Searching for the quark--diquark systematics of baryons composed by light quarks q=u,d
A.V. Anisovich, V.V. Anisovich, M.A. Matveev, V.A. Nikonov, A.V. Sarantsev, T.O. Vulfs
aa r X i v : . [ h e p - ph ] J a n Searching for the quark–diquark systematics of baryonscomposed by light quarks q = u, d A.V. Anisovich, V.V. Anisovich ∗ , M.A. Matveev,V.A. Nikonov, A.V. Sarantsev and T.O. VulfsOctober 25, 2018 Abstract
Supposing quark–diquark structure of baryons, we look for systematics of baryonscomposed of light quarks ( q = u, d ). We systematize baryons using the notion oftwo diquarks: (i) axial–vector state, D , with the spin S D = 1 and isospin I D = 1and (ii) scalar one, D , with the spin S D = 0 and isospin I D = 0. We consider severalschemes for the composed baryons: (1) with different diquark masses, M D = M D , (2)with M D = M D and overlapping qD and qD states (resonances), (3) with/without SU (6) constraints for low-lying states (with quark–diquark orbital momenta L = 0).In the high-mass region the model predicts several baryon resonances at M ∼ . − . M but different Im M ) in many amplitudes at masses M > ∼ . L = 0), the SU (6) constraint isneeded. The experiment gives us much lesser number of highly excited baryons than the modelwith three constituent quarks predicts. One of the plausible explanation is that the excitedbaryons do not prefer to be formed as three-body systems of spatially separated coloredquarks. Instead, similarly to mesons, they are two-body systems of quark and diquark: q α D α = q α (cid:20) ε αβγ q β q γ (cid:21) . (1)Here ε αβγ is the three-dimensional totally antisymmetrical tensor which works in the colorspace. Below we omit color indices, imposing the symmetry anzatz for the spin–flavor–coordinate variables of wave functions.It is an old idea that a qq -system inside the baryon can be regarded as a specific object– diquark. Thus, interactions with a baryon may be considered as interactions with quark, ∗ [email protected] , and two-quark system, ( qq ): such a hypothesis was used in [1] for the description of thehigh-energy hadron–hadron collisions. In [2, 3, 4], baryons were described as quark–diquarksystems. In hard processes on nucleons (or nuclei), the coherent qq state (composite diquark)can be responsible for interactions in the region of large Bjorken- x values, at x ∼ /
3; deepinelastic scatterings were considered in the framework of such an approach in [5, 6, 7, 8, 9].More detailed considerations of the diquark and the applications to different processes maybe found in [10, 11, 12].Here we suppose that excited baryons are quark–diquark systems. It means that in thespace of three colors ( c ) the excited baryons, similarly to excited mesons, are (cid:18) ¯c ( D ) c ( q ) (cid:19) or (cid:18) ¯c ( D ) c ( q ) (cid:19) systems.The two-particle system has considerably less degrees of freedom than three-particleone and, consequently, much less excited states. At the same time, the comparison ofexperimental data with model calculations [13, 14, 15] demonstrates that the number ofpredicted three-quark states is much larger than the number of observed ones. The aimof this paper is to analyze mechanisms which may reduce the number of predicted excitedstates. Generally, it is the main motivation for the developing of quark–diquark models, seediscussion in [16, 17].Now let us have a look at what type of states appears in qD and qD systems. The qD systems with total spin S = 1 / I = 1 / J P at different orbital momenta L (we restrict ourselves by L ≤ : L = 0 :
