Bottom (70,1-) baryon multiplet

. The aim of this paper is to derive systems of equations for the amplitudes for the case of negative parity (70,1-) bottom nonstrange baryons and to calculate the masses of these particles. In order to calculate masses of (70,1-) bottom baryons we use the relativistic quark model. The relativistic three-quark equations of the (70,1-) bottom baryon multiplet are derived in the framework of the dispersion relation technique. The relativistic three-quark equations of the (70,1-) bottom baryon multiplet are derived. The approximate solutions of these equations using the method based on the extraction of leading singularities of the amplitude are obtained. The masses of 21 baryons are predicted.

Hadron spectroscopy has always played an important role in the revealing mechanisms underlying the dynamic of strong interactions.At low energies, typical for baryon spectroscopy, QCD does not admit a perturbative expansion in the strong coupling constant.In 1974 't Hooft [1] suggested a perturbative expansion of QCD in terms of the parameter 1/N c where N c is the number of colors.This suggestion together with the power counting rules of Witten [2] has lead to the 1/N c expansion method which allows to systematically analyse baryon properties.
In the series of papers [3][4][5][6][7] a practical treatment of relativistic three-hadron systems have been developed.The physics of three-hadron system is usefully described in term of the pairwise interactions among the three particles.The theory is based on the two principles of unitarity and analyticity, as applied to the two-body subenergy channels.The linear integral equations in a single variable are obtained for the isobar amplitudes.The coupled integral equations are solved in terms of simple algebra.
In our papers [8,9] relativistic generalization of the three-body Faddeev equations was obtained in the form of dispersion relations in the pair energy of two interacting particles.The mass spectrum of S-wave baryons including u, d, s-quarks was calculated by a method based on isolating the leading singularities in the amplitude.We searched for the approximate solution of integral three-quark equations by taking into account two-particle and triangle singularities, all the weaker ones being neglected.If we considered such an approximation, which corresponds to taking into account two-body and triangle singularities, and defined all the smooth functions of the middle point of the physical region of Dalitz-plot, then the problem was reduced to the one of solving a system of simple algebraic equations.
In our paper [10] the construction of the orbital-flavor-spin wave functions for the (70, 1 − ) multiplet are given.We deal with a three-quark system having one unit of orbital excitation.The orbital part of wave function must have a mixed symmetry.The spin-flavor part of wave function must have the same symmetry is order to obtain a totally symmetric state in the orbital-flavor-spin space.The integral equations using the orbital-flavor-spin wave functions was constructed.It allows to calculate the mass spectra for all baryons of (70, 1 − ) multiplet.The 15 resonances are in good agreement with experimental data.We have predicted 15 masses of baryons.In our model the four parameter are used: gluon coupling constants g + and g − for the various parity, cutoff energy parameters λ, λ s for the nonstrange and strange diquarks.
In the paper [11] the relativistic three-quark equations of the excited (70, 1 − ) charmed baryons are found in the framework of the dispersion relation technique.The (70, 1 − ) charmed baryon multiplet has 23 baryons with different masses.The 6 resonances are in good agreement with the experimental data.We have predicted 17 masses of charmed excited baryons.
In the framework of the proposed approximate method of solving the relativistic three-particle problem, we have obtained a spectrum of P -wave bottom baryons.This paper is a generalization of our works [10,11].
The paper is organized as follows.After this introduction, we consider the derivation of relativistic generalization of the Faddeev equations for the example of S-wave bottom state Ω bbb with J P = 3 In Sect.IV the relativistic three-quark equations are obtained in the form of the dispersion relation over the two-body subenergy.
In Sect.V the systems of equations for the reduced amplitudes are derived.Section VI is devoted to the calculation results for the mass spectrum of the (70, 1 − ) bottom multiplet (Tables I-IV).

II. BRIEF INTRODUCTION OF RELATIVISTIC FADDEEV EQUATIONS.
We consider the derivation of the relativistic generalization of the Faddeev equation for the example of the S-wave bottom state Ω bbb with J P = 3 2 + .This is the simplest example because this state consists of the three similar quarks.
