Masses and magnetic moments of hadrons with one and two open heavy quarks: heavy baryons and tetraquarks

In this work, we compute masses and magnetic moments of the heavy baryons and tetraquarks with one and two open heavy flavors in a unified framework of MIT bag model. Using the parameters of MIT bag model, we confirm that an extra binding energy, which is supposed to exist between heavy quarks ($c$ and $b$) and between heavy and strange quarks in literatures, is required to reconcile light hadrons with heavy hadrons. Numerical calculations are made for all light mesons, heavy hadrons with one and two open heavy flavors, predicting the masses of doubly charmed baryons to be $ M(\Xi _{cc})=3.604$ GeV, $ M(\Xi _{cc}^{\ast })=3.714$ GeV, and that of the strange isosinglet tetraquark $ud\bar{s}\bar{c}$ with $J^{P}=0^{+}$ to be $ M\left( ud\bar{s}\bar{c},0^{+}\right) =2.934$ GeV. The state mixing due to chromomagnetic interaction is shown to be sizable for the strange scalar tetraquark $nn\bar{s}\bar{c}$.

In identifying and/or finding these DH hadrons experimentally, it is helpful to have a systematic estimate of masses and other properties of them within an unified framework. For instance, a mass predictions [14,16] of the doubly charmed baryon Ξ cc , which are larger about 100 MeV than that measured in 2002 by the SELEX Collaboration at Fermilab [30](awaiting confirmation), helps LHCb Collaboration to search the Ξ ++ cc [1] eventually. In this work, we apply MIT bag model [31,32] with chromomagnetic interaction and a strong coupling α s running with the bag radius to systematically study the open heavy baryons and tetraquarks with one and two heavy flavors and compute the masses and other static properties(magnetic moments, electric charge radii) of them. It is confirmed that an extra binding energy between heavy quarks (c and b) and between heavy and strange quarks is required to reconcile light hadrons with heavy hadrons. Computed results are compared to other calculations and in consistent with the measured masses and other properties of light hadrons and singly heavy baryons in their ground states(except for π). For the J P = 1 2 + states of heavy baryons Ξ c , Ξ b , Ξ bc , Ω bc and the heavy tetraquarks, the chromomagnetic mixing is taken into account, and the respective mass splittings are computed variationally.
It is well known that bag model [31] embodies two primary features of quantum chromodynamics (QCD): asymptotic freedom at short distance and confinement at long distance. The simple structure of the model enables us to describe mesons(qq), baryons(qqq) and even hadrons made of multiquarks. In the past few decades, bag model has been applied to describe the doubly heavy baryons [11][12][13] and multiquark hadrons, including light exotic baryons with five and seven nonstrange quarks [33]. In order to evaluate the masses of doubly heavy baryons, a large running strong coupling α s was applied in Ref. [12]. A Coulomb-like interaction is derived between heavy quarks in a bag in Refs. [34,35]. This paper will be organized as follows. In Sec. II, we review some basic relations of MIT bag model, including chromomagnetic interaction (CMI) among the quarks in bag. In Sec. III, a systematic numerical calculation is performed for the established light and singly heavy(SH) baryons, with the optimal set of parameters obtained and the results for masses and other properties reproduced. In Sec. IV, we present detailed predictions for masses and other properties for doubly heavy baryons and the tetraquarks with one and two open heavy quarks. The paper ends with summary and conclusions in Sec. V.