12 + L = 2 :
32 + ,
52 + L = 4 :
72 + ,
92 + L = 6 :
112 + ,
132 + L = 1 : − , − L = 3 : − , − L = 5 : − , − (2)The qD systems have quark–diquark total spins S = 1 / , / I = 1 / I = 3 / J P at orbital2omenta L ≤ : L = 0, S = :
12 + L = 0, S = :
32 + L = 2, S = :
32 + ,
52 + L = 2, S = :
12 + 32 + ,
52 + 72 + L = 4, S = :
72 + ,
92 + L = 4, S = :
52 + ,
72 + ,
92 + 112 + , L = 6, S = :
112 + ,
132 + L = 6, S = :
92 + ,
112 + ,
132 + 152 + L = 1, S = : − , − L = 1, S = : − , − , − , L = 3, S = : − , − L = 3, S = : − , − , − − , L = 5, S = : − , − L = 5, S = : − , − , − − (3)Symmetry properties, such as those of the SU (6), lead to certain constraints in the realizationof these states.In Eqs. (2) and (3), the basic states are included only. Actually, every state in (2) and(3) is also characterized by its radial quantum number n = 1 , , , . . . . So, in (2) and (3)every state labelled by J P represents a set of baryons: J P → ( n, J P ) , n = 1 , , , . . . (4)The states with different L and S but with the same ( n, J P ) can mix with each other.However, the meson systematics tell us that L may be considered as a good quantum numberfor q ¯ q systems. Below, in the consideration of quark–diquark models, we use the samehypothesis and characterize qD and qD systems by the orbital momentum L . We alsoconsider total spin S in qD systems as another conserved quantum number, though werealize that it should be regarded as a rough approximation only.Now let us present in a more detail the arguments in favor of a possible realization of thequark–diquark structure of highly excited baryons. We use as a guide the spectral integral(or Bethe–Salpeter equation) for understanding quark–diquark systems considered here –the equation is shown schematically in Figs. 1a,b. If the interaction (a helix-type line inFig. 1b) is flavor-neutral (gluonic or confinement singularity exchange), diquarks retain theirquantum numbers, qD → qD and qD → qD , and the states qD and qD do not mix.In the equation shown in Figs. 1a,b, it was supposed that three-quark intermediate statesare absent. It means that, first, the diquarks should be effectively point-like (diquark formfactors lead to qqq state, see Fig. 1c) and, second, the quark-exchange processes, Fig. 1d, aresuppressed (these processes include three-quark states). Both requirements can be fulfilled,if the diquark size is much less than the baryon one, R diquark ≪ R baryon , that may happenfor highly excited states. Regretfully, we do not know for which states R diquark ≪ R baryon ,thus we consider here several variants. 3 q Dq a) b) Dq Dq c) Dq qDq d) Figure 1: a,b) Equation for quark–diquark system (the flavor-neutral interaction denoted byhelix-type line). c,d) Processes considered as negligibly small in the quark–diquark modelfor highly excited states.The paper is organized as follows.In section 2, we consider wave functions for quark–diquark systems in the nonrelativisticapproximation (the relativization of vertices B → qqq is not difficult, it can be found, forexample, in [17, 18]). In this section, we also demonstrate the way to transform the quark–diquark wave function into three-quark SU (6)-symmetrical one.In sections 3, 4 and 5 we consider different variants of the classification of baryon states.First, on the basis of the absence of N L =0
