Heavy baryons in a pion mean-field approach: A brief review
IINHA-NTG-05/2018
Heavy baryons in a pion mean-field approach: A brief review
Hyun-Chul Kim
1, 2, ∗ Department of Physics, Inha University, Incheon 22212, Republic of Korea School of Physics, Korea Institute for Advanced Study (KIAS), Seoul 02455, Republic of Korea (Dated: November 8, 2018)We review in this paper a series of recent works on properties of singly heavy baryons, basedon a pion mean-field approach. In the limit of an infinitely heavy-quark mass, the heavy quarkinside a heavy baryon can be regarded as a static color source. In this limit, a heavy baryon can beviewed as N c − N c valence quarks. We show that this mean-field approach can successfullydescribe the masses and the magnetic moments of the lowest-lying singly heavy baryons, using allthe parameters fixed in the light-baryon sector except for the hyperfine spin-spin interactions. Wealso review a recent work on identifying the newly found excited Ω c baryons reported by the LHCbCollaboration. We discuss possible scenarios to identify them. Finally, we give a future perspectiveon this pion mean-field approach. Keywords: heavy baryons, pion mean fields, chiral quark-soliton model, flavor SU(3) symmetry breaking ∗ E-mail: [email protected] a r X i v : . [ h e p - ph ] A p r I. INTRODUCTION
An ordinary heavy baryon constitutes a pair of light quarks and a heavy quark. Since the charm and bottomquarks are very heavy in comparison with the light quarks, it is plausible to take the limit of the infinitely heavymass of the heavy quark, i.e. m Q → ∞ . In this limit, the physics of heavy baryons become simple. The spin ofthe heavy quark is conserved, because of its infinitely heavy mass. It results in the conservation of the total spin oflight quarks: J L ≡ J − J Q , where J L , J Q , and J denote the spin of the light-quark pair, that of the heavy quark,and the total spin of the heavy baryon. This is called the heavy-quark spin symmetry that allows J L to be a goodquantum number. Moreover, the physics is kept intact under the placement of heavy quark flavors. This is calledthe heavy-quark flavor symmetry [1–4]. Then a heavy quark becomes static, so that it can be considered as a staticcolor source. Its importance is only found in making the heavy baryon a color singlet, and in giving higher-ordercontributions arising from 1 /m Q corrections. Consequently, the dynamics inside a heavy baryon is mainly governedby the light quarks.The flavor structure of the heavy baryon is also determined by them. Since there are two light quarks inside theheavy baryon, we have two different flavor SU f (3) irreducible representations, i.e. ⊗ = ⊕ . In the languageof a quark model, the spatial part of the heavy-baryon ground state is symmetric due to the zero orbital angularmomentum, and the color part is totally antisymmetric. Since the flavor anti-triplet ( ) is antisymmrtric, the spinstate corresponding to should be antisymmetric. Thus, the baryons belonging to the anti-triplet should be J L = 0.Similarly, the flavor-symmetric sextet ( ) should be symmetric in spin space, i.e. J L = 1. This leads to the fact thatthe baryon antitriplet has spin J = 1 /
2, while the baryon sextet carries spin J = 1 / J = 3 /
2, with the spin ofthe light-quark pair being coupled with the heavy quark spin J Q = 1 /
2. So, we can classify 15 different lowest-lyingheavy baryons as shown in Fig. 1 in the case of charmed baryons.
FIG. 1. The anti-triplet ( ) and sextet ( ) representations of the lowest-lying heavy baryons. The left panel draws the weightdiagram for the anti-triplet with the total spin . The centered panel corresponds to that for the sextet with the total spin1 / / Recently, there has been a series of new experimental data on the spectra of heavy baryons [5–13], which renewedinterest in the physics of the heavy baryons. The lowest-lying singly heavy baryons are now almost classified except forΩ ∗ b . In the meanwhile, the LHCb Collaboration has announced the first finding of two heavy pentaquarks, P c (4380)and P c (4450) [14–17]. Very recently, the five excited Ω c baryons were reported [18], among which the four of themwas confirmed by the Belle experiment [19]. Interestingly the two of the excited Ω c s, i.e. Ω c (3050) and Ω c (3119),have very narrow widths: Γ Ω c (3050) = (0 . ± . ± .
1) MeV and Γ Ω c (3119) = (1 . ± . ± .
