Masses of doubly heavy-quark baryons in an extended chromomagnetic model
Xin-Zhen Weng Xiao-Lin Chen Wei-Zhen Deng
hep-ph




 
xzhweng@pku.edu.cn 

School of Physics, Peking University, Beijing 100871,
  China
State Key Laboratory of Nuclear Physics and Technology,
  Peking University, Beijing 100871, China

 
chenxl@pku.edu.cn 

School of Physics, Peking University, Beijing 100871,
  China

 dwz@pku.edu.cn 

School of Physics, Peking University, Beijing 100871,
  China
State Key Laboratory of Nuclear Physics and Technology, 
  Peking University, Beijing 100871, China



  We extend the chromomagnetic model by further considering the effect
  of color interaction.  The effective mass parameters between quark
  pairs (m_qq or m_qq̅) are introduced to account for both
  the effective quark masses and the color interaction between the two
  quarks. Using the experimental masses of hadrons, the quark pair
  parameters are determined between the light quark pairs and the
  light-heavy quark pairs.  Then the parameters of heavy quark pairs
  (cc, cb, bb) are estimated based on simple quark model
  assumption. We calculate all masses of doubly and triply
  heavy-quark baryons.  The newly discovered doubly charmed baryon
  Ξ_cc fits into the model with an error of 12.


12.39.Pn, 14.20.-c, 14.20.Lq, 14.20.Mr, 12.40.Yx

Masses of doubly heavy-quark baryons in an extended
  chromomagnetic model
    Wei-Zhen Deng
    
==========================================================================



empty




§ INTRODUCTION



In 2002, the SELEX Collaboration <cit.> reported the
first observation of a doubly charmed baryon Ξ_cc^+ in the
decay mode Ξ_cc^+→Λ_c^+K^-π^+.

Its mass was determined to be 3519±1.

Further works identified its isospin partner Ξ_cc^++(3460)
<cit.> and an excited state
Ξ_cc^++(3780) <cit.>.

Later the SELEX Collaboration confirmed the Ξ_cc^+ state in
the Ξ_cc^+→pD^+K^- <cit.> and
Ξ_cc^+→Ξ_c^+π^+π^- <cit.>
decay modes.

However, none of these states were confirmed by other experimental
collaborations
<cit.> 


in the subsequent searches.

Recently, the LHCb Collaboration <cit.> reported the
observation of Ξ_cc^++ in the Λ_c^+K^-π^+π^+ decay
mode.

But its mass was determined to be 3621.40±0.72(stat.)
±0.27(syst.) ±0.14(Λ_c^+).

In contrast to the rarity of the experimental observation of the
doubly heavy baryons, there is a vast literature of theoretical
studies concerning the doubly and even triply heavy baryons with
different approaches, including

quark models <cit.>,






QCD sum rules <cit.>,



lattice QCD <cit.>,

the bag model <cit.>, 

heavy-quark effective theory <cit.>,

heavy-quark spin symmetry <cit.>,

effective field theory with potential nonrelativistic QCD <cit.>,

the Feynman-Hellmann theorem <cit.>, 

variational method <cit.>, 

the Skyrmion model <cit.> 

and the Regge phenomenology
<cit.>.

The quark model is one of the most used approaches to study the mass
spectra of hadrons <cit.>.

In the nonrelativistic limit, the QCD interaction can be reduced to
the potential interaction between quarks. Usually the potential
interaction in a quark model consists of the spin-independent color
interaction including the linear confinement and Coulomb-type terms,
plus higher order terms such as the spin-spin chromomagnetic
interaction, tensor interaction, and spin-orbit interactions.

When focusing on lowest S-wave states of hadrons, one may adopt the
chromomagnetic model <cit.>.

The chromomagnetic model assumes a mass formula by simply adding a term
of chromomagnetic hyperfine interaction to the effective quark masses.

This simplified model gives a good account of the hyperfine splittings
in hadron mass spectra and produces many useful Gell-Mann–Okubo (GMO)
mass relations.

From the point of view of the quark model, the effective quark masses also
include the chromoelectric effects from the color interaction.