We take into account only pair quark interactions.The state has the three similar channels consisting of the diquark with the quantum number J P = 1 + (in the color state 3c ) and the third quark (in the color state 3 c ).The 3Q baryon state Ω bbb is constructed as color singlet.Suppose that there is a Ω bbb current which produces three b quarks (Fig. 1a).Accounting all possible pair interactions lead to the diagrams, which can be grouped in accordance with the latest interacting pair of particles (Fig. 1b-1d).The total amplitude can be represented as a sum of diagrams.Taking into account the equality of all pair interactions of bottom quarks in the state with J P = 1 + , we obtain the corresponding equation for the amplitudes: Here the s ik are the pair energies squared of particles i and k, and s is the total energy squared of the system.Using the diagrams of Fig. 1, it is easy to write down a graphical equation for the function A 1 (s, s 12 ) (Fig. 2).For this we need in an explicit form of the amplitude of the pair quark interaction.We write this amplitude for the diquark with J P = 1 + in the form: Here G 1 (s 12 ) is the vertex function of a diquark with J P = 1 + .B 1 (s 12 ) is the Chew-Mandelstam function [12] and ρ 1 (s 12 ) is the phase space for a diquark with J P = 1 + .The pair quarks amplitudes QQ → QQ are calculated in the framework of the dispersion N/D method with the input four-fermion interaction [13,14] with the quantum numbers of the gluon.
The four-quark interaction is considered as an input: Here I f is the unity matrix in the flavour space (u, d).λ are the color Gell-Mann matrices.Dimensional constants of the four-fermion interaction g V , g V , g V , g V , g V , g V , g V , g V , g V , and g the flavour SU (3) f symmetry occurs.The s, c, and b quarks violate the flavour SU (3) f symmetry.In order to avoid additional violation parameters we introduce the scale of the dimensional parameters [14]: Here m i and m k are the quark masses in the intermediate state of the quark loop.Dimensionless parameters g and Λ are supposed to be constants which are independent of the quark interaction type.The applicability of Eq. ( 5) is verified by the success of De Rujula-Georgi-Glashow quark model [13].Only the short-range part of Breit potential connected with the gluon exchange is responsible for the mass splitting in hadron multiplets.
We use the energy cutoff Λ(ik) in the integrals to avoid divergences.The equation corresponding to Fig. 2 can be written in the form: In Eq. ( 8) z is the cosine of the angle between the relative momentum of particles 1 and 2 in the intermediate state and the momentum of the third particle in the final state in the c.m.s. of the particles 1 and 2. In our case of equal mass of the quarks 1, 2 and 3, s ′ 13 and s ′ 12 are related by the equation ( 9) The expression for s ′ 23 is similar to (9) with the replacement z → −z.This makes it possible to replace [A 1 (s, s ′ 13 ) + A 1 (s, s ′ 23 )] in (8) by 2A 1 (s, s ′ 13 ).We extract the two-particle singularity from the amplitude A 1 (s, s 12 ).This singularity is the pole corresponding to the final diquark and is not interesting for us: The singularity corresponding to a bound baryon state is contained in the reduced amplitude α 1 (s, s 12 ).The equation for the reduced amplitude α 1 (s, s 12 ) can be written as We construct the approximate solution of Eq. (11).We take into account only two-particle and three-particle singularities and neglect others weaker ones.
The functions G(s 12 ) are practically independent of the two-particle energy and we consider them equal to constants.The function α 1 (s, s 12 ) is weakly dependent on the two-particle energy.For fixed values of s and s ′ 12 the integration is carried out over the region of the variable s ′ 13 corresponding to a physical transition of the current into three quarks (the physical region of Dalitz plot).It is convenient to take the central point of this region, corresponding to z = 0, to determinate the function α 1 (s, s 12 ) and also the Chew-Mandelstam function B 1 (s 12 ) at the point s 12 = s 0 = s 3 + m 2 b .Then the equation for the Ω bbb takes the form: We can obtain an approximate solution of Eq. ( 13): The pole in s corresponds to a bound state of the baryon.
In analogy with the case of the Ω bbb we can obtain the rescattering amplitudes for all S-wave bottom baryons with + , which include quarks of various flavours.These amplitudes will satisfy systems of integral equations.
Then we also solve them by approximate method and find masses of the all S-wave bottom baryons multiplet.
In our calculation we use the following parameters of model: from the previous paper [15] quark masses are given: m u,d = 495 M eV , m s = 770 M eV , and m c = 1655 M eV ; cutoff parameters λ q = 10.7 (q = u, d, s), λ c = 6.5; gluon coupling constants g 0 = 0.70, g 1 = 0.55 for J p = 0 + and 1 + light diquarks, g c = 0.857 for charmed diquarks.As usual we have We consider a three-quark system having one unit of orbital excitation.We take into account u, d, b-quarks.The orbital part of wave function must have a mixed symmetry.The spin-flavor part of wave function possesses the same symmetry in order to obtain a totally symmetric state in the orbital-spin-flavor space.