4) MeV.While there is a great deal of theoretical approaches for the description of heavy baryons, we will focus on a pionmean-field approach in the present short review. This mean-field approach was first proposed by E. Witten in thisseminal papers [20, 21], where he asserted that in the limit of the large number of colors ( N c ) the nucleon can beregarded as a bound state of N c valence quarks in a pion mean field with a hedgehog symmetry [22, 23]. Since a baryonmass is proportional to N c whereas the quantum fluctuation around the saddle point of the pion field is suppressedby 1 /N c , the mean-field approach is a rather plausible method for explaining properties of baryons. The presenceof N c valence quarks in this large N c limit, which consist of the lowest-lying baryons, produce the pion mean fieldsby which they are influenced self-consistently . This picture is very similar to a Hartree approximation in many-bodytheories. Witten also showed how to construct the mean-field theory for the baryon schematically in two-dimensionalquantum chromodynamics (QCD). Though his idea was criticized sometimes ago by S. Coleman [24] because of itstechnical difficulties, it is worthwhile to pursue it to see how far we can describe the structure of the baryon in thepion mean-field approach.The chiral quark-soliton model ( χ QSM) [25–27] has been constructed based on Witten’s argument. The χ QSMstarts from the effective chiral action (E χ A) that was derived from the instanton vacuum [28, 29]. The E χ A respectschiral symmetry and its spontaneous breakdown, in which the essential physics of the lowest-lying hadrons consists.One can derive the classical energy of the nucleon by computing the nucleon correlation function in Euclidean space,taking the Euclidean time to go to infinity. Minimizing the classical energy self-consistently in the large N c limitwith the 1 /N c meson quantum fluctuations suppressed, we obtain the classical mass and the self-consistent profilefunction of the chiral soliton. While we ignore the 1 /N c quantum fluctuations around the saddle point of the solitonfield, we need to take into account the zero modes that do not change the soliton energy. Since the soliton withhedgehog symmetry is not invariant under translational, rotational and isotopic transformations, we impose thesesymmetry properties on the soliton and obtain a completely new solution with the same classical energy. Because ofthe hedgehog symmetry, an SU(2) soliton needs to be embedded into the isospin subgroup of the flavor SU(3) f [21],which was already utilized by various chiral soliton models [30–32]. This collective quantization of the chiral solitonleads to the collective Hamiltonian with effects of flavor SU(3) f symmetry breaking. The χ QSM has one salient feature:the right hypercharge is constrained to be Y (cid:48) = N c / N c valence quarks. This right hyperchargeselects allowed representations of light baryons such as the baryon octet ( ), the decuplet ( ), etc. The χ QSM wassuccessfully applied to the properties of the lowest-lying light baryons such as the mass splittings [33, 34], the formfactors [35–37], the magnetic moments [38–41], hyperon semileptonic decays [42, 43], parton distributions [44, 45],transversities of the nucleon [46–48], generalized parton distributions [49], and so on.
FIG. 2. Schematic picture of a heavy baryon. The N c − K P = 0 + with the heavy quark stripped off. K P denotes the grand spin which we will explain later and P is the corresponding parity ofthe level. The presence of the valence quarks will interact with the sea quarks filled in the Dirac sea each other. This interactionwill bring about the pion mean field. Very recently, Ref. [50] extended a mean-field approach to describe the masses of singly heavy baryons, beingmotivated by Ref. [51]. A singly heavy baryon constitutes a heavy baryon and N c − m Q → ∞ , the heavy quark can be considered as a static color source. Thus, the dynamics inside aheavy baryon is governed by the N c − N c − Y (cid:48) = ( N c − / ), the baryon sextet ( ), the baryon anti-decapentaplet ( ). The model reproduced successfully the mass splittingof the baryon anti-triplet and sextet in both the charm and bottom sectors. In addition, the mass of the Ω ∗ b baryon,which has not yet found, was predicted. The model was further extended by including the second-order perturbativecorrections of flavor SU f (3) symmetry breaking [52]. The magnetic moments baryons [55] and electromagnetic formfactors [56] of the singly heavy baryons were also studied within the same framework. The χ QSM was also used tointerpret the five Ω c baryons newly found by the LHCb Collaboration [57, 58]. Within the present framework, two ofthe Ω c s with the smaller widths are classified as the members of the baryon , whereas all other Ω c ’s belong to theexcited baryon sextet. The widths were quantitatively well reproduced without any free parameter. In the presentwork, we will review briefly these recent investigations on the singly heavy baryons.We sketch the present work as follows: In Section II, we review the general formalism of the χ QSM for singly heavybaryons. In Section III, we examine the mass splittings of the heavy baryons, emphasizing the discussion of the effectsof SU(3) f breaking. In Section IV, we discuss the recent results of the magnetic moments and electromagnetic formfactors of the heavy baryons. In Section V, we briefly introduce a theoretical interpretation of the excited Ω c baryonsfound by the LHCb, based on the present mean-field approach. The final Section is devoted to the conclusions andoutlook. II. THE CHIRAL QUARK-SOLITON MODEL FOR SINGLY HEAVY BARYONS
In the present approach, a heavy baryon is considered as a bound state of the N c − N c − B (0 , T ) = (cid:104) J B (0 , T / J † B (0 , − T / (cid:105) = 1 Z (cid:90) D U D ψ † D ψJ B (0 , T / J † B (0 , − T / e (cid:82) d x ψ † ( i / ∂ + iMU γ + i ˆ m ) ψ , (1)where J B denotes the light-quark current with the N c − BJ B ( x , t ) = 1( N c − ε β ··· β Nc − Γ { f } J (cid:48) J (cid:48) ,T T Ψ β f ( x , t ) · · · Ψ β Nc − f Nc − ( x , t ) . (2) β i stand for color indices and Γ { f ··· f Nc − } J (cid:48) J (cid:48) ,T T represents a matrix with both flavor and spin indices. J (cid:48) and T are thespin and isospin of the heavy baryon, respectively. J (cid:48) and T are their third components, respectively. The notation (cid:104)· · · (cid:105) in Eq. (1) is the vacuum expectation value, M the dynamical quark mass, and the chiral field U γ is defined as U γ = U γ U † − γ U = exp( iπ a λ a ) . (4)Here, π a represents the pseudo-Goldstone boson field and ˆ m denotes the flavor matrix of the current quarks, writtenas ˆ m = diag( m u , m d , m s ). We assume isospin