A. Mass Formula
Treating hadron as a spherical bag, MIT bag model provides an approach to estimate masses and other properties of hadrons in their ground states [31,32], in which the chromomagnetic interaction is derived from the energy of a sphere-like gluon field interacting with quark fields in bag 3 [31]. The mass formula of hadron in MIT bag model is, where the first term is the kinematic energy of all quarks in bag with radius R, the second is the volume energy of bag with bag constant B, the third is the zero-point-energy (ZPE) with coefficient Z 0 , and ∆H is the short-range interaction among quarks in bag, which we will address in this work. Here in Eq. (1), n i is number of quark or antiquark in bag with mass m i and flavor i, where i can be the light nonstrange quarks n = u, d, the strange quark s, the charm quark c and the bottom quark b. The value of R is to be determined variationally, and the dimensionless parameters x i = x i (mR) are related to the bag radius R by an transcendental eigen-equation The interaction energy ∆H = B EB + M CMI is composed of two energy terms: (1) The spin-independent binding energy B EB , due mainly to the short-range chromoelectric interaction between quarks (and/or antiquarks). Owing to its smallness for the relativistic light quarks n(= u, d), this energy, scales mainly as − α s /r i j , becomes sizable when both of two quarks i and j are massive and moving nonrelativistically. In present work, we treat this energy as sum of the pair binding energies B QQ ′ (B Qs ) between heavy quarks and between heavy quark Q and strange quark s [14,36,37]. The net effect for this chromoelectric interaction amounts to introduction of five binding energies B cs , B cc , B bs , B bb and B bc for any quark pair in color configuration3 c , which are extractable from heavy mesons and can be scaled to other color configurations.
(2) The chromomagnetic interaction energy, due to perturbative gluon exchange between quarks (antiquarks) i and j, with λ i the Gell-Mann matrices, σ i the Pauli matrices, and C i j the CMI parameter. In MIT bag model, the parameters C i j are given by where α i ≡ ω i R, λ i ≡ m i R, α s (R) is the running strong coupling,μ i is the reduced magnetic moment without electric charge, and I i j and F(x i , x j ) are rational functions of x i and x j , given explicitly by [31]. where (3) for a given m i R, and Bag radius R ranges from 3 GeV −1 to 6 GeV −1 , and the standard radius is set to be 5 GeV −1 (≈ 1 fm) for checking. The solid line represents our result (11) while the dashed line corresponds to Eq. (10) with γ = 2.847, Λ QCD = 0.281 GeV and w = 2πn/9. The dotted line shows Eq. (10) with γ = 1, Λ QCD = 0.281 GeV and w = 2πn/9. The dotdashed line indicates that of Ref. [12]. All four behaviors adopt n = 1.
In some applications of bag model [11,12,34,35,38], the parameter α s takes a logarithmic form where w and n(= 1 or 2) are the parameters, Λ QCD is the QCD scale(0.2 ∼ 0.5 GeV), and γ is prefactor used to avoid infrared divergence. Similar to Ref. [12], we take w = 0.296, Λ QCD = 0.281 GeV, γ = 1 and n = 1 to set α s (R) = 0.296 which is plotted in Fig. 1. Among four lines showing the running of α s in the plot, the solid line shows notable variation and corresponds to a relative lower value of α s . Given the parameter values of the quark mass m i , bag constant B, the ZPE coefficient Z 0 and strong coupling constant α s (R) depending on bag radius R, one can apply variational method to determine the respective bag radius R for each hadron and the respective x i through Eq. (3). Then, it is straightforward to use Eqs. (1),(2) and (4) to compute the ground-state masses and other static properties (magnetic moments, the charge radius) of the hadrons ranging from the light hadrons to heavy tetraquarks. The computed results for the light hadrons are listed in Table I, compared to that predicted by original MIT bag model.
We stress that for a given hadronic state there is in principle a unique set of the solution x i and R corresponding to respective bag dynamics, as indicated by our computation. Owing to (x i , R)-dependence of the ∆H , a simple and analytic mass formula is lacking for the hadrons with chromomagnetic-mixing since for that purpose one has to first diagonalize the CMI matrices before the variational analysis, which amounts to a higher-order algebraic equations.

B. Chromomagnetic Interaction
In evaluating the spin-dependent mass due to the CMI (4), in which λ i should be replaced by −λ * i for an antiquark, one has to diagonalize the CMI matrix for given hadron multiplets with 6 certain spin-parity J P to give the respective mass splittings [7] within the multiplets. For this, we list all the flavor-spin-color wavefunctions of hadrons including tetraquarks considered in this work, and present relevant formulas of the color and spin factors for them. Mesons: The color wavefunction φ M = |q 1q2 can be one of two spin states (of vector and scalar like): where subscript J = 0 or 1 outside the bracket denotes the total spin of hadron. The spin-color wavefunctions with spin J = 0 and 1 are then Baryons: The color wavefunctions φ B = (q 1 q 2 )¯3q 3 can be in one of three spin states where (q 1 q 2 ) stands for a diquark with spin J = 1 or 0 in color configuration3 c .