However it is difficult to account for the two-body chromoelectric
effects in all relevant mesons and baryons by the effective quark
masses, which are one-body type. 

In Ref. <cit.>,
Høgaasen et al.  generalized the chromomagnetic model by
including a chromoelectric term
H_CE = -∑_i,j A_ijλ̃_i·λ̃_j.

Similarly, Karliner et al. introduced the color-singlet binding
energies B(cc̅)=-242.7 and
B(bb̅)=-532.2 <cit.>.

In this paper, we use the extended chromomagnetic model with the
chromoelectric term to study the mass spectra of all the lowest
S-wave doubly and triply heavy-quark baryons systematically.

In Sec. <ref> we introduce the extended chromomagnetic
model and construct the model wave functions of mesons and baryons.

In Sec. <ref> we determine the model parameters.

The numerical results are presented and discussed in
Sec. <ref>.

We conclude in Sec. <ref>.




§ THE EXTENDED CHROMOMAGNETIC MODEL






 §.§ The Hamiltonian




In the quark model, the quark effective Hamiltonian reads
<cit.>


    H=H_0 + ∑_i<j V_ij,


where 


    H_0 = ∑_i √(_i^2+ m_i^2)


is the relativistic mass term and V_ij is the quark interaction
potential between ith and jth quarks.

In a nonrelativistic reduction,


    H_0 →∑_i ( m_i + _i^2/2m_i),


and


    V_ij→ V_ij^conf +  V_ij^hyp +  V_ij^so,


where


    V_ij^conf = - [ 3/4 c + 3/4 br 
          - α_s(r)/r] F_i·F_j


includes the color linear confinement and the Coulomb-type interaction,
V_ij^hyp is the color hyperfine interaction, and
V_ij^so is the spin-orbit interaction.

For the S-wave hadron, V_ij^so has no contribution, 
and V_ij^hyp can be simply replaced by 
the chromomagnetic interaction


    V_ij^cm = -8π/3α_s(r)/m_i m_jδ^3(r) S_i·S_j F_i·F_j,


where S_i and F_i are the ith quark's spin 
operator and color operator respectively,


    F_i =
      
        +λ_i/2   for quarks, 
    
        -λ_i^*/2   for antiquarks.


In the case of the lowest S-wave hadron, one may further simplify the
chromomagnetic interaction by ignoring its spatial dependency. Then
the chromomagnetic model Hamiltonian reads


    H = ∑_i m_i + ∑_i<j V_ij^cm ,


where the effective mass m_i of ith constituent quark (or
antiquark) should include the constituent quark mass and the kinetic
energy and chromoelectric effects from V_ij^conf. The
chromomagnetic interaction reads


    V^cm_ij = - v_ijS_i·S_jF_i·F_j .


The coefficient v_ij depends on the quark masses and the spatial
wave function of the hadron

    v_ij= 8π/3m_i m_j⟨α_s(r)δ^3(r) ⟩.


However it is difficult to adsorb all the two-body chromoelectric
effects into the one-body effective quark masses if we want to study
all lowest S-wave mesons and baryons together
<cit.>. 

In Ref. <cit.>, Høgaasen et al.  generalized
the chromomagnetic model by including a chromoelectric term


    H_CE = -∑_i,j A_ijλ̃_i·λ̃_j,


where λ̃_i=2F_i. We use this extended
chromomagnetic model to study all lowest S-wave mesons and baryons
systematically.

Since

    ∑_i<j( m_i + m_j ) F_i·F_j
    
    =   1/2∑_i,j( m_i + m_j ) F_i·F_j 
    - ∑_i m_i F_i^2 
    
    =   (∑_i m_iF_i) ·(∑_i F_i) 
    - 4/3∑_i m_i  ,

and the color operator ∑_iF_i nullifies any colorless
physical state,

we can introduce a new mass parameter of quark pair

    m_ij = ( m_i + m_j) + 16/3 A_ij .