symmetry, i.e. m u = m d . Since the strange current quark mass issmall enough, we will treat it perturbatively.Integrating over the quark fields, we derive the correlation function asΠ B (0 , T ) = 1 Z Γ { f } J (cid:48) J (cid:48) ,T T Γ { g }∗ J (cid:48) J (cid:48) ,T T (cid:90) D U N c − (cid:89) i =1 (cid:28) , T / (cid:12)(cid:12)(cid:12)(cid:12) D ( U ) (cid:12)(cid:12)(cid:12)(cid:12) , − T / (cid:29) e − S eff ( U ) , (5)where the single-particle Dirac operator D ( U ) is defined as D ( U ) = iγ ∂ + iγ k ∂ k + iM U γ + i ˆ m (6)and S eff is the effective chiral action written as S eff = − N c Tr log D ( U ) . (7)Equation (5) can be schematically depicted as Fig. 2. It consists of two different terms: The first and second ones arerespectively called the valence-quark contribution and sea-quark contribution within the χ QSM. When the Euclideantime T is taken from −∞ to ∞ , the correlation function picks up the ground-state energy [25, 26]lim T →∞ Π B ( T ) ∼ exp[ − { ( N c − E val + E sea } T ] , (8)where E val and E sea the valence and sea quark energies. Minimizing self-consistently the energies around the saddlepoint of the chiral field UδδU [( N c − E val + E sea ] (cid:12)(cid:12)(cid:12)(cid:12) U c = 0 , (9) FIG. 3. Correlation function for a heavy baryon we get the classical soliton mass M sol = ( N c − E val ( U c ) + E sea ( U c ) . (10)Note that a singly heavy baryon has a heavy quark, so its classical is expressed as the sum of the classical andheavy-quark masses M cl = M sol + m Q . (11)We want to mention that m Q is the effective heavy quark mass that is different from that of QCD and will be absorbedin the center mass of each representation.The rotational excitations of the soliton with N c − U c ( r ) will be embedded into SU(3) [21] U ( r ) = (cid:18) U c ( r ) 00 1 (cid:19) . (12)As mentioned in Introduction, we consider explicitly the rotational zero modes. Assuming that the soliton U ( r ) inEq.(12) rotates slowly, we apply the rotation matrix A ( t ) in SU f (3) space U ( r , t ) = A ( t ) U ( r ) A † ( t ) . (13)Then, we can derive the collective Hamiltonian for heavy baryons H = H sym + H (1)sb + H (2)sb , (14)where H sym represents the flavor SU(3) symmetric part, H (1)sb and H (2)sb the SU(3) symmetry-breaking parts respectivelyto the first and second orders. H sym is expressed as H sym = M cl + 12 I (cid:88) i =1 ˆ J i + 12 I (cid:88) a =4 ˆ J a , (15)where I and I are the moments of inertia of the soliton and the operators ˆ J i denote the SU(3) generators. We getthe eigenvalue of the quadratic Casimir operator (cid:80) i =1 J i in the ( p, q ) representation, given as C ( p, q ) = 13 (cid:2) p + q + pq + 3( p + q ) (cid:3) , (16)which leads to the eigenvalues of H sym E sym ( p, q ) = M cl + 12 I J L ( J L + 1) + 12 I [ C ( p, q ) − J L ( J L + 1)] − I Y (cid:48) . (17)The right hypercharge Y (cid:48) is constrained by the N c − Y (cid:48) = ( N c − / ψ ( R ) B ( JJ , J L ; A ) = (cid:88) m = ± / C JJ J Q m J L J L χ m (cid:112) dim( p, q )( − − Y (cid:48) + J L D ( R ) ∗ ( Y,T,T )( Y (cid:48) ,J L , − J L ) ( A ) , (18)wheredim( p, q ) = ( p + 1)( q + 1) (cid:18) p + q (cid:19) . (19) J and J in Eq. (18) are the spin angular momentum and its third component of the heavy baryon, respectively. J L and J Q represent the soliton spin and heavy-quark spin, respectively. J L and m are the corresponding thirdcomponents, respectively. Since the spin operator for the heavy baryon is given as J = J Q + J L , (20)the relevant Clebsch-Gordan coefficients appear in Eq.(18). The SU(3) Wigner D function in Eq.(18) means just thewave-function for the quantized soliton with the N c − χ m is the Pauli spinor for the heavyquark. R designates a SU(3) irreducible representation corresponding to ( p, q ). Since the soliton is coupled to theheavy quark, we finally obtain the three lowest-lying representations illustrated in Fig. 1. In the limit of m Q → ∞ ,the two sextet representations are degenerate. One needs to introduce a hyperfine spin-spin interaction to lift thisdegeneracy. As will be discussed soon, this hyperfine interaction will be determined by using the experimental dataon the masses of heavy baryons.In the present zero-mode quantization scheme, we find the following the two important selection rule. The allowedSU(3) representations must contain states with Y (cid:48) = ( N c − / T of the states with Y (cid:48) = ( N c − K = T + J L = , where K is called the grand spin. Thelowest-lying heavy baryons have the grand spin K = 0, that is, we must have always J L = T with Y (cid:48) = ( N c − / FIG. 4. The baryon anti-triplet has the J L = T = 0 state with Y (cid:48) = 2 / J L = T = 1state with Y (cid:48) = 2 / An observable of the heavy baryon can be expressed in general as a three-point correlation function (cid:104)
B, p (cid:48) | J µ (0) | B, p (cid:105) = 1 Z lim T →∞ exp (cid:18) ip T − ip (cid:48) T (cid:19) (cid:90) d xd y exp( − i p (cid:48) · y + i p · x ) × (cid:90) D U (cid:90) D ψ (cid:90) D ψ † J B ( y , T / ψ † (0) γ Γ O ψ (0) J † B ( x , − T /
2) exp (cid:20) − (cid:90) d zψ † iD ( U ) ψ (cid:21) , (21)where Γ and O represent respectively generic Dirac spin and flavor matrices. Computing Eq. (21), one can studyheavy baryonic observables such as form factors, magnetic moments, axial-vector constants, etc. For the detailedformalism, we refer to Refs. [26, 35]. III. MASS SPLITTINGS OF THE SINGLY HEAVY BARYONS
We first discuss the mass splittings of the singly heavy baryons. In order to obtain the mass splittings, one shouldinclude the symmetry-breaking part of the collective Hamiltonian [26, 33] H (1)sb = αD (8)88 + β ˆ Y + γ √ (cid:88) i =1 D (8)8 i ˆ J i , (22)where α = (cid:18) − Σ πN m + K I Y (cid:48) (cid:19) m s , β = − K I m s , γ = 2 (cid:18) K I − K I (cid:19) m s . (23)The parameters α , β , and γ are the essential ones in determining the masses of the lowest-lying singly heavy baryons,which are expressed in terms of the moments of inertia I , and K , . However, we do not need to fit them, sincethey are related to α , β , and γ in the light-baryon sector. The valence parts are only different from those in the lightbaryon sector by the color factor N c −
1. So, we need to replace N c by N c − πN is just the πN sigma termwith different N c factor: Σ πN = ( N c − N − c Σ πN , where Σ πN = ( m u + m d ) (cid:104) N | ¯ uu + ¯ dd | N (cid:105) = ( m u + m d ) σ . On theother hand, the sea parts should be kept intact as in the light baryon sector.The dynamical parameters α , β and γ have been fixed by using the experimental data on the baryon octet massesand a part of the baryon decuplet and anti-decuplet masses with isospin symmetry breaking effects [53]. The valuesof α , β , and γ have been obtained by the χ fit [34] α = − . ± .
