Excited state mass spectra of doubly heavy $\Xi$ baryons

In this paper, the mass spectra are obtained for doubly heavy $\Xi$ baryons, namely, $\Xi_{cc}^{+}$, $\Xi_{cc}^{++}$, $\Xi_{bb}^{-}$, $\Xi_{bb}^{0}$, $\Xi_{bc}^{0}$ and $\Xi_{bc}^{+}$. These baryons are consist of two heavy quarks($cc$, $bb$ and $bc$) with a light($d$ or $u$) quark. The ground, radial and orbital states are calculated in framework of Hypercentral constituent quark model with coul- omb plus linear potential. Our outcomes are also compared with other predictions, thus, the average possible range of excited states masses of these $\Xi$ baryons can be determined. The study of the Regge trajectories are performed in (n, $M^{2}$) and (J, $M^{2}$) planes and their slopes and intercepts are also determined. Lastly, the ground state magnetic moments of these doubly heavy baryons are also calculated.

We have used the QCD inspired Hypercentral constituent quark model(hCQM) with coloumb plus linear potential. The first order correction is also taken into account to the potential and calculation has been performed by solving six dimensional hyper radial Schrödinger equation numerically [1, 28,29]. We have calculated the mass spectra of radial excited states upto 5S and orbital excited states for 1P-5P, 1D-4D and 1F-2F states. As per our knowledge, all the theoretical approaches have considered the m u =m d so far but the light quark masses are different in our model. Thus, we have obtained the mass spectra with u & d quarks combinations for these baryons. Obtained masses were used in formation of Regge trajectories in (n, M 2 ) and (J, M 2 ) planes. The determination of slope and intercept of Regge trajectories of these baryons are very important as it provides better understanding of the dynamics of strong interactions in production of charmed and bottom baryons at high energies.
The paper is organized as follows. We give brief introduction in sect.1 and explained our Hypercentral Constituent Quark Model in sect. 2. We present our mass spectra results of all doubly heavy Ξ baryons in sect. 3. Regge trajectories and magnetic moments are discussed in sect. 4. At last, our conclusion is in sect. 5.

The Model
The methodology for the determination of excited masses follow the same pattern as in our previous work [1] and Refs. their-in. Therefore, we discuss model very briefly in present paper. Starting with the Jacobi coordinates of three quark baryons that are given in terms of mass(m i ) and co ordinates(r i ) below [30]. The quark masses are taken in calculations as m u = 0.338, m d = 0.350, m c = 1.275 and m b = 4.67 (in GeV). The co-ordinates ρ and λ are with the respective reduced masses are given by The Hamiltonian of three body baryonic system in the hCQM is then expressed as The hyper radius x = ρ 2 + λ 2 is a collective co ordinate and therefore the hypercentral potential contains also the three-body effects. where, m = 2mρm λ mρ+m λ , is the reduced mass and x is the six dimensional radial hyper central coordinate of the three body system. In present paper, the confining three-body potential is chosen within a string-like picture, where the quarks are connected by gluonic strings and the potential strings increases linearly with a collective radius r 3q as mentioned in [31]. Accordingly the effective two body interactions can be written as In the hypercentral approximation, the potential is only depends on hyper radius(x). More details can be seen in references [31,32]. We consider a reduced hypercentral radial function, is the hypercentral wave function and γ is the grand angular quantum number. Thus, six dimensional hyperradial Schrodinger equation reduces to, −1 2m For the present study we consider the hypercentral potential V (x) as the color coulomb plus linear potential with first order correction [33,34] is given below.
Here, τ is the hyper-Coulomb strength corresponds to the strong running coupling constant α s . β is the string tension of the confinement part of potential. C F and C A are the Casimir charges of the fundamental and adjoint representation with values 2 3 and 3. The spin-dependent part, V SD (x) contains three types of the interaction terms [35]: the spin-spin term V SS (x), the spin-orbit term V γS (x) and tensor term V T (x). The detail of the terms are given in [28]. We begin with calculating the ground state masses of doubly heavy Ξ cc , Ξ bb & Ξ bc baryons 1 . The masses are computed for both parities 1 2 + and 3 2 + mentioned in Table 1. As we know, the ground state of Ξ + cc is found experimentally as Ξ cc (3520) + ; but its J P value is still undefined. Our prediction suggests that it would be J P = 1 2 + ; similar suggestion given by refs. [10,23,25].
The other ground state, with J P = 3 2 + is found as 3.695 GeV by us. The value is nearer to other predictions [10,22,24] and lattice [11,12] calculations. In case of Ξ bb baryon, our ground state outcomes(both parities) are matched(with [9,16]) very well. Our predicted ground states of Ξ bc are very close to refs. [12,20,24]. We have also calculated the ground state spectra of ccu, bbu and bcu baryons. They are close to (8MeV, 5MeV and 6MeV difference, respectively) the results of d quark combinations.
For Ξ bb , our 1P state J P = 1 2 − and J P = 3 2 − are only 14 and 7 MeV higher than ref. [17] whereas Ref. [9] masses for J P = 1 2 − and J P = 3 2 − are 31, 26 MeV lower than our prediction. Our 2P state is few MeV higher than [9,17]. Our 1D-2D states have ≈35 MeV and ≈178 MeV difference with [17]. The P and D states of Ξ 0 bc baryons are given in table 5 and it follow the same description mentioned in [1] and refs. therein. We have compare our results with recent paper [10] for 1P and 1D. Their values are higher than us. The rest spectra (2P-5P and 2D-4D) is performed by us for complete-ness.
F state masses for all three doubly heavy baryons are given in table 6. Apart from us, Ref. [10] has also calculated the 1F state of Ξ cc and Ξ bb for J P = 7 2 − and 9 2 − . For Ξ cc , their masses are 73 and 280 MeV higher while for other system 212 and 328 MeV higher than our predictions. We did not find any other F state calculations for Ξ bc systems.