32 + ( ∼ L =0
32 + ( ∼ L = 0) states to obeythe SU (6) symmetry rules (section 3).In section 4 the model with qD and qD systems is considered in general form and theoverall predictions are given. With an exception for ( L = 0) states, we suggest in section 4the setting of quark–diquark baryons which are in a qualitative agreement with data. Still,some uncertainties exist in the ∆ − sector owing to certain contradictions in data. To beillustrative, we present in this section the ( J, M ) and ( n, M ) plots.In section 5 we consider the model with qD resonances overlapping with those of qD ,with the same J, L, S = 1 /
2. It reduces the number of easily visible bumps, though notsubstantially.We see that even in the quark–diquark model the number of resonances is noticeablylarger than presently observed in the experiment. Also, the model predicts a set of over-lapping resonances, resulting in a hide of some of them in visible bumps. It is a commonprediction inherent in all considered schemes, being therefore a challenge for the experiment.In Conclusion, we summarize the problems which appear in the consideration of thequark–diquark scheme. 4
Baryons as quark–diquark systems
Here, to be illustrative, we consider wave functions of quark–diquark systems, qD and qD ,in the non-relativistic limit. Relativistic generalization of the B → q vertices may be found,for example, in Chapter 7 of [17]. S -wave diquarks and baryons Recall that we have two S -wave diquarks with color numbers ¯c = 3: scalar diquark D andaxial–vector one, D I Z S Z . The diquark spin–flavor wave functions with I D = 1 , S D = 1 and I D = 0 , S D = 0 read as follows: D ( ij ) = u ↑ ( i ) u ↑ ( j ) ,D ( ij ) = 1 √ (cid:18) u ↑ ( i ) u ↓ ( j ) + u ↓ ( i ) u ↑ ( i ) (cid:19) ,D ( ij ) = 1 √ (cid:18) u ↑ ( i ) d ↑ ( j ) + d ↑ ( i ) u ↑ ( j ) (cid:19) ,D ( ij ) = 12 (cid:18) u ↑ ( i ) d ↓ ( j ) + u ↓ ( i ) d ↑ ( j ) + d ↑ ( i ) u ↓ ( j ) + d ↓ ( i ) u ↑ ( j ) (cid:19) ,D ( ij ) = 12 (cid:18) u ↑ ( i ) d ↓ ( j ) − u ↓ ( i ) d ↑ ( j ) − d ↑ ( i ) u ↓ ( j ) + d ↓ ( i ) u ↑ ( j ) (cid:19) . (5) L = 0 In general case, we have the following sets of baryon states:( qD ) J ± ,L,S (=1 / , / ,n , ( qD ) J ± ,L,S (=1 / ,n . (6)Recall that positive and negative parities P = ± are determined by the orbital momentum L between quark and diquark: (+) and ( − ) for even and odd L . The total spin of the quark–diquark states runs S = 1 / , /
2. The states with the same (