82 MeV , β = − . ± .
20 MeV , γ = − . ± .
33 MeV , (24)While β and γ are not required to be changed in the heavy-baryon sector, α should be modified by α = ρα, (25)where ρ = ( N c − /N c . However, there is a caveat when one uses the values of Eq. (24). As mentioned above, onlythe valence parts should be modified, while the scaling in Eq. (25) changes the sea part too. To compensate this wechoose ρ ≈ .
9. If one computes the parameters α , β , and γ in a self-consistent way, we do not have this problem [52].Considering the first-order perturbative corrections of m s , one can express the masses of the singly heavy baryonsin representation R as M QB, R = M Q R + M (1) B, R (26)with M Q R = m Q + E sym ( p, q ) . (27)Here, M Q R is the center mass of a heavy baryon in representation R . E sym ( p, q ) is the eigenvalue energy of thesymmetric part of the collective Hamiltonian defined in Eq. (17). Note that the lower index B designates a certainbaryon in a specific representation R . The upper index Q denotes either the charm sector ( Q = c ) or the bottomsector ( Q = b ). Then the center masses for the anti-triplet and sextet representations are obtained as M Q = M cl + 12 I , M Q = M Q + 1 I , (28)where M cl was defined in Eq. (11). The second term in Eq. (26), which arises from the linear-order m s corrections,is proportional to the hypercharge of the soliton with the light-quark pair M (1) B, R = (cid:104) B, R| H (1)sb | B, R(cid:105) = Y δ R , (29)where δ = 38 α + β, δ = 320 α + β − γ. (30)Finally, we arrive at the expressions for the masses of the lowest-lying baryon anti-triplet and sextet as follows M QB, = M Q + Y δ , M QB, = M Q + Y δ , (31)with the linear-order m s corrections taken into account.Since the baryon sextet with spin 1/2 and 3/2 are degenerate, we need to remove the degeneracy by introducingthe hyperfine spin-spin interaction Hamiltonian [54]. Typically, the hyperfine Hamiltonian is written as H LQ = 23 κm Q M sol J L · J Q = 23 κ m Q J L · J Q , (32)where κ stands for the flavor-independent hyperfine coupling. M sol has been incorporated into an unknown coefficient κ that will be fixed by using the experimental data. . The Hamiltonian H LQ does not affect the states with J L = 0.On the other hand, the baryon sextet acquire additional contribution from H LQ which bring about the splittingbetween different spin states M QB, / = M QB, − κ m Q , M QB, / = M QB, + 13 κ m Q , (33)which leads to the splitting M QB, / − M QB, / = κ m Q . (34)The numerical values of κ /m Q were determined by using the center values of the masses of the baryon sextet [50] κ m c = (68 . ± .
1) MeV , κ m b = (20 . ± .
0) MeV . (35)Note that κ is flavor-independent. So, knowing the ratio m c /m b , one can extract the value of κ from Eq. (35).We now present the numerical results of the masses of the heavy baryons [50]. Using the values of α , β , and γ , wecan immediately determine the values of δ and δ defined in Eq. (30) δ = ( − . ± .
5) MeV , δ = ( − . ± .
3) MeV . (36)Including the results of κ /m c and κ /m b , we can obtain the numerical results of the heavy baryon masses. In Table Iand Table II the numerical results of the charmed and bottom baryon masses are presented, respectively. They arein good agreement with the experimental data taken from Ref. [60]. The mass of Ω ∗ b is still experimentally unknown.Thus, the prediction of its mass is given as M Ω ∗ b = (6095 . ± .
4) MeV . (37)The uncertainties in Tables I and II are due to those in α , β , γ , and κ /m Q . R QJ B c Mass Experiment c / Λ c . ± . . ± . c . ± . . ± . c / Σ c . ± . . ± . (cid:48) c . ± . . ± . c . ± . . ± . c / Σ ∗ c . ± . . ± . ∗ c . ± . . ± . ∗ c . ± . . ± . R QJ B b Mass Experiment b / Λ b . ± . . ± . b . ± . . ± . b / Σ b . ± . . ± . (cid:48) b . ± . . ± . b . ± . . ± . b / Σ ∗ b . ± . . ± . ∗ b . ± . . ± . ∗ b . ± . − TABLE II. The results of the masses of the bottom baryons in comparison with the experimental data [60].
