Doubly heavy baryon spectra guided by lattice QCD
Abstract
This paper provides results for the ground state and excited spectra of three-flavored doubly heavy baryons, and . We take advantage of the spin-independent interaction recently obtained to reconcile the lattice SU(3) QCD static potential and the results of nonperturbative lattice QCD for the triply heavy baryon spectra. We show that the spin-dependent potential might be constrained on the basis of nonperturbative lattice QCD results for the spin splittings of three-flavored doubly heavy baryons. Our results may also represent a challenge for future lattice QCD work, because a smaller lattice error could help in distinguishing between different prescriptions for the spin-dependent part of the interaction. Thus, by comparing with the reported baryon spectra obtained with parameters estimated from lattice QCD, one can challenge the precision of lattice calculations. The present work supports a coherent description of singly, doubly and triply heavy baryons with the same Cornell-like interacting potential. The possible experimental measurement of these states at LHCb is an incentive for this study.
pacs:
14.40.Lb,12.39.Pn,12.40.-yI Introduction
In a recent publication Vij14 () we have pointed out that the static three-quark potential with parameters determined from SU(3) lattice QCD Tak02 () does not reproduce the triply heavy-baryon and spectra measured also in lattice QCD Mei10 (); Mei12 (); Pad13 (). We argued several possible reasons for such disagreement. In a subsequent work Vij15 () we demonstrated that the spectra of baryons containing three identical heavy quarks, or , could be reproduced by means of a Cornell-like interaction, a simple Coulomb plus linear confining potential. As it happens in the heavy meson spectra, a larger value of the Coulomb strength than predicted by SU(3) lattice QCD was concluded. The phenomenological strengths of the Coulomb potential reproducing the heavy meson, , and the triply-heavy baryon spectra, , was found to satisfy , slightly different from the rule as the one-gluon exchange result Ric12 (). The strength concluded for the linear confining interaction was also slightly larger than the results from SU(3) lattice QCD. The spectra obtained in Ref. Vij15 () supported a coherent description of the and heavy baryon spectra with the same Coulomb and confining strengths in a constituent quark model approach.
In Ref. Vij15 () it was also pointed out that the description of the spectra is improved with the additional contribution of a spin-spin term, because relativistic effects are more important than in the case and spin-dependent contributions start playing a significant role. Although the spin-spin interaction comes suppressed by , it helped to correctly allocate the negative parity excitations with respect to the radial excitations of the ground state. However, due to the identity of the three quarks, in a system there cannot exist a good diquark, a couple of quarks with total spin 0 in a relative wave, where the spin-spin term is attractive and its contribution becomes relevant.
Thus, being the spin-independent part of the quark-quark interaction constrained by the triply heavy baryon spectra, it remains to analyze the spin-dependent part of the quark-quark potential. The best testing ground for this purpose are baryons made of three distinguishable quarks, with one of them light. As compared to singly heavy baryons, they are free of the uncertainties of the interaction between light quarks Val08 (), whose spin-dependent part would be dominant, and the light quark kinematics Val14 (). Unlike triply heavy baryons, there exist pairs of quarks with total spin 0 in a relative wave, whose contribution will become crucial to study the spin splittings recently reported by nonperturbative lattice QCD Bro14 (); Pad15 (); Mat16 ().
In this work we aim to analyze doubly heavy baryons with non-identical heavy quarks within a constituent quark model framework by means of a simple Cornell-like potential guided by lattice QCD. The spin-independent part of the quark-quark interaction has been determined from the triply heavy baryon spectra recently calculated by means of nonperturbative lattice QCD techniques and inspired by the static potentials derived within SU(3) lattice QCD. The spin-dependent part will be analyzed in comparison with the recent spin-splitting results derived in nonperturbative lattice QCD Bro14 (); Pad15 (); Mat16 (). We will show how the spin-dependent part might be constrained on the basis of nonperturbative lattice QCD results for three-flavored doubly heavy baryons, and . Analogously, we will emphasize the importance of having lattice QCD results with smaller lattice errors, what could help in distinguishing between different prescriptions for the spin-dependent part of the interaction. For our purposes, we will present an exact calculation solving the Faddeev equations for three non-identical particles.