Regge Trajectories and Magnetic Moments
As discussed in section 3, we have calculated the 1S-5S, 1P-5P, 1D-4D and 1F-2F state masses for all doubly heavy Ξ baryons. The obtained masses were very much useful in constructing Regge trajectories in (n, M 2 ) and (J, M 2 ) planes. Where, n is principal quantum number and J is a total spin. The Regge trajectories are presented in Figs. 1-5 for Ξ cc (ccd), Ξ bb (bbd) and Ξ bc (bcd) baryons. Similar trajectories can also be plotted for the rest of the baryons. Straight lines were obtained by the Where, β and β 0 are slope and intercept, respectively. The fitted slopes and intercepts are given in Table 7. We use natural parity J P = 1 2 − ) parity masses and plotted graphs for Ξ cc and Ξ bb baryon states[See Fig. 4 and 5]. For that we use, Where, α and α 0 are slope and intercept, respectively. The fitted slopes and intercepts are given in Table 8. We observe that the square of the calculated masses fit very well to the linear trajectory and almost parallel    and equidistant in S, P, D and F states. We can determine the possible quantum numbers and prescribe them to particular Regge trajectory with the help of our obtained results.
To obtain magnetic moments of the Ξ family, we need to calculate their effective masses first. As the combination of quarks in baryons changes, its binding interaction affects and m ef f where e i is a charge and σ i is the spin of the respective constituent quark corresponds to the spin flavor wave function of the baryonic state. Using these equations and our obtained ground state masses(mentioned in Table  1), we calculated magnetic moments of all six Ξ baryons. The spin flavor wave function and magnetic moments are given in Table 9. Our obtained ground state magnetic moments are also compared with others shown in Table 9.

Conclusion
The Hypercentral constituent quark model is used to construct the mass spectra of doubly heavy Ξ baryons. Ground states as well as excited state masses are obtained successfully. Mass difference between the light quarks (u and d) is 12 MeV in our model. So, it is obvious that when we move towards the calculations of the excited states the baryon masses would also have a very small mass difference. For the sake of completeness we calculated whole mass spectrum for all six doubly heavy baryon and noticed that it hardly differs less than ≈10 MeV; which can be observed in Tables 1-6. The ground state of Ξ cc is experimental known and while comparing our ground states of Ξ ++ cc ,Ξ + cc baryons we define the state with parity J P = 1 2 + .
We successfully plotted Regge trajectories of present work in both (n, M 2 ) and (J,M 2 ) planes and fortunately assigned the quantum number for each cases of six Ξ baryons. The magnetic moments of the ground states are also calculated using obtained masses. We can observe that our obtained results are close to other predictions(except Ξ ++ cc ,Ξ + * cc , Ξ 0 * bb baryons). 10 This study will definitely help future experiments and other theoretical models to identify the baryonic states from resonances. We would like to extend this model to calculate the mass spectra and other properties of triply heavy baryons.