As an example we derived the wave functions for the decuplets (10, 2).The fully symmetric wave function for the decuplet state can be constructed as: Then we obtain: here M A and M S define the mixed antisymmetric and symmetric part of wave function, MA , ϕ MS .
The functions ϕ MS are given: ↑ and ↓ determine the spin directions. 1 and 0 correspond to the excited or nonexcited quarks.The three projections of orbital angular momentum are l z = 1, 0, −1.The (10, 2) multiplet with J p = 3 2 − can be obtained using the spin S = 1 2 and l z = 1, but the (10, 2) multiplet with J p = 1 2 − is determined by the spin S = 1 2 and l z = 0. We construct the SU (3)-function for each particle of multiplet.For instance, the SU (3)-function for Σ + b -hyperon of decuplet have following form: We obtain the SU (6) × O(3)-function for the Σ + b of the (10, 2) multiplet: Here the parenthesis determine the symmetrical function: The wave functions of Σ 0 b -and Σ − b -hyperons can be constructed by similar way.For the Ξ 0,− bb state of the (10, 2) multiplet the wave function is similar to the Σ +,− b state with the replacement by u ↔ b or d ↔ b.The wave function for the Ω bbb of (10, 2) decuplet is determined as: The wave functions and the method of the construction for the multiplets (8, 2), (8,4) and (1, 2) are similar.
By the construction of (70, 1 − ) bottom baryon multiplet integral equations we need to using the projectors for the different diquark states.The projectors to the symmetric and antisymmetric states can be obtained as: (q 1 q 2 + q 2 q 1 ) , 1 2 (q 1 q 2 − q 2 q 1 ) . (24) One can obtain the four types of totally symmetric projectors: We use these projectors for the consideration of various diquarks: u ↑ b ↓ : u ↑ b ↑ : Here the lower index determines the value of spin projection, and the upper index corresponds to the value of orbital angular momentum.
We use the diquark projectors and consider the particle Σ b 3 2 − of the (10, 2) multiplet.This wave function contains the contribution u 1 ↓ u ↑ b ↑, which includes three diquarks: The diquark projectors allow us to obtain the equations (39) -(41) (with the definition ρ J (s ij ) ≡ k ij ).
Then all members of wave function can be considered.After the grouping of these members we can obtain: We group the same members and obtain the system integral equations for the Σ b state with the J p = 3 2 − (10, 2) multiplet: Here function L J (s ik ) has the form The integral operator K J (s ik ) is: The function b J (s ik ) is the truncated function of Chew-Mandelstam.z is the cosine of the angle between the relative momentum of particles i and k in the intermediate state and the momentum of particle j in the final state, taken in the c.m. of the particles i and k.Let some current produces three quarks (first diagram Fig. 1) with the vertex constant λ.This constant do not affect to the spectra mass of excited baryons.
By analogy with the Σ b 3 2 − (10, 2) state we obtain the rescattering amplitudes of the three various quarks for all P -wave states of the (70, 1 − ) multiplet which satisfy the system of integral equations.The Ω bbb -hyperon masses of multiplet (70, 1 − ).
Multiplet Baryon Mass (GeV ) Mass (GeV ) (exp.)In this section we will give a brief explanation of some symbols.In accordance with the theorem on the connection of spin and statistics fermion states should have an antisymmetric wave functions.Three-quark wave functions of S-wave baryons have a structure corresponding to the symmetry  αJ βJ δJ
where multipliers are gropes of flavor, spin, and color symmetry respectively.The part of the structure corresponding to the SU (3) c is antisymmetric, other part corresponding to the SU (6) = SU (3) f ×SU (2) is symmetric.Combining three fundamental representations of SU (6) yields 6 ⊗ 6 ⊗ 6 = 56 ⊕ 70 ⊕ 70 ⊕ 20, here 56 is the multiplet possessing symmetric wave function, 70 are the multiplets possessing mixed symmetric and mixed antisymmetric wave functions, and 20 is the multiplet possessing antisymmetric wave function.In the case of S-wave baryons we have the 56 multiplet.If we consider P -wave baryons, that is with the orbital excitatin L = 1, then we will have two 70 multiplets because in the according with the oscillator model we will have two O(3) mixed symmetry wave functions.So in the case of L = 1 we have SU (6) × O(3) × SU (3) c and write (70, 1 − ).This multiplet has negative parity.

TABLE V :
Coefficient of Ghew-Mandelstam functions for the different diquarks.