The road we outline in this work is similar to the path went through to study the heavy meson spectra. Once charmonium and bottomonium spectra were understood within a constituent quark model framework by means of simple Cornell-like potentials Qui79 (); Eic80 (), the question of predicting and trying to understand the structure of open-flavor mesons with a heavy-quark was soon posed God85 (). Compared to the heavy meson case we have the great advantage of the guidance of nonperturbative lattice QCD results and the static potentials derived within SU(3) lattice QCD, which combined with the exact method to solve the three-body problem, makes the difference between our work and other studies of doubly heavy baryons Fle89 (); Kis02 (); Ebe97 (); Ton00 (); Ron95 (); Kor94 (); Ito00 (); Kau00 (); Ebe02 (); Vol01 (); Kar13 (); Yos15 (); Mah16 (); Sha16 (). Our results may also serve for a future analysis of the validity of the so-called superflavor symmetry, relating the spectra and properties of singly heavy mesons and doubly heavy baryons Sav90 (); Bra05 (), broken by the smallness of the heavy quark masses, which makes the size of the heavy diquark not small enough compared to . Additional symmetries including excitations of the heavy diquark Eak12 () could also be tested against our results.
A substantial basis for optimism in the observation of ( stands for a light or quark) and doubly heavy baryons is the large number of doubly heavy mesons measured at the LHCb Aai14 (), indicating that simultaneous production of and pairs which are close to each other in space and in rapidity and can coalesce to form doubly heavy hadrons is not too rare. The cross section of pair doubly heavy diquark production in high energy proton-proton collisions has been already estimated Tru16 (). It has also been recently discussed the production of doubly heavy flavored hadrons in colliders Zhe16 () as well as the doubly heavy baryon photoproduction in the future International Linear Collider (ILC) within the framework of non-relativistic QCD Che14 ().
The paper is organized as follows. In the next section we will briefly review the parametrization of Cornell potential we have determined to get a unified description of the nonperturbative lattice QCD and spectra. We will use Sec. III to discuss the solution of the non-relativistic Faddeev equations for three non-identical particles. In Sec. IV we will present and discuss the results of our work. Finally, in Sec. V, we will summarize the main conclusions of this study.
Ii A potential model for doubly heavy baryons
The spin-independent part of the quark-quark interaction in a baryon should be the analog of the famous Cornell potential for quarkonium. The short-distance behavior is expected to be described by the two-body Coulomb potential as the one-gluon exchange (OGE) result in perturbative QCD. It should be extended for the baryon case, with a factor in front of its strength due to color factors Ric12 (). As for the case, the characteristic signature of the long-range non-Abelian dynamics is believed to be a linear rising of the static interaction. Moreover, the general expectation for the baryonic case is that, at least classically, the strings meet at the so-called Fermat (or Torricelli) point, which has minimum distance from the three sources (shape configuration) Tak04 (); Bor04 (). The confining short-range potential could be also described as the sum of two-body potentials (shape or linear configuration) Tak04 (); Bor04 (); Cor04 (); Ale03 (). We have shown in Ref. Vij15 () the equivalence of both prescriptions for the case of triply heavy baryons (see Table II of that reference) for different values of the heavy-quark mass. Thus, a minimal model to study doubly heavy baryons comes given by,
(1) |
The value of the confinement strength, , is usually fixed to reproduce that obtained from the linear Regge trajectories of the pseudoscalar and mesons, (4292) MeV Bal01 (). In the case of baryons, the linear string tension is considered to be of the order of a factor of the case. The reduction factor in the string tension can be naturally understood as a geometrical factor rather than a color factor, due to the ratio between the minimal distance joining three quarks and the perimeter length of a triangle, suggesting Tak02 (). For the particular case of quarks in an equilateral triangle Bor04 (). When the linear ansatz is adopted for the two-body potential, still the same relation holds for the strength of the Coulomb potential , due to color factors. The ansatz (linear potential) has been widely adopted in the nonrelativistic quark model because of its simplicity Val08 (); Gar07 (); Isg78 (); Oka81 (); Sil96 (); Kle10 (); Cre13 ()
