aa r X i v : . [ h e p - ph ] J u l S-wave bottom baryons
S.M. Gerasyuta and E.E. MatskevichDepartment of Theoretical Physics, St. Petersburg State University, 198904,St. Petersburg, RussiaandDepartment of Physics, LTA, 194021, St. Petersburg, Russia
Abstract
The masses of S -wave bottom baryons are calculated in the framework of coupled-channel formalism. The relativistic three-quark equations for the bottom baryons usingthe dispersion relation technique are found. The approximate solutions of these equa-tions based on the extraction of leading singularities of the amplitude are obtained.The calculated mass values of S -wave bottom baryons are in good agreement with theexperimental ones.e-mail address: [email protected] address: [email protected]: 11.55.Fv, 12.39.Ki, 12.40.Yx, 14.20.-c. I. Introduction.
Hadron spectroscopy has always played an important role in the revealing mecha-nisms underlying the dynamic of strong interactions.The heavy hadron containing a single heavy quark is particularly interesting. Thelight degrees of freedom (quarks and gluons) circle around the nearby static heavyquark. Such a system behaves as the QCD analogue of familar hydrogen bound by theelectromagnetic interaction.The heavy quark expansion provides a systematic tool for heavy hadrons. When theheavy quark mass m Q → ∞ , the angular momentum of the light degree of freedom is agood quantum number. Therefore heavy hadrons form doublets. For example, Ω b andΩ ∗ b will be degenerate in the heavy quark limit. Their mass splitting is caused by thechromomagnetic interaction at the order O (1 /m Q ), which can be taken into accountsystematically in the framework of heavy quark effective field theory (HQET) [1 – 3].Recently CDF Collaboration observed four bottom baryons Σ ± b and Σ ∗± b [4]. D0 andCDF have seen candidates for Ξ − b [5, 6].In the part two decades, various phenomenological models have been used to studyheavy baryon masses [7 – 12]. Capstick and Isgur studied the heavy baryon systemin a relativized quark potential model [7]. Roncaglia et al. predicted the massesof baryons containing one or two heavy quark using the Feynman-Hellman theoremand semiempirical mass formulas [8]. Jenkins studied heavy baryon masses using acombined expansion of 1 /m Q and 1 /N c [9]. Mathur et al. predicted the masses ofcharmed and bottom baryons from lattice QCD [10]. Ebert et al. calculated themasses of heavy baryons with the light-diquark approximation [11]. Stimulated byrecent experimental progress, there have been several theoretical papers on the masses f Σ b , Σ ∗ b and Ξ b using the hyperfine interaction in the quark model [12 – 16]. Recentlythe strong decays of heavy baryons were investigated systematically using P modelin Ref. [17].QCD sum rule has been applied to study heavy baryon masses [18 – 22].In our papers [23 – 25] relativistic generalization of the three-body Faddeev equa-tions was obtained in the form of dispersion relations in the pair energy of two inter-acting particles. The mass spectra of S -wave baryons including u , d , s , c quarks werecalculated by a method based on isolating of the leading singularities in the amplitude.We searched for the approximate solution of integral three-quark equations by takinginto account two-particle and triangle singularities, the weaker ones being neglected. Ifwe considered such an approximation, which corresponds to taking into account two-body and triangle singularities, and defined all the smooth functions at the middlepoint of the physical region of Dalitz-plot, then the problem was reduced to the one ofsolving a system of simple algebraic equations.In the present paper the relativistic three-particle amplitudes in the coupled-channelsformalism are considered. We take into account the u , d , s , c , b quarks and constructthe flavor-spin functions for the 35 baryons with the spin-parity J p =