IV. MAGNETIC MOMENTS OF HEAVY BARYONS
In this Section, we briefly summarize a recent work on the magnetic moments of the heavy baryons [55]. Startingfrom Eq. (21), one can derive the general expressions of the collective operator for the magnetic momentsˆ µ = ˆ µ (0) + ˆ µ (1) , (38)where ˆ µ (0) and ˆ µ (1) denote the leading and rotational 1 /N c contributions, and the linear m s corrections respectivelyˆ µ (0) = w D (8) Q + w d pq D (8) Q p · ˆ J q + w √ D (8) Q ˆ J , ˆ µ (1) = w √ d pq D (8) Q p D (8)8 q + w (cid:16) D (8) Q D (8)88 + D (8) Q D (8)83 (cid:17) + w (cid:16) D (8) Q D (8)88 − D (8) Q D (8)83 (cid:17) . (39) d pq is the SU(3) symmetric tensor of which the indices run over p = 4 , · · · ,
7. ˆ J and ˆ J p denote the third and the p th components of the spin operator acting on the soliton with the light-quark pair. D (8) Q arises from the rotation ofthe electromagnetic current D (8) Q = 12 (cid:18) D (8)33 + 1 √ D (8)83 (cid:19) . (40)The coefficients w i in Eq. (39) are independent of baryons involved, which encode the interaction of light quarks withthe electromagnetic current. Each term has a physical meaning: w represents the leading-order contribution, a partof the rotational 1 /N c corrections, and linear m s corrections, whereas w and w describe the rest of the rotational1 /N c corrections. w includes the m s -dependent term, which is not explicitly involved in the breaking of flavor SU(3)symmetry. So, we need to treat w as if it had contained the SU(3) symmetric part. On the other hand, w , w , and w are the SU(3) symmetry breaking terms. There are yet another m s corrections, which arise from the collectivewave functions. Though w i can be determined within a specific chiral solitonic model such as the χ QSM [35, 38], wewill use the values of w i , which have been already fixed from the experimental data on the magnetic moments of thebaryon octet.The baryon wave function given in Eq. (18) is not enough to compute the magnetic moments, because the collectivewave functions should be revised when the perturbation coming from the strange current quark mass is considered.In this case, the baryon is no more in a pure state but is mixed with higher representations. In Ref. [56], the collectivebaryon wave functions for the heavy baryons have been already derived. Those for the baryon anti-triplet ( J L = 0)and the sextet ( J L = 1) are expressed respectively as [56] | B (cid:105) = | , B (cid:105) + p B | , B (cid:105) , | B (cid:105) = | , B (cid:105) + q B | , B (cid:105) + q B | , B (cid:105) , (41)with the mixing coefficients p B = p (cid:20) −√ / − √ / (cid:21) , q B = q √ / √ / , q B = q −√ / −√ / −√ / , (42)respectively, in the basis [Λ Q , Ξ Q ] for the anti-triplet and (cid:2) Σ Q (cid:0) Σ ∗ Q (cid:1) , Ξ (cid:48) Q (cid:0) Ξ ∗ Q (cid:1) , Ω Q (cid:0) Ω ∗ Q (cid:1)(cid:3) for the sextets. Theparameters p , q , and q are written by p = 34 √ αI , q = − √ (cid:18) α + 23 γ (cid:19) I , q = 45 √ (cid:18) α − γ (cid:19) I . (43)Combining Eq. (41) with the heavy-quark spinor as in Eq. (18), one can construct the collective wave functions forthe heavy baryon states [55].Computing the baryon matrix elements of ˆ µ in Eq. (38), we get the magnetic moments of the heavy baryons µ B = µ (0) B + µ (op) B + µ (wf) B (44)where µ (0) B is the part of the magnetic moment in the chiral limit and µ (op) B comes from ˆ µ (1) in Eq. (38), which include w , w , and w . µ (wf) B is derived from the interference between the O ( m s ) and O (1) parts of the collective wavefunctions in Eq. (41).0Since the soliton with the light-quark pair for the baryon anti-triplet has spin J L = 0, , the magnetic moments ofthe baryon anti-triplet vanish. In this case 1 /m Q contributions are the leading ones. However, we will not includethem, since we need to go beyond the mean-field approximation to consider the 1 /m Q contributions within the presentframework.Since w contains both the leading-order contributions and the 1 /N c rotational corrections, we have to decomposethem. Following the argument of Ref. [58], we can separately consider each contribution. The coefficients w , w , and w are expressed in terms of the model dynamical parameters w = M − M ( − )1 I (+)1 , w = − M ( − )2 I (+)2 , w = − M (+)1 I (+)1 , (45)where the explicit forms of M , M ( ± )1 , M ( − )2 are given in Refs. [35, 61]. I (+)1 and I (+)2 are the moments of inertia withthe notation of Ref. [61] taken. In the limit of the small soliton size, the parameters in Eq. (45) can be simplified as M → − N c K, M ( − )1 I (+)1 → K, M (+)1 I (+)1 → − K, M ( − )2 I (+)2 → − K. (46)These results yield the expressions of the magnetic moments in the nonrelativistic (NR) quark model. For example,the ratio of the proton and magnetic moments can be correctly obtained as µ p /µ n = − /
