Model independent determination of the spins of the $P_{c}$(4440) and $P_{c}$(4457) from the spectroscopy of the triply charmed dibaryons
Ya-Wen Pan Ming-Zhu Liu Fang-Zheng Peng Mario Sánchez Sánchez Li-Sheng Geng Manuel Pavon Valderrama
hep-ph hep-ex hep-lat




School of Physics and
Nuclear Energy Engineering & International Research Center for Nuclei and Particles in the Cosmos &
Beijing Key Laboratory of Advanced Nuclear Materials and Physics,  Beihang University, Beijing 100191, China


School of Physics and
Nuclear Energy Engineering & International Research Center for Nuclei and Particles in the Cosmos &
Beijing Key Laboratory of Advanced Nuclear Materials and Physics,  Beihang University, Beijing 100191, China


School of Physics and
Nuclear Energy Engineering & International Research Center for Nuclei and Particles in the Cosmos &
Beijing Key Laboratory of Advanced Nuclear Materials and Physics,  Beihang University, Beijing 100191, China


Centre d'Études Nucléaires, CNRS/IN2P3, Université de Bordeaux, 33175 Gradignan, France


[E-mail: ]lisheng.geng@buaa.edu.cn School of
Physics and Nuclear Energy Engineering & International Research
Center for Nuclei and Particles in the Cosmos & Beijing Key
Laboratory of Advanced Nuclear Materials and Physics,  Beihang
University, Beijing 100191, China
School of Physics and Engineering, Zhengzhou University, Zhengzhou, Henan 450001, China

[E-mail: ]mpavon@buaa.edu.cn
School of Physics and Nuclear Energy Engineering &
International Research Center for Nuclei and Particles in the
Cosmos & Beijing Key Laboratory of Advanced Nuclear Materials and
Physics,  Beihang University, Beijing 100191, China



  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 D̅Σ_c and D̅^* Σ_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) D̅^* Σ_c
  pentaquarks and the corresponding splitting for the 0^+ and 1^+
  Ξ_ccΣ_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 Ξ_cc^(*)Σ_c^(*) molecules,
  which, if confirmed in the lattice, will largely expand our understanding
  of the non-perturbative strong interaction in the heavy-quark sector.


13.60.Le, 12.39.Mk,13.25.Jx

Model independent determination of the spins of the  P_c(4440) and P_c(4457)  from the spectroscopy of the triply charmed dibaryons

    Manuel Pavon Valderrama
    Received: date / Accepted: date
====================================================================================================================================




Recently the LHCb collaboration has reported the observation of
three pentaquark states <cit.> — the P_c(4312),
P_c(4440) and P_c(4457) — very close to two meson-baryon
thresholds, the D̅Σ_c in the case of the P_c(4312)
and the D̅^∗Σ_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 <cit.>
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 <cit.>.

In addition to  the molecular picture,
there are other explanations, e.g., hadro-charmonium <cit.>
or compact pentaquark states <cit.>.

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) <cit.>.


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 <cit.>,
while recent theoretical works prefer the opposite
identification <cit.>.




Given this situation [Even 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 <cit.>,
there have been only two lattice QCD studies on J/ψ N and η_c N
interactions <cit.>,
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 D̅^(*)Σ_c^(*) pentaquarks from
the Ξ_cc^(*)Σ_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) <cit.>,
which can be exploited to predict unobserved
states <cit.>
or explain relations among experimentally known
states <cit.>.

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 <cit.>
(a pair of heavy-quarks behaves like a heavy antiquark).

Regarding the pentaquark states, HQSS predicts the existence of seven molecular states <cit.>.

Namely, from HQSS and the LHCb pentaquarks <cit.>
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 D̅^(*) and the Ξ_cc^(*)  to be almost
  identical: if the D̅^(*)Σ_c^(*) system binds,
  the Ξ_cc^(*)Σ_c^(*) system should also bind.
  Specifically we can update the triply charmed dibaryon spectrum,
from the four dibaryons originally predicted in Ref. <cit.>
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 Ξ_cc^(*)Σ_c^(*) systems
than the D̅^(*)Σ_c^(*)  systems,

which up to now lattice QCD studies have
not been able to simulate.

On the other hand, a recent lattice QCD study <cit.>
has reported the likely existence of a triply charmed Ξ_ccΣ_c
dibaryon with spin-parity J^P = 1^+,  isospin I=12 and
a binding energy of B_2 = (8 ± 17)   MeV,
where Ref. <cit.> 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 ∼ (15-30)  MeV,
which is compatible within its large error with 
the results of Ref. <cit.>, but clearly leans towards
the conclusion that the 1^+ Ξ_ccΣ_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^+ Ξ_ccΣ_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. <cit.>.

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 <cit.>.

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

    ℒ_ contact   =    
      C_a   (q_L^† q_L)   (d_L^† d_L)
      
       +    C_b (q_L^† σ⃗_L q_L)  · 
      (d_L^† S⃗_L d_L)   ,

where σ⃗_L are the Pauli matrices for the q_L subfield and
S⃗_L the spin-1 matrices for the d_L subfield.