12 + and J p =
32 + : J p =
12 + J p =
32 + Σ b uub, udb, ddb Σ b uub, udb, ddb Λ b udb Ξ sb usb, dsb Ξ Asb usb, dsb Ω ssb ssb Ξ Ssb usb, dsb Ξ cb ucb, dcb Ω ssb ssb Ω scb scb Ξ Acb ucb, dcb Ω ccb ccb Ξ Scb ucb, dcb Ξ bb ubb, dbb Λ Ascb scb Ω sbb sbb Λ Sscb scb Ω cbb cbb Ω ccb ccb Ω bbb bbb Ξ bb ubb, dbb Ω sbb sbb Ω cbb cbb (1)In the paper [25] the relativistic equations were obtained and the mass spectrum of S -wave charmed baryons was calculated.In the present paper we will be able to use the similar method. In this case weconsider 35 baryons with the spin-parity J p =
12 + and J p =
32 + , which include one, twoand three bottom quarks. We have considered the 23 baryons with different masses.The paper is organized as follows. In Section II we obtain the relativistic three-particle equations which describe the interaction of the quarks in baryons. In SectionIII the coupled systems of equations for the reduced amplitudes are derived. SectionIV is devoted to a discussion of the results for the mass spectrum of S -wave bottombaryons (Tables I, II). In the conclusion the status of the considered model is discussed.In Appendix A we write down the three-particle integral equations for the J p =
12 + and J p =
32 + bottom baryon multiplets. In Appendix B the coupled systems ofapproximate equations for lowest bottom baryons are given. I. The three-quark integral equations for the S-wave bottom baryons.
We calculate the masses of the bottom baryons in a relativistic approach usingthe dispersion relation technique. The relativistic three-quark integral equations areconstructed in the form of the dispersion relations over the two-body subenergy.We use the graphic equations for the functions A J ( s, s ik ) [23 – 25]. In order torepresent the amplitude A J ( s, s ik ) in the form of dispersion relations, it is necessary todefine the amplitudes of quark-quark interaction a J ( s ik ). The pair amplitudes qq → qq are calculated in the framework of the dispersion N/D method with the input four-fermion interaction with quantum numbers of the gluon [26]. We use results of ourrelativistic quark model [27] and write down the pair quark amplitudes in the followingform a J ( s ik ) = G J ( s ik )1 − B J ( s ik ) , (2) B J ( s ik ) = Λ J ( i,k ) Z ( m i + m k ) ds ′ ik π ρ J ( s ′ ik ) G J ( s ′ ik ) s ′ ik − s ik , (3) ρ J ( s ik ) = ( m i + m k ) π α J s ik ( m i + m k ) + β J + δ J s ik ! ×× q ( s ik − ( m i + m k ) )( s ik − ( m i − m k ) ) s ik . (4)Here G J is the vertex function of a diquark, which can be expressed in terms ofthe N -function of the bootstrap N/D method as G J = √ N J ; B J ( s ik ) is the Chew-Mandelstam function [28], and ρ J ( s ik ) is the phase space for a diquark. s ik is the pairenergy squared of diquark, the index J p determines the spin-parity of diquark. Thecoefficients of Chew-Mandelstam function α J , β J and δ J in Table III are given. Λ J ( i, k )is the pair energy cutoff. In the case under discussion the interacting pairs of quarks donot form bound states. Therefore the integration in the dispersion integral (3) is carriedout from ( m i + m k ) to Λ J ( i, k ) (i,k=1,2,3). Including all possible rescatterings of eachpair of quarks and grouping the terms according to the final states of the particles, weobtained the coupled systems of integral equations. For instance, for the Σ + b with J p =