Then the model Hamiltonian reads

    H_CM = -3/4∑_i<j m_ij V^C_ij 
      - ∑_i<j v_ij V^CM_ij,

where we have briefly introduced two operators to represent the color
and chromomagnetic (CM) interactions between quarks,

    V^C_ij =   F_i·F_j , 
    
      V^CM_ij =   S_i·S_jF_i^a·F_j^a .


For the mesons the Hamiltonian is simplified to

    H_CM = m_qq̅
      - v_qq̅ V^CM_qq̅,

and for the baryons

    H_CM = 1/2∑_i<j m_ij
      - ∑_i<j v_ij V^CM_ij .

Since the quark model parameters of the baryon system usually are
different from that of the meson system, we assume that the pair
parameters m_qq and v_qq are different from their partners
m_qq̅ and v_qq̅ respectively. 

Their relations are studies in the next section, 
based on the numerical analysis and
the quark model consideration.




 §.§ Mesons



A meson is a color-singlet hadron composed of a quark and an antiquark. 
Its total spin is either 1 or 0.

The corresponding spin wave functions are denoted by

    χ_1m   = {|↑↑⟩ , 
        1/√(2)( |↑↓⟩ 
          + |↓↑⟩)  , 
        |↓↓⟩} , 
    
      χ_00   = 1/√(2)( |↑↓⟩
        - |↓↑⟩)  ,

where m is the third component of the total spin.

The masses of the pseudoscalar and vector mesons are given by


    M_J=0   = m_qq̅ - v_qq̅ , 
    
      M_J=1   = m_qq̅ + 1/3 v_qq̅ .





 §.§ Baryons





Baryons are composed of three quarks. 

Since we only consider the lowest S-wave baryons

and the color wave function is antisymmetric,

we only have to construct the symmetric spin⊗flavor wave functions.

The total spin of the baryon can be either 3/2 or 1/2. The spin
wave functions are classified according to the permutation symmetry,


    χ_3/2m^S   = {|↑↑↑⟩ , 
        1/√(3)( |↑↑↓⟩
          + |↑↓↑⟩
          + |↓↑↑⟩)  , 
        1/√(3)( |↑↓↓⟩
          + |↓↑↓⟩
          + |↓↓↑⟩)  , 
        |↓↓↓⟩} , 
    
      χ_1/2m^MS   = { - 1/√(6)( 
          |↑↓↑⟩
          + |↓↑↑⟩
          - 2 |↑↑↓⟩)  , 
        1/√(6)( |↑↓↓⟩
          + |↓↑↓⟩
          - 2 |↓↓↑⟩) } , 
    
      χ_1/2m^MA   = {1/√(2)( 
          |↑↓↑⟩
          - |↓↑↑⟩)  , 
        1/√(2)( |↑↓↓⟩
          - |↓↑↓⟩) } ,

where the superscript MS (MA) suggests the symmetric (antisymmetric)
property of the wave functions under the exchange of the first two
quarks.

Next, we combine the flavor wave functions |q_1q_2q_3⟩ with the
spin wave functions.  We get the following spin⊗flavor base wave functions:

    J=3/2:      ϕ_3/2m^{q_1q_2q_3} = |{qqq}⟩⊗χ_3/2m^S , 
    
      J=1/2:      ϕ_1/2m^{q_1q_2}q_3 = |{q_1q_2}q_3⟩⊗χ_1/2m^MS + permutations , 
    
         ϕ_1/2m^[q_1q_2]q_3 = |[q_1q_2]q_3⟩⊗χ_1/2m^MA + permutations ,

where we use the brace {⋯} to symmetrize the quark flavors and
the bracket [⋯] to antisymmetrize the flavors.

The mass of the spin-3/2 baryon is given by

    M_J=3/2 = 1/2(m_q_1q_2+m_q_1q_3 + m_q_2q_3) 
      + 1/6(v_q_1q_2+v_q_1q_3 + v_q_2q_3).