I, J P ) may have differentradial excitation numbers n .Here we consider the wave functions of quark–diquark systems with L = 0, namely, ∆ ++3 / ,∆ +3 / and p , as well as ∆ ++1 / , N +3 / and corresponding radial excitations. These examples giveus a guide for writing other wave functions of the quark–diquark states composed by lightquarks ( u, d ). ∆ isobar: quark–diquark wave function for arbitrary n and its trans-formation into the SU (6) wave function The wave function, up to the normalizing coefficient, for Ψ(∆ ++ ↑↑↑ ) with arbitrary n reads u ↑ (1) D (23)Φ (1; 23) + u ↑ (2) D (13)Φ (2; 13) + u ↑ (3) D (12)Φ (3; 12)= u ↑ (1) u ↑ (2) u ↑ (3)Φ (1; 23) + u ↑ (2) u ↑ (1) u ↑ (3)Φ (2; 13) + u ↑ (3) u ↑ (1) u ↑ (2)Φ (3; 12) ≡ Ψ(∆ ++ ↑↑↑ ) . (7)5ere the indices (1,2,3) label the momenta (or coordinates) of quarks. The wave functionsof quarks are symmetrical in the spin, coordinate and flavor spaces. Momentum/coordinatecomponent of the wave function is normalized in a standard way: R | Ψ(∆ ++ ↑↑↑ ) | d Φ = 1,where d Φ is the three-particle phase space.Let us consider n = 1. If the momentum/coordinate wave function of the basic state issymmetrical, Φ (1; 23) = Φ (2; 13) = Φ (3; 12) ≡ ϕ ( sym )1 (1 , , , (8)we have the SU (6) symmetry for ∆ + ↑↑↑ :Ψ SU (6) (∆ ++ ↑↑↑ ) = h u ↑ (1) u ↑ (2) u ↑ (3) + u ↑ (2) u ↑ (1) u ↑ (3) + u ↑ (3) u ↑ (1) u ↑ (2) i ϕ (1 , , u ↑ (1) u ↑ (2) u ↑ (3) 3 ϕ ( sym )1 (1 , , ≡ { u ↑ u ↑ u ↑ } ϕ (1 , , . (9)Here and below we omit the index ( sym ) , i.e. ϕ ( sym )1 (1 , , → ϕ (1 , , ++ ↑ wave function with arbitrary n readsΨ(∆ ++ ↑ ) = C
32 32
12 12 C
32 12
12 12 h u ↑ (1) D (23)Φ (1; 23) + u ↑ (2) D (31)Φ (2; 31)+ u ↑ (3) D (12)Φ (3; 12) i + C
32 32
12 12 C
32 12 − h u ↓ (1) D (23)Φ (1; 23) + u ↓ (2) D (31)Φ (2; 31)+ u ↓ (3) D (12)Φ (3; 12) i . (10)If at n = 1, as previously, the momentum/coordinate wave function is symmetrical, see Eq.(8), we deal with the SU (6) symmetry for basic ∆ + ↑ :Ψ SU (6) (∆ ++ ↑ ) = 1 √ h u ↓ (1) u ↑ (2) u ↑ (3) + u ↑ (1) u ↑ (2) u ↓ (3) + u ↑ (1) u ↓ (2) u ↑ (3) i × ϕ (1 , , . (11)In a more compact form, it readsΨ SU (6) (∆ ++ ↑ ) = 1 √ h u ↓ u ↑ u ↑ + u ↑ u ↑ u ↓ + u ↑ u ↓ u ↑ i ϕ (1 , , ≡ { u ↓ u ↑ u ↑ } ϕ (1 , , . (12)The ∆ + ↑ wave function, Ψ(∆ + ↑ ), is proportional to C
32 12
12 12 C
32 12
12 12 h u ↑ (1) D (23)Φ (1; 23) + u ↑ (2) D (31)Φ (2; 31)+ u ↑ (3) D (12)Φ (3; 12) i + C
32 12
12 12 C
32 12 − h u ↓ (1) D (23)Φ (1; 23) + u ↓ (2) D (31)Φ (2; 31)+ u ↓ (3) D (12)Φ (3; 12) i + C
32 12 − C
32 12
12 12 h d ↑ (1) D (23)Φ (1; 23) + d ↑ (2) D (31)Φ (2; 31) +6 d ↑ (3) D (12)Φ (3; 12) i + C
32 12 − C
32 12 − h d ↓ (1) D (23)Φ (1; 23) + d ↓ (2) D (31)Φ (2; 31)+ d ↓ (3) D (12)Φ (3; 12) i , (13)so we have:Ψ(∆ + ↑ )= 23 (cid:16) u ↑ (1) D (23)Φ (1; 23) + u ↑ (2) D (31)Φ (2; 31) + u ↑ (3) D (12)Φ (3; 12) (cid:17) + √ (cid:16) u ↓ (1) D (23)Φ (1; 23) + u ↓ (2) D (31)Φ (2; 31) + u ↓ (3) D (12)Φ (3; 12) (cid:17) + √ (cid:16) d ↑ (1) D (23)Φ (1; 23) + d ↑ (2) D (31)Φ (2; 31) + d ↑ (3) D (12)Φ (3; 12) (cid:17) + 13 (cid:16) d ↓ (1) D (23)Φ (1; 23) + d ↓ (2) D (31)Φ (2; 31) + d ↓ (3) D (12)Φ (3; 12) (cid:17) . (14)The SU (6)-symmetrical wave function readsΨ SU (6) (∆ + ↑ ) = (cid:16)s { u ↑ u ↑ d ↓ } + s { u ↑ u ↓ d ↑ } (cid:17) ϕ (1 , , , (15)where { q i q j q ℓ } = 1 √ { q i q j q ℓ + q i q ℓ q j + q j q i q ℓ + q j q ℓ q i + q ℓ q i q j + q ℓ q j q i } (16)for q j = q i = q ℓ .Above, to simplify the presentation, we transformed the wave functions to the SU (6)-symmetry ones for n = 1. One can present certain examples with an easy generalization to n >