On the other hand, potential models are also less accurate for baryons containing light quarks, because relativistic effects are more important and spin-dependent contributions may start playing a significant role. Although of small importance in heavy quark systems for being suppressed as , the spin-spin interaction derived from the one-gluon exchange helps to improve the description of the nonperturbative lattice QCD results Vij15 (). Thus, an spin-spin term must be considered in the interacting potential for those systems where spin-dependent corrections may play a role, having the quark-quark interaction the final form,
(2) |
The spin-spin interaction arising from the one-gluon exchange potential has the same strength as the Coulomb term Ric12 (). Its radial structure has to be regularized in order to avoid an unbound spectrum Bha80 (). The determination of the strength of the spin-dependent part in heavy baryons with three-identical quarks is not efficient. As mentioned above, the identity of the quarks doe not allow for the existence of a good diquark, a couple of particles with total spin 0 in a relative wave, where the spin-spin term is attractive and its contribution significant. This is why doubly heavy baryons with non-identical heavy quarks are ideal systems to test the spin-dependent part of the quark-quark potential. On one hand, the problem is free of the uncertainties of chiral symmetry breaking effects related to pairs of light quarks and, on the other hand, the distinguishability of the quarks allows for the existence of all pairs in a relative wave with spin 0.
Finally, to make contact with our previous studies of the heavy baryon spectra Val08 (); Val14 (), the linear potential is screened at long distances,
(3) |
such that the same linear strength is guaranteed at short-range, ^{1}^{1}1As shown in Ref. Vij04 () for the light baryon spectra, the long distance screening would just provide with a better understanding of low-spin highly excited baryons and high-spin baryons, with no significant effect on the states studied on this work..
Iii Faddeev equations for three non-identical particles
After partial-wave decomposition, the Faddeev equations are integral equations in two continuous variables as shown in Ref. Vac05 (). They can be transformed into integral equations in a single continuous variable by expanding the two-body matrices in terms of Legendre polynomials as shown in Eqs. (32)(36) of Ref. Ter06 (). One obtains the final set of equations,
(4) |
with
(5) | |||||
and are Legendre polynomials, , , and a scale parameter. are the coefficients of the expansion of the two-body matrices in terms of Legendre polynomials defined by Eq. (34) of Ref. Ter06 (). and are the spin and isospin of the pair while and are the total spin and isospin. is the orbital angular momentum of the pair , is the orbital angular momentum of particle with respect to the pair , and is the total orbital angular momentum.
(6) |
are the usual reduced masses. For a given set of values of the integral equations (4) couple the amplitudes of the different configurations . The spin-isospin recoupling coefficients are given by,
(7) |
with and the spin and isospin of particle , and is the Racah coefficient. The orbital angular momentum recoupling coefficients are given by
(8) |
with if is odd and
(9) |
if is even. The angles , , and can be obtained in terms of the magnitudes of the momenta by using the relations
(10) |
where is a cyclic pair. The magnitude of the momenta and , on the other hand, are obtained in terms of , , and using Eqs. (10) as
(11) |
The integral equations (4) couple the amplitude to the amplitudes and . When the three particles are different, by substituting the equation for into the corresponding equations for and , one obtains at best integral equations that involve two independent amplitudes which means that in that case the numerical calculations are more time consuming. If one represents in Eq. (4) the integration over by a numerical quadrature Abr72 (), then for a given set of the conserved quantum numbers , , and , Eq. (4) can be written in the matrix form
(12) |
where is a vector whose elements correspond to the values of the indices , , , , , and , i.e.,
(13) |
with the abscissas of the integration quadrature. The matrix is given by,
(14) |
where the vertical direction is defined by the values of the indices , , , , , and while the horizontal direction is defined by the values of the indices , , , , , and . and are the abscissas and weights of the integration quadrature.