To write wavefunction for a hadron, the flavor symmetry has to be considered. For a flavorsymmetric wavefunction of (q 1 q 2 ), with isospin I = 1 or identical flavors, we use a symbol δ S 12 = 1. For a flavor-asymmetric wavefunction with I = 0, a symbol δ A 12 = 1 will be used. For two quarks q 1 and q 2 with different flavors which goes beyond isospin symmetry, one can use δ S 12 = δ A 12 = 1. With the help of Pauli principle, one can write three flavor-spin-color wavefunctions for baryons Owing to the non-diagonal chromomagnetic interaction (4), some hadronic states with same J P but different spin-color wavefunctions can mix(CMI mixing). For example, for the doubly heavy baryons Ξ bc and Ξ ′ bc with flavor structure (bc)s the use of δ S 12 = δ A 12 = 1 is not enough to distinguish the two configurations φ B χ B 2 and φ B χ B 3 solely in terms of their J P quantum numbers. Thus, the physical state must be one of the mixing states of them. See Sect. IV for the details of chromomagnetic mixing.
Tetraquarks: A tetraquark can have the color structure of whether 6 c ⊗6 c or3 c ⊗ 3 c , with the respective color wavefunctions, and it can be one of the following six states χ T 1 = (q 1 q 2 ) 1 (q 3q4 ) 1 2 , χ T 2 = (q 1 q 2 ) 1 (q 3q4 ) 1 1 , χ T 3 = (q 1 q 2 ) 1 (q 3q4 ) 1 0 , χ T 4 = (q 1 q 2 ) 1 (q 3q4 ) 0 1 , 7 which lead to twelve basis wavefunctions We list all relevant color wavefunctions in Appendix A and spin wavefunctions in Appendix B. With them, one can evaluate the color and spin factors in Eq. (4) with the help of the following formulas where n, m and x, y indicate the specific color and spin states respectively, i and j are the indexes of quarks (antiquarks), and the functions c and χ are the respective basis vectors in the color and spin spaces. Table II lists a set of non-mixed hadronic states with their respective CMI's. Now, we are in the position to construct the matrix formula of CMI energy (4) and diagonalize it so as to minimize the obtained mass formula. Adding the binding energy (for heavy quark pair and for a pair of one heavy quark and one strange quark) to the bag energy, one can solve the dynamical parameters x i and R, and thereby obtain the wavefunctions of a given hadron.

C. Hadronic Properties
Given the parameters x i and R describing a hadronic state, mass and other properties(e.g.,the charge radius and magnetic moment) can be evaluated. Following the standard method, one can firstly calculate the contribution of a quark or an antiquark i with electric charge Q i to charge The sum of Eq. (21) then gives the charge radius of a hadronic state [40] We note that Eq. (22) also holds true for the chromomagnetic-mixing systems having the identical quark constituents. For magnetic moment, the following equations [31,41], which are computed relative to the magnetic moment of proton and has the unit of µ p , are useful: III: Sum rule for magnetic moments of spin states of mesons (q 1q2 ), baryons (q 1 q 2 ) q 3 , and tetraquarks (q 1 q 2 ) (q 3q4 ) and their spin-mixed systems.
where g i = 2, and S iz is the third component of spin for an individual quark or antiquark. In all Tables for the results of magnetic moments, obtained from Eq. (23) and Eq. (24), we transform them into that in the unit of the nuclear magneton µ N , with the help of the measured data µ p = 2.79285µ N . If the chromomagnetic mixing enters, the total spin wavefunction becomes by which Eq. (24) gives with µ tr the cross-term standing for transition moment [13] and (C 1 , C 2 ) the eigenvector of the given mixing state. We list all spin wavefunctions in Appendix B, and derive magnetic moments for them and their possibly-mixed systems involved in this work. The results for the spin wavefunctions and the respective magnetic moments are listed in Table III collectively. Note that the cross-terms in CMI-mixing systems are not symmetric under the exchange between quarks q 1 and q 2 or, between q 3 and q 4 in the flavor space. While the expression of crossterm for the diquark (ud) differs a sign for (ud) and (du) within the symmetric or asymmetric flavor wavefunctions when I 3 = 1 or -1 in isospin space, respectively, the explicit computation via these wavefunctions can offset such cross-term. Similar conclusions also apply as the hadron systems respect the S U(2) isospin symmetry.