From this Lagrangian, we end up with the non-relativistic potential

    V_C = C_a + C_b  σ⃗_L ·S⃗_L   .

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

    ⟨D̅ | σ⃗_L | D̅⟩   =    0 
         ,   ⟨D̅^* | σ⃗_L | D̅^* ⟩ = S⃗_1   ,

 where S⃗_1 are the spin-1 matrices as applied to
 the D̅^* meson.
 
For the doubly heavy baryons

    ⟨Ξ_cc | σ⃗_L | Ξ_cc⟩   =   
      -1/3 σ⃗_1 
         ,   ⟨Ξ_cc^* | σ⃗_L | Ξ_cc^* ⟩ =
      +2/3 Σ⃗_1   ,

where σ⃗_1 and Σ⃗_1 are the Pauli and
spin-32 matrices as applied to
the Ξ_cc and Ξ_cc^* baryons.

Finally for the singly heavy hadrons


    ⟨Σ_c | S⃗_L | Σ_c⟩   =   
      +2/3 σ⃗_2 
         ,   ⟨Σ_c^* | S⃗_L | Σ_c^* ⟩ =
      +2/3 Σ⃗_2   ,

with σ⃗_2 and Σ⃗_2 as in Eq. (<ref>),
but applied to the Σ_c and
Σ_c^* baryons.

With these rules and the contact-range potentials of Eq. (<ref>)
we arrive at the potentials of Table <ref>.

We only consider contact-range interactions, as
previous studies of heavy hadron-hadron
systems <cit.>,

indicate that pion exchanges are perturbative

in the charm sector.


The potentials of Table <ref> 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 →∞.

For finite m_Q we expect small violations, which for HQSS are of the order
of Λ_ QCD / m_Q with Λ_QCD∼ (200-300)  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 Λ_QCD / (m_Q v) <cit.>, with v the expected velocity for the heavy-quark pair.

From the estimation of Ref. <cit.>, m_Q v ∼ 0.8  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 <cit.>.

But recent lattice QCD calculations <cit.> of the mass splitting between
the J=1/2 and J=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 <ref>.

If we work in momentum space we simply solve the Lippmann-Schwinger equation
for the bound state pole, which reads

    ϕ(k)+∫d^3p/(2π)^3⟨
    k|V|p⟩ϕ(p)/B+p^2/2μ=0,

where ϕ(k) is the vertex function, B the binding energy and μ
the reduced mass.

To solve this equation 

we first regularize the potential

    ⟨ p| V_Λ|
    p^'⟩=C(Λ)f(p/Λ)f(p^'/Λ),

with Λ 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(Λ), 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 Λ=0.5-1 GeV,
where the average within this range 
corresponds to the ρ meson mass.

For the masses of heavy hadrons, we use
m_D=1867 MeV, m_D^∗=2009 MeV, m_Σ_c=2454 MeV,
m_Σ_c^∗=2518 MeV,
m_Ξ_cc=3621 MeV, and m_Ξ_cc^∗=3727 MeV, where
the mass of Ξ_cc^∗ has been deduced from the HADS relation
m_Ξ_cc^∗-m_Ξ_cc= 3 (m_D^∗-m_D)/4 <cit.>.[
Ref. <cit.> has calculated perturbative corrections
to the above mass relation, which happen to be small (≈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
D̅^* Σ_c bound states, their contact-range potentials read

    V(D̅^* Σ_c, J = 12)    =    C_a - 4/3 C_b   , 
    
      V(D̅^* Σ_c, J = 32)    =    C_a + 2/3 C_b   .

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 12 state and the P_c(4457)
the spin 32 one (scenario A),
or vice versa (scenario B) <cit.>.

This in principle give us a consistency test in terms of the postdiction
of the P_c(4312) as a D̅Σ_c bound state,
see Table <ref> 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  MeV, 
  see Fig. <ref> 
for a graphical representation of the full dibaryon
spectrum and more details can be found in Table <ref>.


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^+ Σ_cΞ_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 Ξ_ccΣ_c (0^+,1^+) multiplet and an increase
in the Ξ_cc^*Σ_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

    M(1^+)-M(0^+)    =    - 4/9( M(32^-)-M(12^-)
      )
      
    
         ≈   ∓  (7.6 ± 1.9) MeV  ,

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 
Λ = 0.5  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 (Ξ_cc^(*)Σ_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)  MeV range, where the details
depend on the cutoff and the spins of the pentaquarks.

Among them, we predict a 1^+ Ξ_ccΣ_c dibaryon
with a binding energy of 15-30  MeV, to be compared with 
(8 ± 17)  MeV in the lattice <cit.>.
The spin-splitting of the dibaryon masses together with HADS,
i.e., Eq. (<ref>), 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. <cit.> implies that
such studies are within reach of current state of the art
lattice QCD simulations.

 












§ 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.