Substituting Eq. (12) for into the corresponding equations for and one obtains,
(15) |
so that the binding energies of the system are the zeroes of the Fredholm determinant
(16) |
where
(17) |
Iv Results and discussion
^{1}^{1}1Each Faddeev amplitude may exist with and for the pair . is fixed, either or . | ||
^{2}^{2}2Each Faddeev amplitude is only compatible with for the pair . is fixed, either or | ||
^{1}^{1}1Each Faddeev amplitude may exist with and for the pair . is fixed, either or . | ||
^{2}^{2}2Each Faddeev amplitude is only compatible with for the pair . is fixed, either or | ||
^{1}^{1}1Each Faddeev amplitude may exist with and for the pair . is fixed, either or . | ||
^{1}^{1}1Each Faddeev amplitude may exist with and for the pair . is fixed, either or . | ||
^{2}^{2}2Each Faddeev amplitude is only compatible with for the pair . is fixed, either or |
To obtain the predictions of the Cornell-like potential of Eq. (3) for the and baryon spectra to compare with the results measured in nonperturbative lattice QCD Bro14 (); Pad15 (); Mat16 (), we solve the three-body problem for non-identical quarks by means of the Faddeev method described in Sec. III. We solve the nonrelativistic Schrödinger equation
where is the free part of quarks without center-of-mass-motion
and is the mass of quark . The mass of the heavy baryon will be finally given by . The quarks masses are taken as in Ref. Val08 (): 5.034 GeV, 1.659 GeV, GeV, and 0.313 GeV, as well as the long-distance screening parameter 0.7 fm.
We show in Table 1 the Faddeev amplitudes that we consider to solve the three-body problem for each state, indicating the channel giving the lowest energy. As indicated in the table, for those cases with intrinsic spin , each Faddeev amplitude would contribute twice, with the two possible spin couplings of the pair, 0 and 1. The isospin of the pair is fixed.
Before proceeding to analyze the results, we present in Table 2 the convergence of our calculation with respect to the number of the Faddeev amplitudes considered in our calculation, indicated in Table 1. As we can see the results are fully converged with three Faddeev amplitudes (times its degeneracy, an additional factor two for spin and another factor two for negative parity states because they are reached with non-identical pairs of and , see Table 1), i.e., when all Faddeev amplitudes with have been considered. For the sake of completeness, our results have been obtained with all amplitudes quoted in Table 1, which guarantees full convergence.
1 | 1005 | 4 | 1226 | 2 | 953 | 2 | 987 |
---|---|---|---|---|---|---|---|
2 | 996 | 8 | 1203 | 4 | 937 | 4 | 978 |
3 | 995 | 12 | 1199 | 6 | 934 | 6 | 977 |
4 | 995 | 16 | 1197 | 8 | 933 | 8 | 977 |
5 | 995 | 20 | 1196 | 10 | 932 | 10 | 976 |
6 | 995 | 12 | 932 | 12 | 976 |
We present in Fig. 1 the excitation spectra of baryons with the potential of Eq. (3) and the parameters used in Ref. Vij15 () to reproduce the nonperturbative lattice QCD results of triply heavy baryons and : 0.1875, 0.1374 GeV, and an almost constant regularization, , for the spin-spin term following the line of the model AL1 in Ref. Sil96 (). The spin-independent part of the quark-quark interaction fixes in a unique manner the position of the radial and orbital excitations. As explained above, these systems provide with an additional advantage, as compared to triply heavy baryons, that they allow to scrutinize the strength of the spin-spin interaction. In Ref. Vij15 () it was shown how the addition of the spin-dependent part of the quark-quark interaction, maintaining the strength of the Coulomb and the confining potential determined from the spectra, allows for a better agreement in the case. The idea behind this improvement is that potential models probably are also less accurate for than for baryons, because the system is more relativistic and spin-dependent contributions may start playing a significant role.