2. In the NR limit, we alsoderive the relation M ( − )1 = − M (+)1 . Furthermore, we have to assume that this relation can be also applied to thecase of the realistic soliton size. Then, we can write the leading-order contribution M in terms of w and w M = w + w . (47)Since a heavy baryon constitutes N c − M is modified by introducing ( N c − /N c . Asmentioned previously, only the valence part of M should be changed by this scaling factor. Since, however, we havedetermined the values of w i using the experimental data, we can not fix separately the valence and sea parts. Thus,we introduce an additional scaling factor σ to express a new coefficient ˜ w ˜ w = (cid:20) N c − N c ( w + w ) − w (cid:21) σ. (48) σ compensates also possible deviations from the NR relation M ( − )1 = − M (+)1 assumed to be valid in the realisticsoliton case. The value of σ is taken to be σ ∼ . w i ˜ w = − . ± . ,w = 4 . ± . ,w = 8 . ± . ,w = − . ± . ,w = − . ± . ,w = − . ± . . (49)Before we carry on the calculation of the magnetic moments, we examine the general relations between them. First,we find the generalized Coleman and Glashow relations [63], which arise from the isospin invariance µ (Σ ++ c ) − µ (Σ + c ) = µ (Σ + c ) − µ (Σ c ) ,µ (Σ c ) − µ (Ξ (cid:48) c ) = µ (Ξ (cid:48) c ) − µ (Ω c ) , µ (Σ + c ) − µ (Ξ (cid:48) c )] = µ (Σ ++ c ) − µ (Ω c ) . (50)Similar relations were also found in Ref. [62]. However, there is one very important difference. While the Coleman-Glashow relations are known to be valid in the chiral limit, the relations in Eq. (50) are justified even when the effectsof SU(3) flavor symmetry breaking are considered. We also find the relation according to the U -spin symmetry µ (Σ c ) = µ (Ξ (cid:48) c ) = µ (Ω c ) = − µ (Σ + c ) = − µ (Ξ (cid:48) + c ) = − µ (Ω c ) , (51)1 TABLE III. Numerical results of the magnetic moments for the charmed baryon sextet with J = 1 / µ N . µ (cid:104) / , B c (cid:105) µ (0) µ (total) Σ ++ c . ± .
09 2 . ± . + c . ± .
02 0 . ± . c -1 . ± .
05 -1 . ± . (cid:48) + c . ± .
02 0 . ± . (cid:48) c -1 . ± .
05 -1 . ± . c -1 . ± .
05 -0 . ± . J = 3 / µ N . µ (cid:104) / , B c (cid:105) µ (0) µ (total) Σ ∗ ++ c . ± .
14 3 . ± . ∗ + c . ± .
04 0 . ± . ∗ c − . ± . − . ± . ∗ + c . ± .
04 0 . ± . ∗ c − . ± . − . ± . ∗ c -1 . ± .
07 -1 . ± . which are only valid in the SU(3) symmetric case. We derive also the sum rule given as (cid:88) B c ∈ sextet µ ( B c ) = 0 (52)in the SU(3) symmetric case.In Tables III and IV, we list the numerical results of the charmed baryon sextet with spin 1/2 and 3/2, respectively.We obtain exactly the same results for the bottom baryons because of the heavy-quark symmetry in the m Q → ∞ limit. In Ref. [55], a detailed discussion can be found, the present results being compared with those from many othermodels. V. EXCITED Ω c BARYONS
The present mean-field approach was applied to the classification of the excited Ω c ’s that were recently reportedby the LHCb Collaboration [18]. The masses and decay widths of the Ω c ’s, which were reported by the LHCbCollaboration, are listed in Table V. The Belle Collaboration has confirmed the four of them [19] (see Table VI). TheBelle data unambiguously confirmed the existence of the Ω c (3066) and Ω c (3090), and Ω c (3000) and Ω c (3050) are alsoconfirmed with reasonable significance. On the other hand the narrow resonance Ω c (3119) was not seen in the Belleexperiment but the nonobservation of Ω c (3119) is not in disagreement because it is due to the small yield.When one examines the excited heavy baryons in the present work, we need to consider states with the grand spin K = 1. Since we have the quantization rule K = J L + T , the possible values of the spin are determined by J L = | K − T | , · · · , K + T. (53)Thus, In the case of T = 0 which corresponds to the anti-triplet with Y (cid:48) = 2 /
3, we must have J L = 1 because of K = 1. Combining it with the heavy-quark spin 1/2, we have two excited baryon anti-triplet. Similarly, T = 1corresponds to the sextet. In this case J L can have the values of 0, 1, and 2. Being coupled with the heavy-quark spin1 /
2, we get five excited baryon sextets: (1 / / , / / , / J L = 0, and J L = 1, and J L = 2. In each sextet representation, we have a isosinglet Ω c . Thus, is is natural to think that the newly found fiveΩ c ’s are those in the excited baryon sextets. Note that the representations for each value of J are degenerate in thelimit of m Q → ∞ . So, we need to introduce an additional hyperfine spin-spin interaction as done for the ground-state2 TABLE V. Experimental data on the five Ω c baryons reported by the LHCb Collaboration [18].Resonance Mass (MeV) Decay width (MeV)Ω c (3000) . ± . ± . +0 . − . . ± . ± . c (3050) . ± . ± . +0 . − . . ± . ± . c (3066) . ± . ± . +0 . − . . ± . ± . c (3090) . ± . ± . +0 . − . . ± . ± . c (3119) . ± . ± . +0 . − . . ± . ± . c (3188) 3188 ± ±