1. Assuming for n = 1 that ϕ (1 , , ≡ ϕ ( n =1)1 (1 , ,
3) = A (1)1 exp( − b (1) s ) , (17)where s is the total energy squared s = ( k + k + k ) , one may have for n = 2: ϕ ( n =2)1 (1 , ,
3) = A (1)2 exp( − b (1) s )( s − B (1)2 ) . (18)Here B (1)2 is chosen to introduce a node into the ( n = 2) wave function. Likewise, we canwrite down wave functions for higher n . It is not difficult to construct models with wavefunctions of the type (17), (18) – the variants of corresponding models are duscussed insection 2.4. I = 3 / , J = 1 / at L = 0 and S = 1 / , ∆ J P =
12 + ( L = 0 , S = 1 / ↑ J P =
12 + ( L = 0 , S = 1 / I Z = and S Z = as follows: C
32 32
12 12 C
12 12
12 12 h u ↑ (1) D (23)Φ (1; 23) + u ↑ (2) D (31)Φ (2; 31) +7 u ↑ (3) D (12)Φ (3; 12) i + C
32 32
12 12 C
12 12 − h u ↓ (1) D (23)Φ (1; 23) + u ↓ (2) D (31)Φ (2; 31)+ u ↓ (3) D (12)Φ (3; 12) i = − √ h u ↑ (1) u ↑ (2) u ↓ (3) + u ↓ (2) u ↑ (3) √ (1; 23) + (1 ⇀↽
2) + (1 ⇀↽ i + s h u ↓ (1) u ↑ (2) u ↑ (3)Φ (1; 23) + (1 ⇀↽
2) + (1 ⇀↽ i . (19)In the SU (6) limit, the wave functions for ∆ ↑ J P =
12 + ( L = 0 , S = 3 / s only,are equal to zero: − √ h u ↑ (1) u ↑ (2) u ↓ (3) + u ↓ (2) u ↑ (3) √ ⇀↽
2) + (1 ⇀↽ i ϕ ( s )+ s h u ↓ (1) u ↑ (2) u ↑ (3) + (1 ⇀↽
2) + (1 ⇀↽ i ϕ ( s ) = 0 . (20)Radial excitation wave functions in the SU (6) limit, if they depend on s only (for example,see Eq. (18)), are also equal to zero. So, in the SU (6) limit we have for L = 0 the state∆ J P =
32 + only. N / : quark–diquark wave function for arbitrary n and itstransformation into the SU (6) one The S -wave functions for N + ↑ / ( qD ) state with arbitrary n readsΨ + ↑ J =1 / ( qD ) = h u ↑ (1) D (23)Φ (1; 23) + u ↑ (2) D (31)Φ (2; 31)+ u ↑ (3) D (12)Φ (3; 12) i . (21)For the symmetrical momentum/coordinate wave function,Φ (1; 23) = Φ (2; 31) = Φ (3; 12) ≡ ϕ (1 , , , (22)we have: Ψ + ↑ SU (6) ( qD ) = (cid:16)s { u ↑ u ↑ d ↓ } − s { u ↑ u ↓ d ↑ } (cid:17) ϕ (1 , , . (23)Likewise, we can construct a nucleon as qD system – the wave function of N + ↑ J =1 / ( qD )is written at arbitrary n asΨ + ↑ J =1 / ( qD ) = 13 (cid:16) u ↑ (1) D (23)Φ (1; 23) + u ↑ (2) D (31)Φ (2; 31)+ u ↑ (3) D (12)Φ (3; 12) (cid:17) +8 ( − √
23 ) (cid:16) u ↓ (1) D (23)Φ (1; 23) + u ↓ (2) D (31)Φ (2; 31)+ u ↓ (3) D (12)Φ (3; 12) (cid:17) + ( − √
23 ) (cid:16) d ↑ (1) D (23)Φ (1; 23) + d ↑ (2) D (31)Φ (2; 31)+ d ↑ (3) D (12)Φ (3; 12) (cid:17) + 23 (cid:16) d ↓ (1) D (23)Φ (1; 23) + d ↓ (2) D (31)Φ (2; 31)+ d ↓ (3) D (21)Φ (3; 12) (cid:17) . (24)In the limit of Eq. (8), which means the SU (6) symmetry for qD states, we haveΨ + ↑ SU (6) ( qD ) = (cid:16)s { u ↑ u ↑ d ↓ } − s { u ↑ u ↓ d ↑ } (cid:17) ϕ (1 , , . (25)One can see that, if ϕ (1 , , = ϕ (1 , , , (26)we have two different nucleon states corresponding to two different diquarks, D and D .If we require ϕ (1 , ,
3) = ϕ (1 , , , (27)it makes possible to have one level only, not two, that means the SU (6) symmetry.Recall that in the SU (6) limit the nucleon can be presented as a mixture of both diquarks:to be illustrative, we rewrite the spin–flavour part of the proton wave function as follows: s { u ↑ u ↑ d ↓ } − s { u ↑ u ↓ d ↑ } = 1 √ u ↑ (1) D (23) + 13 √ u ↑ (1) D (23) − d ↑ (1) D (23) − u ↓ (1) D (23) + √ d ↓ (1) D (23) . (28)So, the nucleon in the SU (6) limit is a mixture of qD and qD states in equal proportion. I = 1 / , J = 3 / , L = 0 and S = 3 / , N J P =