In a three-quark baryon, any pair of quarks must be in a color state to couple to a color singlet with the color state of the other quark. For identical particles, the color state is antisymmetric. Thus for a symmetric relative wave between the quarks, , if they are identical they can only exist in a symmetric spin state, . In other words, in a triply heavy baryon there are no good diquarks, where the spin-dependent part of the interaction is attractive and significant. A pair of identical quarks could only exist in an antisymmetric spin state, , with a unit of orbital angular momentum, which reconciles the symmetry of the state with the Pauli principle. As the spin-spin interaction is very short-ranged, the relative wave shields its effect. The benefit of this term in the case of triply heavy baryons can be simply understood. In the ground state there are no pairs of quarks with spin zero, while there are in negative parity states. The attraction induced by the spin-spin term allows to relax the value of the quark masses diminishing the repulsive effect of the centrifugal barrier. This was the main effect observed in Ref. Vij15 ().
By using an almost constant regularization for the spin-spin interaction, as suggested by the model AL1 of Ref. Sil96 (), it is obtained a spin-splitting that it is small as compared to the central value recently obtained by nonperturbative lattice QCD Bro14 (): MeV, and MeV^{2}^{2}2 stands for a state with two quarks in a spin 0 state, the ground state; stands for a state with two quarks in a spin 1 state; stands for a state and thus any two quark pair is in a spin 1 state. The same notation is valid for states that are denoted by .. The predictions obtained for these spin-splittings in Fig. 1 are and MeV, respectively. Let us however note that these results are within 2 sigma of the lattice central values which highlights the importance of having smaller lattice errors to help in clearly distinguishing between different prescriptions for the spin-dependent part of the interaction. A similar situation is observed with the mass difference between the first two states with or , that are predicted to be almost degenerate, although in this case we have no lattice results to compare with. This possible underestimation of the spin-spin effects by prescriptions as that in Ref. Sil96 () had already been noted in Ref. Val08 () in the study of singly heavy baryons in a constituent quark model approach, although, as we have mentioned in the introduction, in that case the presence of two light quarks did not allow a clear cut between the spin-independent and spin-dependent effects as in the present case, due to involved dynamics of the two light-quark subsystem.
Latt. Bro14 () | ||||
---|---|---|---|---|
12 | Sil96 () | |||
63 | Val08 () | |||
56 | Val08 () | Sil96 () | Sil96 () | |
22 | Sil96 () | Val08 () | Sil96 () | |
14 | Sil96 () | Sil96 () | Val08 () |
To illustrate the relevance of the spin-dependent terms in three-flavored doubly heavy baryons, we evaluate the mass difference following different prescriptions for the regularization of the term in the spin-spin interaction. The results are shown in Table 3. In the first case we use the almost constant regularization of Ref. Sil96 () (see Eq. (2) of Ref. Sil96 (), fm). In the second case we use the flavor dependent regularization of Ref. Val08 () (see Table 5 of Ref. Val08 (), fm)^{3}^{3}3A similar recipe was used long-ago for a simultaneous study of the meson and baryon spectra in Ref. Ono82 ().. In the last three cases we identify which interaction is responsible for the spin splitting, that as could have been expected is the spin-spin interaction between the lightest flavors , those between and being rather small. At the light of the results of the first and the last files of Table 3 one can easily understand the results of Ref. Vij15 () regarding the spin-dependent part of the interaction, for larger masses of the quarks the spin-splitting is almost the same independently of the prescription used for the regularization. As explained above, this is the reason why triply-heavy baryons are not adequate to analyze the spin-dependent part of the quark-quark interaction.
