(x)$ a regulator, and $C$ the linear combination of the $C_{a}$ and $C_{b}$ couplings that corresponds to the particular molecular state we are interested in. Notice that $C=C(\Lambda)$, i.e., the couplings depend on the cutoff, which is necessary if we want to properly renormalize the calculations, i.e., we want the predictions to depend on the cutoff only moderately, where cutoff variations represent the uncertainty coming from subleading order terms that we have not explicitly taken into account. We will choose a Gaussian regulator of the type $f(x)=e^{-x^{2}}$ and for the cutoff we will use the range $\Lambda=0.5-1$ GeV, where the average within this range corresponds to the $\rho$ meson mass.

For the masses of heavy hadrons, we use $m_{D}=1867$ MeV, $m_{D^{\ast}}=2009$ MeV, $m_{\Sigma_{c}}=2454$ MeV, $m_{\Sigma_{c}^{\ast}}=2518$ MeV, $m_{\Xi_{cc}}=3621$ MeV, and $m_{\Xi_{cc}^{\ast}}=3727$ MeV, where the mass of $\Xi_{cc}^{\ast}$ has been deduced from the HADS relation $m_{\Xi_{cc}^{\ast}}-m_{\Xi_{cc}}=3\,(m_{D^{\ast}}-m_{D})/4$ Hu and Mehen (2006).222 Ref. Mehen and Mohapatra (2019) has calculated perturbative corrections to the above mass relation, which happen to be small ($\approx 10\%$) and we will ignore here.

Now we still have to determine the couplings $C_{a}$ and $C_{b}$. For this, we notice that if the $P_{c}(4440)$ and $P_{c}(4457)$ are indeed $\bar{D}^{*}\Sigma_{c}$ bound states, their contact-range potentials read

[TABLE]

Since the spins of the hidden-charm pentaquarks are not experimentally known yet, there are two possible identifications: the $P_{c}(4440)$ is the spin $\tfrac{1}{2}$ state and the $P_{c}(4457)$ the spin $\tfrac{3}{2}$ one (scenario A), or vice versa (scenario B) Liu et al. (2019a). This in principle give us a consistency test in terms of the postdiction of the $P_{c}(4312)$ as a $\bar{D}\Sigma_{c}$ bound state, see Table 2 for details. The outcome is inconclusive though: scenario A is marginally preferred over scenario B. In both scenarios, we arrive at the conclusion that the ten possible triply charmed dibaryons bind, with binding energies of $10-50\,{\rm MeV}$, see Fig. 1 for a graphical representation of the full dibaryon spectrum and more details can be found in Table 3.

One immediately notices that there is a strong correlation between the ordering of the triply charmed baryons and that of the pentaquark states, particularly that of $P_{c}(4440)$ and $P_{c}(4457)$. For instance, if the $0^{+}$ $\Sigma_{c}\Xi_{cc}$ state had a larger mass than its $1^{+}$ counterpart, then scenario A would be preferred. The difference in their binding energies is about 10 MeV, which might be achievable in the lattice. In addition, one can study (one of) the three extra multiplets to confirm the conclusion. For instance, a decrease of the mass as a function of the spin in the $\Xi_{cc}\Sigma_{c}$ $(0^{+},1^{+})$ multiplet and an increase in the $\Xi_{cc}^{*}\Sigma_{c}$ $(1^{+},2^{+})$ multiplet will both be unambiguous signals that scenario A is preferred.

More concretely, from Table I and assuming that the $C_{b}$ coupling is perturbative, we arrive at the relation

[TABLE]

with the $-/+$ sign respectively for scenarios A and B and where we have used the experimental pentaquark masses to obtain the number on the second line (with the error corresponding to the HADS uncertainty). It is worth noticing that in our numerical study with $\Lambda=0.5\,{\rm GeV}$, we obtain $M(1^{+})-M(0^{+})=-8.4$ MeV and $+9.7$ MeV for scenarios A and B, respectively.

In summary, the experimental discovery of the three pentaquark states — the $P_{c}(4312)$, $P_{c}(4440)$ and $P_{c}(4457)$ — which are conjectured to be hadronic molecules, not only makes it possible to predict the full spectrum of S-wave triply charmed dibaryons ($\Xi_{cc}^{(*)}\Sigma_{c}^{(*)}$ molecules) but also opens the possibility of a model-independent determination of the spins of the pentaquarks from lattice QCD simulations of the dibaryons, which are definitively feasible in the near future. The basis for the prediction of the triply charmed dibaryons are HQSS and HADS. We exploited these two symmetries within the framework of a contact-range EFT, which offers us a series of theoretical advantages including systematicity and controlled error estimations. The binding energies of the triply charmed dibaryons are predicted to lie in the $(10-50)\,{\rm MeV}$ range, where the details depend on the cutoff and the spins of the pentaquarks. Among them, we predict a $1^{+}$ $\Xi_{cc}\Sigma_{c}$ dibaryon with a binding energy of $15-30\,{\rm MeV}$, to be compared with $(8\pm 17)\,{\rm MeV}$ in the lattice Junnarkar and Mathur (2019). The spin-splitting of the dibaryon masses together with HADS, i.e., Eq. (10), indicates that a lattice QCD calculation of the dibaryon spectrum will provide a model-independent determination of the quantum numbers of the LHCb pentaquarks. The recent calculation of Ref. Junnarkar and Mathur (2019) implies that such studies are within reach of current state of the art lattice QCD simulations.

I Acknowledgments

This work is partly supported by the National Natural Science Foundation of China under Grant No.11735003, the fundamental Research Funds for the Central Universities, and the Thousand Talents Plan for Young Professionals.