To obtain the masses of the spin-1/2 baryons consisting of three
different quark flavors, we need to diagonalize the following
2×2 matrix in the above basis [Eqs. (<ref>) and
(<ref>)],


    H_J=1/2 = 1/2 (m_q_1q_2+m_q_1q_3 + m_q_2q_3) +
      [ 1/6( v_q_1q_2 - 2 v_q_1q_3 - 2v_q_2q_3)         - 1/2√(3)( v_q_1q_3 - v_q_2q_3);         - 1/2√(3)( v_q_1q_3 - v_q_2q_3)                            -1/2v_q_1q_2 ] ,


which gives us two mixed states, which we denote by
ϕ_1/2m^q_1q_2q_3±, with masses


    M_J=1/2^± = 1/2 (m_q_1q_2+m_q_1q_3 + m_q_2q_3) 
      - 1/6( v_q_1q_2+v_q_1q_3 + v_q_2q_3)
      ±Δ_J=1/2,

respectively,

where


    Δ_J=1/2 = 1/3√(v_q_1q_2^2+v_q_1q_3^2+v_q_2q_3^2 
        - v_q_1q_2v_q_1q_3 - v_q_1q_2v_q_2q_3 - v_q_1q_3v_q_2q_3) .


Note that if the flavors of any two quarks in the baryon are identical,
we can assign q_1=q_2 and the second combination [Eq. (<ref>)]
does not exist. Then we get only one spin-1/2 baryon state
ϕ^{q_1q_1}q_3_1/2 m with mass


    E_J=1/2= 1/2 (m_q_1q_1+2m_q_1q_3)+ 1/6(v_q_1q_1-4v_q_1q_3).


We collect the wave function assignments of all lowest S-wave
baryons in Table <ref>.










§ NUMERICAL RESULTS







 §.§ Parameters




First we consider the mesons.

We can extract the two parameters m_q_1q̅_2 and
v_q_1q̅_2 from the experimental masses of corresponding
q_1q̅_2 pseudoscalar and vector mesons.

For the nn̅ mesons consisting of u,d flavors, we only use the
isovector π and ρ mesons to extract m_nn̅ and
v_nn̅.

We do not consider η and η' mesons to avoid the complexity
of flavor octet-singlet mixing and the chiral anomaly.

Instead, we use the following PCAC (partially conserved axial current) result
<cit.>,


    M_ss̅(^1S_0) = √(2M_K^2-M_π^2) = 687.220  ,

and the experimental mass of the ϕ meson to extract the parameters
m_ss̅ and v_ss̅.

The equation can also be derived in the chiral perturbation
theory <cit.>.

Another difficulty is that only one of the two cb̅ states, that
is, the B_c meson, was observed in experiment.

This state was first reported by CDF and OPAL collaborations in
1998 <cit.>, whose current mass in PDG is
6275.1 <cit.>.

Godfrey et al. had predicted its mass to be 6.27 using
the quark model in 1985 <cit.>; a more detailed study in
2004 gives M_B_c=6271 <cit.>, which is very
closed to the experimental value.

They also predicted M_B_c^*=6338.

Other quark model calculation coincides with their result.

For instance, Ikhdair et al. <cit.> predict
M_B_c^*=6340 and Ebert et al. <cit.>
predict M_B_c^*=6332.

In our work, we use the prediction M_B_c^*=6338 of
Godfrey et al. to determine the parameters of the cb̅ pair.

All the qq̅ pair parameters are presented in
Table <ref>.











Now we turn to the baryon sector.

We can only use the experimental masses of light-quark baryons and
singly heavy-quark baryons to extract the model parameters.


Besides, the Ω_b^* has not yet been observed in experiment.

We perform an unweighted nonlinear least-squares fit of 23 known
baryon masses to extract 13 model parameters, using the 
library <cit.>. Note that, with two
identical quarks, the pair parameters m_qq and v_qq only
appear in the combination m_qq+v_qq/3 in the mass formulas
(<ref>) and (<ref>).  So we can only
determine the value m_ss+v_ss/3 from the experimental data.  

The baryon parameters obtained are presented in
Table <ref>.






The fitting standard deviation is 7.66. Because the
Ω_b^* has not yet been observed in experiment, the parameters
m_sb and v_sb have large statistical errors.

The comparison of the fitted mass values with experimental data is
listed in Table <ref>.