32 + ( L = 0 , S = 3 / N ↑ J P =
32 + ( L = 0 , S = 3 /
2) with I Z = and S Z = : C
12 12
12 12 C
32 32
12 12 h u ↑ (1) D (23)Φ (1; 23) + u ↑ (2) D (31)Φ (2; 31)+ u ↑ (3) D (12)Φ (3; 12) i + C
12 12 − C
32 32
12 12 h d ↑ (1) D (23)Φ (1; 23) + d ↑ (2) D (31)Φ (2; 31)+ d ↑ (3) D (12)Φ (3; 12) i = − √ h u ↑ (1) u ↑ (2) d ↑ (3) + d ↑ (2) u ↑ (3) √ (1; 23) + (1 ⇀↽
2) + (1 ⇀↽ i s h d ↑ (1) u ↑ (2) u ↑ (3)Φ (1; 23) + (1 ⇀↽
2) + (1 ⇀↽ i . (29)In the SU (6) limit, under the constraint of Eq. (8), the wave function for the ∆ ↑ J P =
12 + ( L =0 , S = 3 /
2) is equal to zero: − √ h u ↑ (1) u ↑ (2) d ↑ (3) + d ↑ (2) u ↑ (3) √ ⇀↽
2) + (1 ⇀↽ i ϕ (1 , , s h d ↑ (1) u ↑ (2) u ↑ (3) + (1 ⇀↽
2) + (1 ⇀↽ i ϕ (1 , ,
3) = 0 . (30)Radial excitation wave functions in the SU (6) limit, if they depend on s only (see Eq. (18)for example), are equal to zero too.So, in the SU (6) limit the nucleon state with L = 0 and J P =
32 + does not exist. L = 0 Let us consider, first, the ∆ isobar at I Z = 3 / J, J Z , total spin S and orbitalmomentum L . The wave function for this state at arbitrary n reads X S Z ,m z C J J Z L J Z − S Z S S Z C S S Z S Z − m z m z C
32 32
12 12 (cid:16) u m z (1) D S Z − m z (23) ×| ~k cm | L Y J Z − S Z L ( θ , φ )Φ ( L )1 (1; 23) + (1 ⇀↽
2) + (1 ⇀↽ (cid:17) . (31)Here | ~k cm | and ( θ , φ ) are the momenta and momentum angles of the first quark in thec.m. system.For other I Z , one should include into wave function the summation over isotopic states,that means the following substitution in (31): C
32 32
12 12 u m z (1) D S Z − m z (23) → X j z C I Z I Z − j z j z q m z j z (1) D I Z − j z S Z − m z (23) . (32)One can see that wave functions of neither(31) nor (32) give us zeros, when Φ ( L )1 (1; 23)depends on s only. Indeed, in this limit we have X S Z ,m z C J J Z L J Z − S Z S S Z C S S Z S Z − m z m z X j z C I Z I Z − j z j z × (cid:16) q m z j z (1) D I Z − j z S Z − m z (23) | ~k cm | L Y J Z − S Z L ( θ , φ ) + (1 ⇀↽
2) + (1 ⇀↽ (cid:17) φ ( L )1 ( s ) . (33)The factor | ~k cm | L Y J Z − S Z L ( θ , φ ) and analogous ones in (1 ⇀↽
2) and (1 ⇀↽
3) prevent thecancelation of different terms in big parentheses of Eq. (33), which are present in case of L = 0, see Eq. (20).For nucleon states ( I = 1 /
2) we write: X S Z ,m z C J J Z L J Z − S Z S S Z C S S Z S Z − m z m z X j z C I Z I Z − j z j z (cid:16) q m z j z (1) D I Z − j z S Z − m z (23) ×| ~k cm | L Y J Z − S Z L ( θ , φ )Φ ( L )1 (1; 23) + (1 ⇀↽
2) + (1 ⇀↽ (cid:17) . (34)10he SU (6) limit, as previously, is realized at Φ ( L )1 ( i ; jℓ ) → ϕ ( L )1 ( s ). Then one has insteadof (34): X S Z ,m z C J J Z L J Z − S Z S S Z C S S Z S Z − m z m z X j z C I Z I Z − j z j z × (cid:16) q m z j z (1) D I Z − j z S Z − m z (23) | ~k cm | L Y J Z − S Z L ( θ , φ ) + (1 ⇀↽
2) + (1 ⇀↽ (cid:17) ϕ ( L )1 ( s ) . (35)For qD states the wave function in the general case reads X m z C J J Z L J Z − m z m z (cid:16) q m z I z (1) D (23) | ~k cm | L Y J Z − m z L ( θ , φ )Φ ( L )0 (1; 23)+(1 ⇀↽
2) + (1 ⇀↽ (cid:17) . (36)In the SU (6) limit we have: X m z C J J Z L J Z − m z m z (cid:16) q m z I z (1) D (23) | ~k cm | L Y J Z − m z L ( θ , φ )+(1 ⇀↽