12 + the wave function is ϕ Σ + b = q { u ↑ d ↑ b ↓} − q { u ↑ d ↓ b ↑} − q { u ↓ d ↑ b ↑} .Then the coupled system of equations has the following form: A ( s, s ) = λ b ( s ) L ( s ) + K ( s ) h A b ( s, s ) + A b ( s, s )++ A b ( s, s ) + A b ( s, s ) i A b ( s, s ) = λ b b ( s ) L b ( s ) + K b ( s ) h A ( s, s ) − A b ( s, s )++ A b ( s, s ) + A ( s, s ) − A b ( s, s ) + A b ( s, s ) i A b ( s, s ) = λ b b ( s ) L b ( s ) + K b ( s ) h A ( s, s ) + A b ( s, s )++ A b ( s, s ) + A ( s, s ) + A b ( s, s ) + A b ( s, s ) i . (5) ere the function L J ( s ik ) has the form L J ( s ik ) = G J ( s ik )1 − B J ( s ik ) . (6)The integral operator K J ( s ik ) is K J ( s ik ) = L J ( s ik ) Λ J ( ik ) Z ( m i + m k ) ds ′ ik π ρ J ( s ′ ik ) G J ( s ′ ik ) s ′ ik − s ik Z − dz . (7)The function b J ( s ik ) is the truncated function of Chew-Mandelstam: b J ( s ik ) = ∞ Z ( m i + m k ) ds ′ ik π ρ J ( s ′ ik ) G J ( s ′ ik ) s ′ ik − s ik , (8) z is the cosine of the angle between the relative momentum of particles i and k inthe intermediate state and the momentum of particle j in the final state, taken in thec.m. of the particles i and k . Let some current produces three quarks with the vertexconstant λ . This constant do not affect to the spectra mass of bottom baryons. Byanalogy with the Σ + b state we obtain the rescattering amplitudes of the three variousquarks for the all bottom states (Appendix A). III. Reduced equations for the S-wave bottom baryons.
Let us extract two-particle singularities in A J ( s, s ik ): A J ( s, s ik ) = α J ( s, s ik ) b J ( s ik ) G J ( s ik )1 − B J ( s ik ) , (9) α J ( s, s ik ) is the reduced amplitude. Accordingly to this, all integral equations can berewritten using the reduced amplitudes. The function α J ( s, s ik ) is the smooth functionof s ik as compared with the singular part of the amplitude. We do not extract thethree-body singularities, because they are weaker than the two-particle singularities.For instance, one considers the first equation of system for the Σ + b with J p =
12 + : α ( s, s ) = λ + 1 b ( s ) Λ (1 , Z ( m + m ) ds ′ π ρ ( s ′ ) G ( s ′ ) s ′ − s ×× Z − dz G b ( s ′ ) b b ( s ′ )1 − B b ( s ′ ) 12 α b ( s, s ′ ) + G b ( s ′ ) b b ( s ′ )1 − B b ( s ′ ) 32 α b ( s, s ′ ) ! . (10)The connection between s ′ and s ′ is [29]: s ′ = m + m − ( s ′ + m − s ) ( s ′ + m − m )2 s ′ ±± z s ′ × q ( s ′ − ( m + m ) ) ( s ′ − ( m − m ) ) × r(cid:16) s ′ − ( √ s + m ) (cid:17) (cid:16) s ′ − ( √ s − m ) (cid:17) . (11) he formula for s ′ is similar to (11) with replaced by z → − z . Thus A b ( s, s ′ ) + A b ( s, s ′ ) must be replaced by 2 A b ( s, s ′ ). Λ J ( i, k ) is the cutoff at the large value of s ik , which determines the contribution from small distances.The construction of the approximate solution of coupled system equations is basedon the extraction of the leading singularities which are close to the region s ik = ( m i + m k ) [29].We consider the approximation, which corresponds to the single interaction of theall three particles (two-particle and triangle singularities) and neglecting all the weakerones.The functions α J ( s, s ik ) are the smooth functions of s ik as compared with the sin-gular part of the amplitude, hence it can be expanded in a series at the singulary pointand only the first term of this series should be employed further. As s it is convenientto take the middle