Most fitting deviations of the baryon masses are within 10. 

The only exception is the Σ, whose deviation is 15.0.






In our model, all chromoelectric effects of color interaction are
included in the pair mass parameter m_qq (or m_qq̅).  If
the chromoelectric effects can be absorbed into the quark mass m_q
like in the original chromomagnetic model, we have the relation

    m_q_1q̅_1+m_q_2q̅_2 - 2m_q_1q̅_2≈ 0.

This is not true from our fitting. Typically

    m_nn̅+m_bb̅ - 2m_nb̅≈ -600  .








We also note that the quark pair mass m_qq is quite different from
m_qq̅ of its quark antiquark partner.

We list the difference 
δm_q_1q_2≡m_q_1q_2-m_q_1q̅_2 
in Table <ref>.

Indeed, many authors found that the effective quark masses extracted from
baryons were larger than that from mesons <cit.>.

This mass difference can be also accounted by adjusting the constant
c in the quark interaction [Eq. (<ref>)], if it can be
treated as a constant <cit.>.

Here we assume that


    A_qq̅≈ A_qq,

in Eq. (<ref>)

and the difference of the pair mass parameter becomes


    δ m_q_1q_2≡ m_q_1q_2 - m_q_1q̅_2≈δ m_q_1 + δ m_q_2 ,


where δ m_q = m_q^b - m_q^m is the difference of the effective
quark mass extracted from the baryon and meson.

Then we perform a least-squares fitting to obtain the mass difference
δm_q,

which is listed in Table <ref>.






The reduced chi-squared statistic is χ_ν^2=0.41.

In Table <ref>, we compare the chromomagnetic
interaction strengths in baryons and mesons using their ratio
R_q_1q_2≡ v_q_1q_2/v_q_1q̅_2.






We find that R_nn, R_ns, R_nc, R_sc are very close to
each others. R_sb is relatively small but with large statistical
error due to the lack of experimental data of B_c^*.

This phenomenon was first observed by
Keren-Zur <cit.>. The ratio was interpreted in the quark
model, using the Cornell potential or the Logarithmic potential. The
author also gave a simple interpretation by assuming that the contact
probability in the chromomagnetic interaction
[Eq. (<ref>)] is inversely proportional to the number
of quarks in the hadron.  Since the quark number is 3 in a baryon
and 2 in a meson, this gives a rough estimate of
R_q_1q_2≈ 2/3.


To estimate the heavy quark pair parameters {v_cc , v_cb ,
v_bb}, we assume that


    R_Q_1Q_2=2/3±0.30  ,


where we use the largest statistical error in
Table <ref> (except R_sb whose statistical error is
mainly due to the lack of experimental data) to set the parameter
range.

We should point out that even the estimate causes large standard errors
in {v_cc,v_cb,v_bb}; it does not have so many significant
effects on the mass of doubly and triply heavy-quark baryons as the
absolute values v_Q_1Q_2 are much smaller than v_qQ between
light and heavy quarks.

Using the mass difference Eq. (<ref>) and ratio
relation Eq. (<ref>), we can determine the
parameters between two heavy quarks, as well as m_ss and v_ss.

All the baryon parameters are collected in 
Table <ref>.
















 §.§ Mass spectra of doubly and triply heavy baryons




Substituting the parameters obtained in Sec. <ref> into
the Hamiltonians, we can obtain the masses of doubly and triply
heavy-quark baryons.

They are summarized in Table <ref>.







In our calculation M_Ξ_cc=3633.3±9.3.

It is much heavier than the SELEX's value by approximately 
100 MeV <cit.>, and very closed to the report of 
LHCb <cit.>.

The Ξ_cc^* state lies 62.8 above Ξ_cc.

This splitting is very closed to the one between Σ_c and 
Σ_c^* (64.5), which is consistent with the GMO mass 
relation <cit.>


    M_Ξ_cc^*-M_Ξ_cc = M_Σ_c^*-M_Σ_c .


A similar relation holds if we replace the u, d quarks by the s quark


    M_Ω_cc^*-M_Ω_cc = M_Ω_c^*-M_Ω_c ,


where both sides are approximately 71.