2) + (1 ⇀↽ (cid:17) ϕ ( L )0 ( s ) . (37)Baryons are characterized by I and J P – the states with different S and L can mix. To selectindependent states, one may orthogonalize wave functions with the same isospin and J P .The orthogonalization depends on the structure of momentum/coordinate parts Φ ( L )1 ( i ; jℓ ).But in case of the SU (6) limit the momentum/coordinate wave functions transform in a com-mon factor Φ ( L )1 ( i ; jℓ ) , Φ ( L )0 ( i ; jℓ ) → ϕ ( L ) SU (6) ( s ), and one can orthogonalize the spin factors.Namely, we can present the SU (6) wave function as follows:Ψ ( A ) J P = Q ( A ) J P ϕ ( A ) SU (6) ( s ) , (38)where Q ( A ) J P is the spin operator and A = I, II, III, ... refer to different (
S, L ). The orthogonalset of operators Q ( A ) J P is constructed in a standard way: Q ( ⊥ I ) J P ≡ Q ( ⊥ A ) J P ,Q ( ⊥ II ) J P = Q ( II ) J P − Q ( ⊥ I ) J P (cid:16) Q ( ⊥ I )+ J P Q ( II ) J P (cid:17)(cid:16) Q ( ⊥ I )+ J P Q ( ⊥ I ) J P (cid:17) ,Q ( ⊥ III ) J P = Q ( III ) J P − Q ( ⊥ I ) J P (cid:16) Q ( ⊥ I )+ J P Q ( III ) J P (cid:17)(cid:16) Q ( ⊥ I )+ J P Q ( ⊥ I ) J P (cid:17) − Q ( ⊥ II ) J P (cid:16) Q ( ⊥ II )+ J P Q ( III ) J P (cid:17)(cid:16) Q ( ⊥ II )+ J P Q ( ⊥ II ) J P (cid:17) , (39)and so on. The convolution of operator (cid:16) Q ( A )+ J P Q ( B ) J P (cid:17) includes both the summation overquark spins and integration over quark momenta. Exploring the notion of constituent quark and composite diquark, we propose several schemesfor the structure of low-lying baryons. To be illustrative, let us turn to Fig. 2.11 (r) rground statea) V(r) rexcited stateb) V(r) rL=1 c)
Figure 2: Illustration of the quark–diquark structure of baryon levels. a) Ground state witha complete mixing of constituent quarks: the bound quarks and diquarks, being compressedstates, provide us the three-quark SU (6)-symmetry structure. b) Conventional picture foran excited state in the standard three-quark model with three spatially separated quarks.c) Example of the excited state in the quark–diquark model: the quark–diquark state with L = 1, quark and diquark being spatially separated.Using potential picture, the standard scheme of the three-quark baryon is shown in Figs.2a and 2b. On the lowest level, there are three S -wave quarks — it is a compact system (theradii of constituent quarks are of the order of ∼ . ∼ . SU (6) classification seems rather reliable.As concern the excited states, the quarks of the standard quark model (see the examplein Fig. 2b), are in the average located at comparatively large distances from each other.Such a three-quark composite system is characterized by pair excitations – the number ofpair excitations may be large, thus resulting in a quick increase of the number of excitedbaryons.The quark–diquark structure of levels, supposed in our consideration for L >
0, is demon-strated in Fig. 2c. For excited states, we assume the following quark–diquark picture: twoquarks are at comparatively small distances, being a diquark state, and the third quark isseparated from this diquark. The number of quark–diquark excitations is noticeably lessthan the number of excitations in the three-quark system.