III. DETERMINATION OF PARAMETERS
In MIT bag model, the parameters (nonstrange m n and strange m s quark masses, B, Z 0 and α s ) are determined based on the mass spectra of the light hadrons N, ∆, ω and Ω in their ground states. The results read [31]  We choose Eq. (27) to be the parameters applying to both of light and heavy hadrons, with one exception that the strong coupling α s changes with the size R of hadron around 0.55, as given by Eq. (11). To fix the model parameters it remains two tasks yet.
The first task is to extract the heavy quark masses m c and m b . Given Eq. (27), one can apply Eqs. (1) and (4) The second is to fix the binding energy B QQ ′ (and B QQ ′ , Q, Q ′ = s, c, b here), which is proposed in Ref. [14] to occur in charmed-strange hadrons, bottom-strange hadrons and heavy quadrennia. It can be due to nontrivial short-range interaction between two heavy quarks and between heavy and strange quarks [14,36,37]. Applied to the strange heavy mesons(Qs and QQ′), this binding enters the mass formula through Applying to the case of the vector mesons D * s = cs, this allows one to solve the binding term, with the bag radius R solved variationally for the D * s mesons. Numerically, one finds, B cs = −0.050 GeV by Eq. (30). Note that the short-range color interaction of the quark pair cs in color singlet(1 c ) in heavy meson D * s can be related to that of the cs pair in color antitriplet(3 c ) in a heavy baryon nsc by the factor 1/2, one can reasonably assume, in the short range, that the strength of the cs interaction in3 c is half that of cs in 1 c . This follows that B cs = B cs /2 = −0.025 GeV. Here, the factor half can be extracted from the ratio of the color factor for the quark pair in3 c and the quark-antiquark pair in 1 c , and solve the model (1) for the heavy mesons B * s , J/ψ, Υ and B * c , obtaining, by Eqs. (29) and (31), where the results for the cs pair is also included. Here in computation, we have used the mass M(B * c ) = 6.332 GeV of the heavy meson B * c in Ref. [42], due to lacking of the measured B * c . It can be seen from Eq. (32) that B QQ ′ depends monotonically on the reduced mass µ QQ ′ = m Q m Q ′ /(m Q +m Q ′ ) of two involved quarks Q and Q ′ . The dependence (FIG. 2) can be approximated by  A). The scale factor g([QQ ′ ] R )(=ratio of the color factor for representation R and the color factor for3 c ) for pair QQ ′ (= bb, cc, bc, bs, cs) can be given explicitly by and the binding energy between the pair QQ ′ in R is then  Tables IV, V, VI, VII and Table IX for the states without state mixing due to the CMI. The results for the DH baryons are also presented in Tables IX. Some remarks are in order: (i) Some of the computed masses M bag in Table I deviate Table IV and V as they share the same masses, charge radii but minus magnetic moments in comparison with the mesons in Tables. The anti-particles of the heavy tetraquarks are ignored as well; (iii) In Table IX, our mass predictions 3.714 GeV(for Ξ * cc ) are comparable to the quark-model prediction M Ξ * cc = 3.727 GeV [16], and also to 3706 ± 28 MeV and 3692 ± 28 MeV by the lattice QCD [43,44] respectively. Table X shows comparison of our predictions with other works for DH baryons. The prediction 3.604 GeV for the Ξ cc is in consistent with the measured mass 3.621 GeV, considering the simplicity of the model. (iv) The predicted magnetic moments in Table VI are in good agreement with the measured values, from which the magnetic moments for heavy baryons and tetraquarks are predicted; (v) Our prediction 0.75 fm for the proton charge radius is slightly lower than the newly-measured value 0.83 fm [45,46], similar to original MIT bag model [31,32].