13 60 ± ± c baryons reported by the Belle Collaboration [19].Resonance Mass (MeV)Ω c (3000) . ± . ± . c (3050) . ± . ± . c (3066) . ± . ± . c (3090) . ± . ± . c (3119) –Ω c (3188) 3199 ± ± baryon sextet H LQ = 23 κ (cid:48) m Q J L · J Q , (54)which is very similar to Eq. (32). κ (cid:48) can be fixed by using the experimental data on the masses of the excited baryonanti-triplet.Following Refs. [64, 65], we revise the eigenvalues of the symmetric Hamiltonian for the excited baryons ( K (cid:54) = 0)as follows M ( K ) (cid:48)R = M ( K ) (cid:48) cl + 12 I (cid:20) C ( R ) − T ( T + 1) − Y (cid:48) (cid:21) + 12 I [(1 − a K ) T ( T + 1) + a K J L ( J L + 1) − a K (1 − a K ) K ( K + 1)] , (55)where C ( R ) is the eigenvalue of the SU(3) Casimir operator, which was already defined in Eq. (16). The parameter a K is related to one-quark excitation. The collective wave functions for the soliton are derived asΦ R B,J L ,J L , ( T,K ) = (cid:114) J L + 12 K + 1 (cid:88) T J (cid:48) L K (cid:48) C KK T T J L J (cid:48) L ( − ( T + T ) Ψ ( R ; B )( R ∗ ; − Y (cid:48) T T ) D ( J L ) ∗ J (cid:48) L J L ( S ) χ K (cid:48) , (56)where index ( R ; Y T T ) denotes the SU(3) quantum numbers of a corresponding baryon in representation R , and( R ∗ ; − Y (cid:48) T T ) is attached to a fixed value of Y (cid:48) and is formally given in a conjugate representation to R . Thefunction D ( J L ) represents the SU(2) Wigner D function and χ K is the spinor corresponding to K and K . The wavefunction for the excited baryons can be constructed by coupling Φ R B,J L ,J L , ( T,K ) with the heavy-quark spinor.The SU(3) symmetry-breaking Hamiltonian in Eq. (22) also needs to be extended to describe the mass splittingsof the excited heavy baryons H ( K )sb = αD (8)88 + β ˆ Y + γ √ (cid:88) i =1 D (8)8 i ˆ T i + δ √ (cid:88) i =1 D (8)8 i ˆ K i . (57)The additional parameter δ can be determined by using the mass spectrum of excited baryons.As shown in Fig. 5, the transition from a K P = 1 − Dirac-sea level to an unoccupied K P = 0 + state may correspondto the first excited heavy baryons [51]. Note that such a transition is only allowed in the heavy-baryon sector, not3 FIG. 5. Schematic picture of the first excited heavy baryons. A possible excitation of a quark from the Dirac sea to the valencelevel might have K P = 1 − . in the light-baryon sector. As discussed already, there are two baryon anti-triplets and five baryon sextets. FromEq. (55), we can derive the following expressions M (cid:48) = M (cid:48) cl + 12 I + 1 I ( a ) ,M (cid:48) = M (cid:48) + 1 − a I + a I × − J L = 00 for J L = 12 for J L = 2 . (58)Considering the SU(3) symmetry breaking from Eq. (57), we find the splitting parameters for the and δ (cid:48) = 38 α + β = δ = −
180 MeV ,δ (cid:48) J L = δ − δ × − J L = 00 for J L = 12 for J L = 2 , (59)where we see that δ (cid:48) is just the same as δ given in Eq. (30). δ is given as −
120 MeV. Though we do not know thenumerical value of the new parameter δ , we still can analyze the mass splittings of the newly found Ω c ’s, using thesplittings between the states with different values of J L .We now turn to the hyperfine splittings. The two anti-triplets of spin 1/2 and 3/2 and the two sextets of spin 1/2and 3/2 are split by∆ hf = ∆ hf J L =1 = κ (cid:48) m c , (60)whereas another two sextets of spin 3/2 and 5/2 are split by∆ hf J L =2 = 53 κ (cid:48) m c . (61)One sextet of spin 1/2 from the J L = 0 case has no hyperfine splitting. The results are depicted in Fig. 6. Note thatthe ∆ represent the splittings between the J L = 0 state and the degenerate J L = 1 state, whereas ∆ denote thosebetween degenerate J L = 1 and J L = 2 states∆ = a I + 320 δ, ∆ = 2∆ . (62)We will soon see that the relation ∆ = 2∆ will play an critical role in identifying the excited Ω c ’s within the χ QSM.If one identifies Λ c (2592) and Ξ c (2790) as the members of the excited baryon anti-triplet of spin (1 / − withnegative parity, and Λ c (2592) and Ξ c (2790) as those of the excited baryon anti-triplet of spin (3 / − , then we find4 FIG. 6. Mass splitting of the five excited sextets. δ = −
198 and −
190 MeV, which are more or less in agreement with the value given in Eq. (59). The κ (cid:48) /m c can bealso determined as κ (cid:48) m c = 13 ( M Λ c (2628) + 2 M Ξ c (2818) ) −
13 ( M Λ c (2592) + 2 M Ξ c (2790) ) = 30 MeV , (63)and M is also fixed by M = 29 ( M Λ c (2628) + 2 M Ξ c (2818) ) + 19 ( M Λ c (2592) + 2 M Ξ c (2790) ) = 2744 MeV . (64)We now assert that as a minimal scenario the newly found Ω c baryons by the LHCb Collaboration belong to thefive excited sextets. Then Ω c (3000) can be identified as the state with ( J L = 0 , / − ), which corresponds to thelightest state in Fig. 6. All other four states can be consequently identified as depicted in Fig. 6. Including thehyperfine interactions, we get the results as summarized in Table VII. We find at least three different contradictions TABLE VII. Scenario 1: All five LHCb Ω c states are assigned to the excited baryon sextets. J L S P M [MeV] κ (cid:48) /m c [MeV] ∆ J L [MeV]0 − − − − − arising from the assignment of these Ω c states as the members of the excited sextets within the χ QSM. Firstly, thisassignment requires that the hyperfine splitting should be almost as twice as smaller than in the case. Secondly, therobust relation ∆ = 2∆ given in Eq. (62) is badly broken. Finally, there are two orthogonal sum rules σ = σ = 0derived from the χ QSM σ = 6 Ω c ( J L = 0 , / − ) − Ω c ( J L = 1 , / − ) − c ( J L = 1 , / − ) + 3 Ω c ( J L = 2 , / − ) , (65) σ = − c ( J L = 0 , / − ) + 9 Ω c ( J L = 1 , / − ) − c ( J L = 1 , / − ) − c ( J L = 2 , / − ) + 3 Ω c ( J L = 2 , / − ) , which are also badly broken. Thus, we come to the conclusion that the