point of physical region of the Dalitz plot in which z = 0. In thiscase we get from (11) s ik = s = s + m + m + m m + m + m , where m ik = m i + m k . We define functions α J ( s, s ik ) and b J ( s ik ) at the point s . Such a choice of point s allows us to replaceintegral equations (5) by the algebraic couple equations for the state Σ + b : α ( s, s ) = λ + α b ( s, s ) I b ( s, s ) b b ( s ) b ( s ) + α b ( s, s ) I b ( s, s ) b b ( s ) b ( s ) α b ( s, s ) = λ + α ( s, s ) I b ( s, s ) b ( s ) b b ( s ) − α b ( s, s ) I b b ( s, s ) + α b ( s, s ) I b b ( s, s ) b b ( s ) b b ( s ) α b ( s, s ) = λ + α ( s, s ) I b ( s, s ) b ( s ) b b ( s ) + α b ( s, s ) I b b ( s, s ) b b ( s ) b b ( s ) + α b ( s, s ) I b b ( s, s ) . (12)The function I J J ( s, s ) takes into account singularity which corresponds to thesimultaneous vanishing of all propagators in the triangle diagram. I J J ( s, s ) = Λ J Z ( m i + m k ) ds ′ ik π ρ J ( s ′ ik ) G J ( s ′ ik ) s ′ ik − s ik Z − dz − B J ( s ij ) (13)The G J ( s ik ) functions have the smooth dependence from energy s ik [27] thereforewe suggest them as constants. The parameters of model: g J vertex constant and λ J cutoff parameter are chosen dimensionless. g J = m i + m k π G J , λ J = 4Λ J ( m i + m k ) . (14)Here m i and m k are quark masses in the intermediate state of the quark loop. Wecalculate the coupled system of equations and can determine the mass values of theΣ + b state. We calculate a pole in s which corresponds to the bound state of the threequarks.By analogy with Σ + b -hyperon we obtain the systems of equations for the reducedamplitudes of all bottom baryons (Appendix B).The solutions of the coupled system of equations are considered as: J = F J ( s, λ J ) D ( s ) , (15)where the zeros of the D ( s ) determinate the masses of bound states of baryons. F J ( s, λ J ) are the functions of s and λ J . The functions F J ( s, λ J ) determine the contri-butions of subamplitudes to the baryon amplitude. IV. Calculation results.
The quark masses ( m u = m d = m , m s and m c ) are given similar to the our paperones [25]: m = 0 . GeV , m s = 0 . GeV and m c = 1 . GeV . The strange quarkmass is heavier than the strange quark mass in the some quark models [7 – 11, 23,27]. This value of strange quark mass allows us to describe the spectroscopy of S -wavecharmed baryons well [25]. We use only two new parameters as compared with the S -wave light baryons [23, 24]. The parameters of model are the cutoff energy parameters λ q = 10 . λ c = 6 . u , d , s and charmed quarks, the vertex constants g = 0 . g = 0 .
55, for the light diquarks with J P = 0 + , 1 + and g c = 0 .
857 for thecharmed diquarks. λ qQ = ( q λ q + q λ Q ) are chosen ( q = u, d, s , Q = c ).In the present paper we have used two new parameters: the cutoff of the bb diquark λ b = 5 . g b = 1 .
03. These values have been determined bythe b -baryon masses: M Σ b
12 + = 5 . GeV and M Σ b
32 + = 5 . GeV . In order to fix m b = 4 . GeV we use the b -baryon masses M Σ b
32 + = 5 . GeV . We represent themasses of all S -wave bottom baryons in the Tables I, II. The calculated mass value M Λ b
12 + = 5 . GeV is equal to the experimental data [30], the mass value M Ξ Asb
12 + =5 . GeV is close to the experimental one [5]. But the more precise CDF mass (TableI) lies close to a prediction of Ref. [13]. We neglect with the mass distinction of u and d quarks. The estimation of the theoretical error on the bottom baryon masses is2 − M eV . This result was obtained by the choice of model parameters.