A. Heavy Baryons including the CMI Mixing
Hadrons containing a diquark or antiquark with different flavors, may not respect flavorsymmetry of wavefunction for involved light quark pairs. As such, the states with same J P but different spin-color wavefunctions may mix due to the CMI (4), as mentioned in Sect. II (B). To begin with, we first consider the system of baryons with J P = 1/2 + in which two spin-color states 3 ) can mix. The associated baryons are the Ξ c , the Ξ b , the Ξ bc and the Ω bc . In terms of the wavefunctions in color and spin space (Appendix A and B), one can compute the CMI matrices in the degenerate subspace of the spin-color basis φ B χ B when the chromomagnetic mixing occurs (Appendix C). These CMI matrices depend upon C i j with the subscripts (i, j) of C i j denote the flavor constituents. One can diagonalize the CMI matrix, say (C1), to write mass formulas of the baryons using Eq. (1). This is done by solving the eigenvalues and eigenvectors of the matrix (C1) analytically and using the later to identify(denote) the mixed states. Of course, the relevant binding energies (B cs , B bs and B bc ) are included in the mass formulas.
In Table VIII, we list our computed results of masses and other properties for the CMI-mixed systems of heavy baryons. The net effects of the state mixing (the second column of Table) are not so significant in general and they are somehow negligible in the case of singly heavy baryons. This can be due to the higher S U(3) flavor symmetry and heavy quark symmetry which suppress the off-diagonal elements in matrix (C1). For this reason, we employ still the normal notations of the states for the SH baryons. The computed masses of the SH baryons Ξ c , the Ξ ′ c , the Ξ b and the Ξ ′ b are comparable with the measured data, as seen in the fifth column with reasonable errors. The magnetic moments for Ξ + c , Ξ 0 c , Ξ 0 b and Ξ − b are predicted to be 0.37µ N , 0.50µ N , −0.12µ N and −0.08µ N which are comparable to 0.35µ N , 0.50µ N , −0.045µ N and −0.08µ N in Ref. [47], respec- VIII: Predicted masses (in GeV), magnetic moments(in µ N ) and charge radii of heavy baryons. M exp is the observed mass isospin-averaged [39]. Magnetic moments and charge radii are organized in the order of I 3 =

B. Singly Heavy Tetraquarks
Let us consider the strange tetraquarks nnsc and nnsb containing one heavy quark, one strange quark and two nonstrange light quarks. In such a case, the CMI mixing happens if J 2. We use a combination of the spin-color basis functions φ T χ T to denote the mixed states. For instance, the combination (φ T 2 χ T 3 , φ T 1 χ T 6 ) stands for a mixed state c 1 φ T 2 χ T 3 + c 2 φ T 1 χ T 6 for (J P , I)=(0 + , 1). Similarly, other mixed states can be denoted as (φ T 1 χ T 3 , φ T 2 χ T 6 ) for (J P , I)=(0 + , 0), as (φ T 2 χ T 2 , φ T 2 χ T 4 , φ T 1 χ T 5 ) for (J P , I)=(1 + , 1) and (φ T 1 χ T 2 , φ T 1 χ T 4 , φ T 2 χ T 5 ) for (J P , I)=(1 + , 0). The binding energy matrices become diagonal in mass formula since the mixed states have two color configurations while spin states are orthogonal. Note that diagonalization should be applied to the sum of the interaction matrices before evaluating the hadron mass.
Following the variational principle, we diagonalize the 2 × 2 matrix to solve two analytical eigenvalues and construct the mass formula as usual. Application of the same procedure to the 3 × 3 matrix is, however, not straightforward, for which the eigenvalues are some roots of a cubic equation. For this, we scan three sets of x i and R to solve the cubic equation numerically so that one can obtain the minimized masses within three root eigenvalues.