the five Ω c baryons is unlikely to belong tothe excited sextets. A similar conclusion was drawn by Ref. [66] in a different theoretical framework. Moreover, thecomputed decay widths of the excited Ω c ’s do not match with the experimental data. Therefore, the first scenario isunrealistic in the present mean-field approach.Since the first scenario is not suitable for identifying the five excited Ω c baryons, we have to come up with anotherscenario. Observing that two of them have rather narrower decay widths than other three Ω c ’s, we assert that thesenarrow Ω c (3050) and Ω c (3119) belong to the possible exotic anti-decapentaplet ( ) which is yet another lowest-lyingallowed representation, whereas three of them belong to the excited sextet. We find in this scenario that two othermembers of the excited baryon sextet with J L = 2 have masses above the Ξ D threshold at 3185 MeV. Since they have5 TABLE VIII. Scenario 2. Only three LHCb states are assigned to the sextets. J L S P M [MeV] κ (cid:48) /m c [MeV] ∆ J L [MeV]0 − − − − input input −
24 164 rather broad widths, they are not clearly seen in the LHCb data and may fall into the bump structures appearing inthe LHCb data.The results of the second scenario are summarized in Table VIII except for the Ω c (3050) and Ω c (3119) which willbe discussed separately. The italic numbers correspond to the bump structures from which Ω c (3222) used as input.Scenario 2 provides a much more plausible prediction than scenario 1 does. Interestingly, the value of κ (cid:48) /m c ≈ = 2∆ is nicely satisfied in this scenario. FIG. 7. Representation of the anti-decapentaplet ( ). As in the case of the baryon sextet, there are two baryon anti-decapentaplets with spin 1/2 and 3/2. The Ω c s belong to the isotriplet in the plet.FIG. 8. The allowed representations for the lowest-lying heavy baryons. The anti-decapentaplet ( ) was first suggested by Diakonov [51]. Figure 7 illustrates the representation of the .Since the belongs to the allowed representations for the ground-state heavy baryons, it satisfies the quantizationrule J L + T = , so T = J L = 1 (see Fig. 8). When the light-quark pair with J L = 1 is coupled to the heavy-quarkspin, there are two possible representations that are degenerate in the limit of m Q → ∞ . It means that one needsto consider the hyperfine interaction defined in Eq. (32). As given in Eq. (35), the value of κ /m c is around 68 MeV.6Surprisingly, the mass difference between the Ω c (3050) and the Ω c (3119) is M Ω c (3 / + ) (3119) − M Ω c (1 / + ) (3050) = κ m c ≈
69 MeV (66)which is almost the same as what was determined from the lowest-lying sextet baryons. The decay widths of theexcited Ω c baryons predicted within the present framework further support the plausibility of scenario 2 [58]. Thedecay widths for the Ω c (3050) and Ω c (3119) are predicted to beΓ Ω c (3050)( , / + ) = 0 .
48 MeV , Γ Ω c (3119)( , / + ) = 1 .
12 MeV , (67)which are in good agreement with the LHCb data Γ Ω c (3050) = (0 . ± . ± .
1) MeV and Γ Ω c (3119) = (1 . ± . ± . c , we refer to Ref. [58].In addition to scenarios 1 and 2, we also tried to examine several other scenarios but find that they all turned out tobe inconsistent with the experimental data. Finally, we want to emphasize that the Ω c (3050) and Ω c (3119) assignedto the members of the are isotriplets. It implies that if they indeed belong to the , charged Ω ± c should exist.Knowing that the excited Ω c ’s have been measured in the Ξ + c K − c channel, we propose that the Ξ + c K and Ξ c K − channels need to be scanned in the range of the invariant mass between 3000 MeV and 3200 MeV to find an isovectorΩ c ’s. If they do not exist, this will falsify the present predictions. VI. CONCLUSION AND OUTLOOK
In the present short review, we briefly summarized a series of recent works on the properties of the singly heavybaryons within a pion mean-field approach, also known as the chiral quark-soliton model. In the limit of the infinitelyheavy quark mass ( m Q → ∞ ), the heavy quark inside a heavy baryon can be treated as a mere static color source.Then a heavy baryon is portrayed as a state of N c − N c -counting prefactor by N c − c baryons reported by the LHCb Collaboration. Assigning the three of them to the excitedbaryon sextets and the two of them with narrower decay widths to the possible exotic baryon anti-decapentaplet, wewere able to classify the Ω c ’s successfully. Since the Ω c baryons in the anti-decapentaplet are the isovector baryons,we anticipate that charged Ω c ’s might be found in other channels such as the Ξ + c K and Ξ c K − .The present model can be further applied to future investigations on various properties and form factors of heavybaryons. As already shown in Ref. [56], the electric form factor of the charged heavy baryon indicates that a heavybaryon is an electrically compact object. Transition form factors of heavy baryons will further reveal their internalstructure. Understanding excited heavy baryons is another crucial issue that should be investigated. Related studiesare under way. ACKNOWLEDGMENTS
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