V. Conclusion.
In a strongly bound systems, which include the light quarks, where p/m ∼
1, theapproximation by nonrelativistic kinematics and dynamics is not justified.In our paper the relativistic description of three particles amplitudes of bottombaryons are considered. We take into account the u , d , s , c , b quarks. The massspectrum of S -wave bottom baryons with one, two and three b quarks is considered.We use only two new parameters for the calculation of 23 baryon masses. The othermodel parameters in the our papers [23 – 25] are given. The charge-averaged hyperfinesplitting between the J = and J = states predicted from that for charmed particlesis similar to the Ref. [31].In their paper the spin-averaged mass of the states Ξ ′ b and Ξ ∗ b is predicted to liearound to 150 − M eV above M Ξ b , while the hyperfine splitting between Ξ ′ b and Ξ ∗ b is predicted to lie in the rough range of 20 to 30 M eV .We have predicted the masses of baryons containing b quarks using the coupled-channel formalism. We believe that the prediction for the S -wave bottom baryonsbased on the relativistic kinematics and dynamics allows as to take into account therelativistic corrections. In our consideration the bottom baryon masses are heavier thanthe masses in the other quark models [11, 31 – 34]. In our model the spin-averagedmass of the states Ξ ′ b and Ξ ∗ b is predicted to lie around to 250 M eV above M Ξ b . The elativistic corrections are particularly important for the splitting between Ω + b and Ω b baryons.We will be able to calculate the P -wave bottom baryons in our approach [35, 36] us-ing the new experimental data. The interesting opinions with the S -matrix singularitiesin Ref. [37] are given. Acknowledgments.
The authors would like to thank S. Capstick and S.L. Zhu for useful discussions.The work was carried with the support of the Russion Ministry of Education (grant2.1.1.68.26). ppendix A. Integral equations for three-particle amplitudes A J p ( s , s ik ) of S-wave bottom baryons.The J P =
32 + multiplet.
1. Baryons Σ b , Ω ssb , Ω ccb , Ξ bb , Ω sbb , Ω cbb .The wave functions: ϕ = { x ↑ x ↑ y ↑} . A xx ( s, s ) = λ b xx ( s ) L xx ( s ) + K xx ( s ) [ A xy ( s, s ) + A xy ( s, s )] A xy ( s, s ) = λ b xy ( s ) L xy ( s ) + K xy ( s ) [ A xx ( s, s ) + A xy ( s, s )] . ( A b : x = q ( q = u, d ), y = b ; for the Ω ssb : x = s , y = b ; for the Ω ccb : x = c , y = b ; for the Ξ bb : x = b , y = q ; for the Ω sbb : x = b , y = s ; for the Ω sbb : x = b , y = c .2. Baryons Ξ sb , Ξ cb , Ω scb .The wave functions: ϕ = { x ↑ y ↑ z ↑} . A xy ( s, s ) = λ b xy ( s ) L xy ( s ) + K xy ( s ) [ A xz ( s, s ) + A yz ( s, s )] A xz ( s, s ) = λ b xz ( s ) L xz ( s ) + K xz ( s ) [ A xy ( s, s ) + A yz ( s, s )] A yz ( s, s ) = λ b yz ( s ) L yz ( s ) + K yz ( s ) [ A xy ( s, s ) + A xz ( s, s )] . ( A sb : x = q , y = s , z = b ; for the Ξ cb : x = q , y = c , z = b ; for the Ω scb : x = s , y = c , z = b .3. Baryon Ω bbb .The wave functions: ϕ = { b ↑ b ↑ b ↑} . A bb ( s, s ) = λ b bb ( s ) L bb ( s ) + K bb ( s ) [ A bb ( s, s ) + A bb ( s, s )] . ( A The J P =
12 + multiplet.