Thus, using the flavor-dependent regularization of the spin-spin interaction derived in Ref. Val08 () to study singly heavy baryons, we have recalculated the and excited spectra, that are shown in Fig. 2. As mentioned above, the spin-independent part of the quark-quark interaction determined in the triply heavy baryon spectra fixes in a unique manner the position of the radial and orbital excitations, thus these states are a first challenge for future lattice works and/or experimental searches. Regarding the low-energy and states, the results in Fig. 2 are close to the central values of nonperturbative lattice QCD for the spin splitting between them Bro14 (); Pad15 (); Mat16 (). Thus, the present results can be considered as a useful challenge for future lattice QCD work because one can see how a smaller lattice error could clearly help in distinguishing between the different prescriptions for the spin-dependent part of the interaction.
Let us note the parameter-free nature of our calculation, making use of the spin-independent interaction derived in Ref. Vij15 () from the analysis of triply heavy baryons together with the parametrization of the spin-dependent term obtained in Ref. Val08 () from the analysis of singly heavy baryons. Our results unify the heavy-quark dynamics for the description of triply Vij15 (), doubly and singly Val08 () heavy baryons by means of a simple Cornell potential with a flavor-dependent spin-spin regularization. They are therefore a nice testbench for future works of lattice QCD on the ground and excited spectra of three-flavored doubly heavy baryons. They might be useful in future projects of lattice QCD calculations and also as a guideline in future experiments looking for doubly heavy baryons with non-identical heavy quarks.
V Summary
In brief, the spectra of three-flavored doubly heavy baryons have been calculated by means of a Faddeev approach. The spin-independent part of the quark-quark interaction was taken for grant from a recent study of nonperturbative lattice QCD results for triply heavy baryons. As in the case of the heavy meson spectra, a larger value of the Coulomb strength than predicted by SU(3) lattice QCD is needed. The phenomenological strengths of the Coulomb potential reproducing the heavy meson and the triply heavy baryon spectra satisfy , slightly different from the 1/2 rule as the one-gluon exchange result. It has been shown that the spin-dependent part of the interaction could be fixed by studying the spin splitting of three-flavored doubly-heavy baryons. The adequacy of these systems to determine the regularized -type interaction has been justified, obtaining a reasonable agreement with the central values of the spin-splittings derived by nonperturbative lattice QCD with the same flavor dependent regularization already used for singly heavy baryons. Our results make evident the importance of having at our disposal nonperturbative lattice QCD results with smaller error, what would allow to clearly distinguish between different prescriptions for the spin-dependent part of the interaction. Besides, they constitute a nice testbench for future works of lattice QCD on the excited spectra of doubly heavy baryons with non-identical heavy quarks as well as experimental searches. Let us finally note that by comparing with the reported baryon spectra obtained with parameters estimated from lattice QCD, one can challenge the precision of lattice calculations.
The detailed theoretical investigation presented in our recent works about the heavy baryon spectra based on nonperturbative lattice QCD guidance, may help to improve our understanding of the interaction in many-quark systems containing heavy quarks, of interest to deepen our understanding on intriguing recent experimental results as the so-called exotic states or the LHCb pentaquark Che16 (). Similarly the possible advent of new experimental data Aai14 () as well as the improvements in lattice QCD calculations of the heavy baryon spectra Bro14 (), makes the present calculation timely to scrutinize the quark-quark interaction in systems containing heavy flavors.
Acknowledgements.
We thank to N. Mathur and S. Meinel for valuable information about the present status of nonperturbative lattice QCD calculations of excited heavy baryon states. This work has been partially funded by COFAA-IPN (México), by Ministerio de Educación y Ciencia and EU FEDER under Contracts No. FPA2013-47443 and FPA2015-69714-REDT, by Junta de Castilla y León under Contract No. SA041U16, and by USAL-FAPESP grant 2015/50326-5. A.V. is thankful for financial support from the Programa Propio XIII of the University of Salamanca.References
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