Masses of doubly heavy-quark baryons in an extended chromomagnetic model

We extend the chromomagnetic model by further considering the effect of color interaction. The effective mass parameters between quark pairs ($m_{qq}$ or $m_{q\bar{q}}$) are introduced to account 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 $\Xi_{cc}$ fits into the model with an error of 12 MeV.

The quark model is one of the most used approaches to study the mass spectra of hadrons [12,[57][58][59][60][61][62][63][64][65][66][67][68][69][70]. 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 [12,[71][72][73][74][75][76][77][78][79][80]. 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. [ 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. II we introduce the extended chromomagnetic model and construct the model wave functions of mesons and baryons. In Sec. III A we determine the model parameters. The numerical results are presented and discussed in Sec. III B. We conclude in Sec. IV.

A. The Hamiltonian
In the quark model, the quark effective Hamiltonian reads [66,67] where is the relativistic mass term and V ij is the quark interaction potential between ith and jth quarks. In a nonrelativistic reduction, and where includes the color linear confinement and the Coulomb-type interaction, V hyp ij is the color hyperfine interaction, and V so ij is the spin-orbit interaction. For the S-wave hadron, V so ij has no contribution, and V hyp ij can be simply replaced by the chromomagnetic interaction where S i and F i are the ith quark's spin operator and color operator respectively, 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 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 conf ij . The chromomagnetic interaction reads The coefficient v ij depends on the quark masses and the spatial wave function of the hadron 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 [81,82]. In Ref. [ whereλ i = 2F i . We use this extended chromomagnetic model to study all lowest S-wave mesons and baryons systematically.
and the color operator i F i nullifies any colorless physical state, we can introduce a new mass parameter of quark pair Then the model Hamiltonian reads where we have briefly introduced two operators to represent the color and chromomagnetic (CM) interactions between quarks, For the mesons the Hamiltonian is simplified to and for the baryons 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 qq and v qq respectively. Their relations are studies in the next section, based on the numerical analysis and the quark model consideration.

B. 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 where m is the third component of the total spin.
The masses of the pseudoscalar and vector mesons are given by

C. 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, 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 1 q 2 q 3 with the spin wave functions. We get the following spin ⊗ flavor base wave functions: 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 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. (27) and (28)], which gives us two mixed states, which we denote by respectively, where Note that if the flavors of any two quarks in the baryon are identical, we can assign with mass We collect the wave function assignments of all lowest S-wave baryons in Table I.

A. Parameters
First we consider the mesons. We can extract the two parameters m q 1q2 and v q 1q2 from the experimental masses of corresponding q 1q2 pseudoscalar and vector mesons. For the nn mesons consisting of u, d flavors, we only use the isovector π and ρ mesons to extract m nn and v nn .
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 [83][84][85], and the experimental mass of the φ meson to extract the parameters m ss and v ss . The equation can also be derived in the chiral perturbation theory [86].  bbb Another difficulty is that only one of the two cb states, that is, the B c meson, was observed in experiment. This state was first reported by CDF and OPAL collaborations in 1998 [87,88], whose current mass in PDG is 6275.  In our work, we use the prediction M B * c = 6338 MeV of Godfrey et al. to determine the parameters of the cb pair. All the qq pair parameters are presented in Table II. 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 GSL library [92].
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 (33) and (29). So we can only determine the value m ss + v ss /3 from the experimental data.
The baryon parameters obtained are presented in Table III Theo. 5832.2 ± 6.2 5947.0 ± 6.8 6054.8 ± 11.7 parameter m qq (or m qq ). If the chromoelectric effects can be absorbed into the quark mass m q like in the original chromomagnetic model, we have the relation This is not true from our fitting. Typically We also note that the quark pair mass m qq is quite different from m qq of its quark antiquark partner. We list the difference δm q 1 q 2 ≡ m q 1 q 2 − m q 1q2 in Table V. Indeed, many authors found that the effective quark masses extracted from baryons were larger than that from mesons [18,78,82,93]. This mass difference can be also accounted by adjusting the constant c in the quark interaction [Eq. (5)], if it can be treated as a constant [66,67]. Here we assume that in Eq. (13) and the difference of the pair mass parameter becomes where δm q = m b q − m m q 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 VI. The reduced chi-squared statistic is χ 2 ν = 0.41. In Table VII, we compare the chromomagnetic interaction strengths in baryons and mesons using their ratio R q 1 q 2 ≡ v q 1 q 2 /v q 1q2 . 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 [94]. 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. (6)] 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 1 q 2 ≈ 2/3. To estimate the heavy quark pair parameters {v cc , v cb , v bb }, we assume that where we use the largest statistical error in Table VII (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 1 Q 2 are much smaller than v qQ between light and heavy quarks.
Using the mass difference Eq. (36) and ratio relation Eq. (37), 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 VIII.

B. Mass spectra of doubly and triply heavy baryons
Substituting the parameters obtained in Sec. III A into the Hamiltonians, we can obtain the masses of doubly and triply heavy-quark baryons. They are summarized in Table IX.
In our calculation M Ξcc = 3633.3 ± 9.3 MeV. It is much heavier than the SELEX's value by approximately 100 MeV [1], and very closed to the report of LHCb [11]. The Ξ * cc state lies 62.8 MeV above Ξ cc . This splitting is very closed to the one between Σ c and Σ * c (64.5 MeV), which is consistent with the GMO mass relation [95]  A similar relation holds if we replace the u, d quarks by the s quark where both sides are approximately 71 MeV. 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; where the left-hand side is 19.9 MeV and the right-hand side is 20.2 MeV. 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 where both sides are approximately 31 MeV.
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 (27) and (28). Numerically, the mixing matrix in Eq. (30)  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 [21]. The following GMO mass relations hold approximately: 2M We find that the errors of all those relations are within 5 MeV.

IV. 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 Ξcc = 3633.3 ±9.3 MeV, 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.