1. Baryons Σ b , Λ b , Ω ssb , Ω ccb , Ξ bb , Ω sbb , Ω cbb .The wave functions:for the Σ b : ϕ Σ + b = q { u ↑ d ↑ b ↓} − q { u ↑ d ↓ b ↑} − q { u ↓ d ↑ b ↑} ;for the Λ b : ϕ Λ b = q { u ↑ d ↓ b ↑} − q { u ↓ d ↑ b ↑} ;for the Ω ssb , Ω ccb , Ξ bb , Ω sbb , Ω cbb : ϕ = q { x ↑ x ↑ y ↓} − q { x ↑ x ↓ y ↑} ;here: for the Ω ssb : x = s , y = b ; for the Ω ccb : x = c , y = b ; for the Ξ bb : x = b , y = q ;for the Ω sbb : x = b , y = s ; for the Ω cbb : x = b , y = c . A x ( s, s ) = λ b x ( s ) L x ( s ) + K x ( s ) h A y ( s, s ) + A z ( s, s )++ A y ( s, s ) + A z ( s, s ) i A y ( s, s ) = λ b y ( s ) L y ( s ) + K y ( s ) h A x ( s, s ) − A y ( s, s )++ A z ( s, s ) + A x ( s, s ) − A y ( s, s ) + A z ( s, s ) i A z ( s, s ) = λ b z ( s ) L z ( s ) + K z ( s ) h A x ( s, s ) + A y ( s, s )++ A z ( s, s ) + A x ( s, s ) + A y ( s, s ) + A z ( s, s ) i . ( A b : x = 1 qq , y = 1 qb , z = 0 qb ; for the Λ b : x = 0 qq , y = 0 qb , z = 1 qb ; forthe Ω ssb : x = 1 ss , y = 1 sb , z = 0 sb ; for the Ω ccb : x = 1 cc , y = 1 cb , z = 0 cb ; for the Ξ bb : x = 1 bb , y = 1 qb , z = 0 qb ; for the Ω sbb : x = 1 bb , y = 1 sb , z = 0 sb ; for the Ω cbb : x = 1 bb , y = 1 cb , z = 0 cb .2. Baryons Ξ Asb , Ξ
Ssb , Λ
Ascb , Λ
Sscb , Ξ
Acb , Ξ
Scb .The wave functions:for the Ξ
Asb , Λ
Ascb , Ξ
Acb : ϕ = q { x ↑ y ↑ z ↓} − q { x ↑ y ↓ z ↑} ;here: for the Ξ Asb : x = b , y = s , z = q ; for the Λ Ascb : x = b , y = c , z = s ; for the Ξ Acb : x = b , y = c , z = q .For the Ξ Ssb , Λ
Sscb , Ξ
Scb : ϕ = q { x ↑ y ↑ z ↓} − q { x ↑ y ↓ z ↑} − q { x ↓ y ↑ z ↑} ;here: for the Ξ Ssb : x = q , y = s , z = b ; for the Λ Sscb : x = s , y = c , z = b ; for the Ξ Scb : x = q , y = c , z = b . A x ( s, s ) = λ b x ( s ) L x ( s ) + K x ( s ) h A y ( s, s ) + A z ( s, s )++ A v ( s, s ) + A w ( s, s ) + A y ( s, s ) + A z ( s, s )++ A v ( s, s ) + A w ( s, s ) i A y ( s, s ) = λ b y ( s ) L y ( s ) + K y ( s ) h A x ( s, s ) − A z ( s, s )++ A w ( s, s ) + A x ( s, s ) − A z ( s, s ) + A w ( s, s ) i A z ( s, s ) = λ b z ( s ) L z ( s ) + K z ( s ) h A x ( s, s ) − A y ( s, s )++ A v ( s, s ) + A x ( s, s ) − A y ( s, s ) + A v ( s, s ) i A v ( s, s ) = λ b v ( s ) L v ( s ) + K v ( s ) h A x ( s, s ) + A z ( s, s )++ A w ( s, s ) + A x ( s, s ) + A z ( s, s ) + A w ( s, s ) i A w ( s, s ) = λ b w ( s ) L w ( s ) + K w ( s ) h A x ( s, s ) + A y ( s, s )++ A v ( s, s ) + A x ( s, s ) + A y ( s, s ) + A v ( s, s ) i . ( A Asb : x = 0 qs , y = 0 qb , z = 0 sb , v = 1 qb , w = 1 sb ; for the Ξ Acb : x = 0 qc , y = 0 qb , z = 0 cb , v = 1 qb , w = 1 cb ; for the Λ Ascb : x = 0 sc , y = 0 sb , z = 0 cb , v = 1 sb , w = 1 cb ; for the Ξ Ssb : x = 1 qs , y = 1 qb , z = 1 sb , v = 0 qb , w = 0 sb ; for the Ξ Scb : x = 1 qc , y = 1 qb , z = 1 cb , v = 0 qb , w = 0 cb ; for the Λ Sscb : x = 1 sc , y = 1 sb , z = 1 cb , v = 0 sb , w = 0 cb . Appendix B. Couple systems of approximate equations for the S-wavebottom baryons.The J P =
32 + multiplet.