Our numerical results are shown in Table XI, with a notable tetraquark of an isosinglet nnsc with J P = 0 + , which has two masses 2.934 GeV and 2.513 GeV for its two mixed states. Comparing with the measured mass 2866 ± 7 MeV of X 0 (2900) reported by LHCb [8] and the quark model prediction 2863.4 ± 12 MeV[9], our prediction 2.934 GeV is larger even if the model error 40 MeV is subtracted. If we rather, as Karliner suggested for the color3 ⊗ 3 configuration, ignore the CMI mixing and evaluate directly the masses of the φ T 1 χ T 3 and φ T 2 χ T 6 states, the resulted masses lie around 2.7 GeV, away from the LHCb reported mass of the X 0 (2900). Our calculation suggests that chromomagnetic mixing is strong for the strange tetraquark nnsc with J P = 0 + and yields a mass splitting as large as 420 MeV. XI: Computed mass (in GeV), magnetic moments(in µ N ) and charge radii of singly heavy tetraquarks nnsc and nnsb. Magnetic moments and charge radii are organized in the order of I 3 = 1, 0, −1 for I = 1. Bag radius R 0 is in GeV −1 . Now, let us consider the doubly heavy tetraquarks qqQQ with strangeness S ≤ 2. In this case, hadrons consist of the nonstrange tetraquarks nnQQ and the strange tetraquarks nsQQ and ssQQ. They lie in a larger(compared to baryons) space spanned by more configurations(bases) in which the CMI mixing occurs variously. For the isotriplet tetraquarks with J P = 0 + , the general ground state can be the mixed one, with the wavefunction (φ T 2 χ T 3 , φ T 1 χ T 6 ) . For isosinglet tetraquark nnQQ with J P = 1 + , the wavefunction has the form of (φ T 2 χ T 5 , φ T 1 χ T 4 ). In the case of the strange tetraquark nsQQ with J P = 1 + , the wavefunction can be of (φ T 2 χ T 2 , φ T 2 χ T 5 , φ T 1 χ T 4 ) and the wavefunction of the tetraquark nncb is similar to that of nnsc. Note that the strange DH states nscb with mixing among six spin-color states are not considered for simplicity.
The computation of the mass and other properties of these DH tetraquarks is similar to that for

V. SUMMARY AND DISCUSSIONS
In this work, we have studied systematically masses and other properties of hadrons with one and two open heavy quarks within an unified framework of MIT bag model with chromomagnetic interaction. Masses, magnetic moments and charge radii of heavy baryons and heavy  We also confirm that a term of extra binding energy B QQ ′ , proposed previously to exist among heavy quarks (c and b) and between heavy and strange quarks [14], is required to reconcile light hadron with heavy hadrons, with a useful formula provided for B QQ ′ . This binding effect may rise from the enhanced short-range interaction between two relatively heavy quarks and makes the mass pattern and other properties of heavy hadrons differing from that in light sector. We have also employed a slowly-running strong coupling α s (R) to reflect its dependence upon the hadron sizes proportional to the average distance between two interacted quarks (or antiquark) in a hadron. The strong coupling α s (R) runs from 0.4 to 0.6 as the bag radius R varies between 3 ∼ 6 GeV −1 .
We remark that the MIT bag model can reproduce the measured masses of heavy hadrons within the accuracy of 40-50 MeV, from which we proceed to predict the masses and other properties of the tetraquarks with one and two open heavy quarks. For the DH tetraquarks, we reduce the error limit to about 40 MeV and exclude X 0 (2900) to be an isosinglet tetraquark of nnsc due to the mismatch with the measured data as high as 70 MeV.
Owing to the uncertainty of model computations, we are not able to discuss the near-threshold effect. The mismatch of our predictions with the measured data may come from the limitations of bag model in this work: (1) the bag may deform into elliptic shape in the case of the DH hadrons, and (2) the constant approximation of the short-range binding energy may not be sufficient as the later may depend upon hadrons size R implicitly, for instance, in the form of a Coulomb-like ∼ 1/R, and needs to be determined variationally. These effects go beyond the scope of this work and await the further exploration in the future.
in the subspace of (χ T 1 , χ T 2 , χ T 3 , χ T 4 , χ T 5 , χ T 6 ). One can then use these factors of color and spin space to find the matrix representation of the CMI in Eq. (4) for a given hadronic state. The color and spin factors are the diagonal elements of the matrices in Eq. (A4-A6) and Eq. (B4-B13), respectively. The off-diagonal elements of these matrices lead to chromomagnetic mixing of these basis functions given in Appendix A and at the beginning of this section.