Similar to the Σ_c^(*) (or Ω_c^(*)) case, the splitting 
between Ξ_cc^* and Ξ_cc (or between Ω_cc^* and 
Ω_cc) is too small to induce a transition through the emission 
of the π meson; however, the transition is still possible through γ 
emission.

The situation for bottomed baryons is similar;


    M_Ξ_bb^*-M_Ξ_cc≈ M_Σ_b^*-M_Σ_b ,


where the left-hand side is 19.9 and the right-hand side 
is 20.2.

This splitting is significantly smaller than that of charmed baryons.

The reason is that the hyperfine splitting is reciprocal to the masses 
of quarks, and of course the b quark is much heavier than the c quark.

There is also one GMO mass relation about the triply heavy-quark baryons,
that is


    M_Ω_cbb^*-M_Ω_cbb = 
    M_Ω_ccb^*-M_Ω_ccb


where both sides are approximately 31.

For spin-1/2 doubly heavy-quark baryons composed of three different
quarks, namely the qcb baryon states (q=u,d,s), one should consider
the mixture between two basis states (<ref>) and (<ref>). 

Numerically, the mixing matrix in Eq. (<ref>)
is given by (in )


    [  -6.7   3.3;   3.3 -31.4 ]  and  [  -1.5   6.6;   6.6 -35.3 ] ,


for ncb and scb flavor configurations respectively.

The eigenvalues of ncb states are {-31.8,-6.2}, with 
eigenvectors {-0.13,0.99} and {0.99,0.13}, 

and the eigenvalues of scb states are {-36.6,-0.3} with
eigenvectors {-0.19,0.98} and {0.98,0.19}.

In both cases, the mixing is very small and the mixing mass effects
are within 2.

If one ignores the mixing, then the Ξ_cb and Ξ_cb' can be
treated as states in the flavor SU(2)_nc singlet and triplet
representations and the Ω_cb and Ω_cb' as states in
the SU(2)_sc singlet and triplet representations, respectively
<cit.>. The following GMO mass relations hold
approximately:


    2M_Ξ_cb^*+M_Ξ_cb'-3M_Ξ_cb≈ 
    2(M_Σ_c^*-M_Σ_c)  , 
    
        2M_Ω_cb^*+M_Ω_cb'-3M_Ω_cb≈ 
    2(M_Ω_c^*-M_Ω_c)  , 
    
        2(M_Ξ_cb^*-M_Ξ_cb') - 
    (M_Ω_cbb^*-M_Ω_cbb) ≈ 
    M_Σ_b^*-M_Σ_b , 
    
        2(M_Ω_cb^*-M_Ω_cb') - 
    (M_Ω_cbb^*-M_Ω_cbb) ≈ 
    M_Ω_b^*-M_Ω_b .


We find that the errors of all those relations are within 5.




§ CONCLUSIONS




In this work, we generalized the chromomagnetic model by considering
the effect of color interaction.

According to color algebra, the quark effective mass and the color
interaction between quarks are combined into a new quark pair mass
parameter.

The quark pair parameters between two light quarks and that between
light-heavy quarks are determined using the experimental masses
of lowest S-wave hadrons.

The pair parameters between two heavy quarks are estimated from the
corresponding pair parameters between the quark and antiquark in mesons,
using the mass difference and a ratio relation about the
chromomagnetic interaction.

We have calculated the mass spectra of the lowest S-wave doubly and
triply heavy baryons.

We obtained M_Ξ_cc=3633.3±9.3, which is close
to the report of LHCb.

We hope that future experiments in LHCb, BES-III et al.
confirm the existence of these states.




§ ACKNOWLEDGMENTS



The authors thank S. L. Zhu for helpful comments and discussions.

X. Z. W. is grateful to J. F. Jiang, H. S. Li, L. Meng, G. J. Wang,
S. J. Wu, and B. Zhou for helpful discussions.

This project is supported by the National Natural Science Foundation
of China under Grants No. 11621131001 and 11575008.