1. Baryons Σ b , Ω ssb , Ω ccb , Ξ bb , Ω sbb , Ω cbb . α x ( s, s ) = λ + 2 α y ( s, s ) I x y ( s, s ) b y ( s ) b x ( s ) α y ( s, s ) = λ + α x ( s, s ) I y x ( s, s ) b x ( s ) b y ( s ) + α y ( s, s ) I y y ( s, s ) . ( A b : x = q , y = b ; for the Ω ssb : x = s , y = b ; for the Ω ccb : x = c , y = b ;for the Ξ bb : x = b , y = q ; for the Ω sbb : x = b , y = s ; for the Ω sbb : x = b , y = c . . Baryons Ξ sb , Ξ cb , Ω scb . α x ( s, s ) = λ + α y ( s, s ) I x y ( s, s ) b y ( s ) b x ( s ) + α z ( s, s ) I x z ( s, s ) b z ( s ) b x ( s ) α y ( s, s ) = λ + α x ( s, s ) I y x ( s, s ) b x ( s ) b y ( s ) + α z ( s, s ) I y z ( s, s ) b z ( s ) b y ( s ) α z ( s, s ) = λ + α x ( s, s ) I z x ( s, s ) b x ( s ) b z ( s ) + α y ( s, s ) I z y ( s, s ) b y ( s ) b z ( s ) . ( A sb : x = q , y = s , z = b ; for the Ξ cb : x = q , y = c , z = b ; for the Ω scb : x = s , y = c , z = b .3. Baryon Ω bbb . α bb ( s, s ) = λ + 2 α bb ( s, s ) I bb bb ( s, s ) . ( A The J P =
12 + multiplet.
1. Baryons Σ b , Λ b , Ω ssb , Ω ccb , Ξ bb , Ω sbb , Ω cbb . α x ( s, s ) = λ + α y ( s, s ) I xy ( s, s ) b y ( s ) b x ( s ) + α z ( s, s ) I xz ( s, s ) b z ( s ) b x ( s ) α y ( s, s ) = λ + α x ( s, s ) I yx ( s, s ) b x ( s ) b y ( s ) − α y ( s, s ) I yy ( s, s )+ α z ( s, s ) I yz ( s, s ) b z ( s ) b y ( s ) α z ( s, s ) = λ + α x ( s, s ) I zx ( s, s ) b x ( s ) b z ( s ) + α y ( s, s ) I zy ( s, s ) b y ( s ) b z ( s ) + α z ( s, s ) I zz ( s, s ) . ( A b : x = 1 qq , y = 1 qb , z = 0 qb ; for the Λ b : x = 0 qq , y = 0 qb , z = 1 qb ; forthe Ω ssb : x = 1 ss , y = 1 sb , z = 0 sb ; for the Ω ccb : x = 1 cc , y = 1 cb , z = 0 cb ; for the Ξ bb : x = 1 bb , y = 1 qb , z = 0 qb ; for the Ω sbb : x = 1 bb , y = 1 sb , z = 0 sb ; for the Ω cbb : x = 1 bb , y = 1 cb , z = 0 cb . . Baryons Ξ Asb , Ξ
Ssb , Λ
Ascb , Λ
Sscb , Ξ
Acb , Ξ
Scb . α x ( s, s ) = λ + α y ( s, s ) I xy ( s, s ) b y ( s ) b x ( s ) + α z ( s, s ) I xz ( s, s ) b z ( s ) b x ( s ) + α v ( s, s ) I xv ( s, s ) b v ( s ) b x ( s ) + α w ( s, s ) I xw ( s, s ) b w ( s ) b x ( s ) α y ( s, s ) = λ + α x ( s, s ) I yx ( s, s ) b x ( s ) b y ( s ) − α z ( s, s ) I yz ( s, s ) b z ( s ) b y ( s ) + α w ( s, s ) I yw ( s, s ) b w ( s ) b y ( s ) α z ( s, s ) = λ + α x ( s, s ) I zx ( s, s ) b x ( s ) b z ( s ) − α y ( s, s ) I zy ( s, s ) b y ( s ) b z ( s ) + α v ( s, s ) I zv ( s, s ) b v ( s ) b z ( s ) α v ( s, s ) = λ + α x ( s, s ) I vx ( s, s ) b x ( s ) b v ( s ) + α z ( s, s ) I vz ( s, s ) b z ( s ) b v ( s ) + α w ( s, s ) I vw ( s, s ) b w ( s ) b v ( s ) α w ( s, s ) = λ + α x ( s, s ) I wx ( s, s ) b x ( s ) b w ( s ) + α y ( s, s ) I wy ( s, s ) b y ( s ) b w ( s ) + α v ( s, s ) I wv ( s, s ) b v ( s ) b w ( s ) . ( A Asb : x = 0 qs , y = 0 qb , z = 0 sb , v = 1 qb , w = 1 sb ; for the Ξ Acb : x = 0 qc , y = 0 qb , z = 0 cb , v = 1 qb , w = 1 cb ; for the Λ Ascb : x = 0 sc , y = 0 sb , z = 0 cb , v = 1 sb , w = 1 cb ; for the Ξ Ssb : x = 1 qs , y = 1 qb , z = 1 sb , v = 0 qb , w = 0 sb ; for the Ξ Scb : x = 1 qc , y = 1 qb , z = 1 cb , v = 0 qb , w = 0 cb ; for the Λ Sscb : x = 1 sc , y = 1 sb , z = 1 cb , v = 0 sb , w = 0 cb . able I. Bottom baryon masses of multiplet
12 + .Parameters of model: quark masses m u,d = 495 M eV , m s = 770 M eV , m c = 1655 M eV , m b = 4840 M eV ; cutoff parameters: λ q = 10 . q = u, d, s ), λ c = 6 . λ b = 5 .
4; gluoncoupling constants: g = 0 . g = 0 .
55 with J p = 0 + and 1 + , g c = 0 . g b = 1 . GeV ) Mass (
GeV ) (exp.)Σ b .
808 5 . b .
624 5 . Asb .
761 5 . Ssb .
007 –Ω ssb .
120 –Ξ
Acb .
789 –Ξ
Scb .
818 –Λ
Ascb .
798 –Λ
Sscb .
836 –Ω ccb .
943 –Ξ bb .
045 –Ω sbb .
999 –Ω cbb .
089 –Table II. Bottom baryon masses of multiplet
32 + .Baryon Mass (
GeV ) Mass (
GeV ) (exp.)Σ b .
829 5 . sb .
066 –Ω ssb .
220 –Ξ cb .
863 –Ω scb .
914 –Ω ccb .
973 –Ξ bb .
104 –Ω sbb .
126 –Ω cbb .
123 –Ω bbb .
197 –Table III. Coefficients of Ghew-Mandelstam functions. α J β J δ J + 13 4 m i m k m i + m k ) − − ( m i − m k ) + 12 −
12 ( m i − m k ) ( m i + m k ) eferences.
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