Constituent quarks and diquarks are effective particles. We assume that propagators of thediquark composite systems can be well described using K¨allen–Lehman representation [20].For scalar and axial–vector diquarks, the propagators readΠ ( D ) ( p ) = ∞ Z m min dm D ρ D ( m D ) m D − p − i , Π ( D ) µν ( p ) = − g ⊥ pµν ∞ Z m min dm D ρ D ( m D ) m D − p − i , (40)12here the mass distributions ρ D ( m D ) and ρ D ( m D ) are characterized by the compactnessof the scalar and axial–vector diquarks. The use of mass propagator (40) is definitely neededin the calculation of subtle effects in baryonic reactions. However, in a rough approximationone may treat diquarks, similarly to constituent quarks, as effective particles: ρ D ( m D ) → δ ( m D − M D ) ,ρ D ( m D ) → δ ( m D − M D ) . (41)We expect the diquark mass to be in the region of the 600-900 MeV [21].Mass distributions for three-quark systems in the approximation of (41) at fixed s = s + s + s − m can be shown on the Dalitz-plot. In Fig. 3, we show the Dalitz-plotsfor the approach of short-range diquarks – conventionally, below we use (41). In Figs. 3aand 3b the cases M D = M D and M D = M D are demonstrated, correspondingly. s s s D , D D , D D , D a) s s s D D D D D D b) Figure 3: Dalitz plots for three-quark systems at a) M D = M D and b) M D = M D The model with spatially separated quark and diquark results in a specific orthogonal-ity/normalization condition. The matter is that interference terms with different diquarksprovide a small contribution. For example, Z d Φ | ~k cm | L Y J Z − S Z L ( θ , φ )Φ ( L )1 (1; 23) × (cid:16) | ~k cm | L Y J Z − S Z L ( θ , φ )Φ ( L )1 (2; 13) (cid:17) + ≃ . (42)Below we neglect such interference terms.Therefore, the normalization condition, Z d Φ (cid:12)(cid:12)(cid:12) Ψ ( L,S ) J,J Z (1 , , (cid:12)(cid:12)(cid:12) = 1 , (43)13o, for the qD systems, we re-write (43) as follows: Z d Φ (cid:12)(cid:12)(cid:12) X S Z ,m z C J J Z L J Z − S Z S S Z C S S Z S Z − m z m z X j z C I,I Z I Z − j z j z q m z j z (1) D I Z − j z S Z − m z (23) ×| ~k cm | L Y J Z − S Z L ( θ , φ )Φ ( L )1 (1; 23) (cid:12)(cid:12)(cid:12) + Z d Φ (cid:12)(cid:12)(cid:12) ⇀↽ (cid:12)(cid:12)(cid:12) + Z d Φ (cid:12)(cid:12)(cid:12) ⇀↽ (cid:12)(cid:12)(cid:12) = 1 , (44)while for qD we have Z d Φ (cid:12)(cid:12)(cid:12) X m z C J J Z L J Z − m z / m z q m z j z (1) D (23) | ~k cm | L Y J Z − m z L ( θ , φ )Φ ( L )1 (1; 23) (cid:12)(cid:12)(cid:12) + Z d Φ (cid:12)(cid:12)(cid:12) ⇀↽ (cid:12)(cid:12)(cid:12) + Z d Φ (cid:12)(cid:12)(cid:12) ⇀↽ (cid:12)(cid:12)(cid:12) = 1 . (45)Here we suppose that L and S are good quantum numbers. If no, one should take intoaccount the mixing in each term of (44). Let us emphasize that under hypothesis (44), (45)the mixing of terms with different diquarks is forbidden. L = 0 and the SU (6) sym-metry This section is devoted to the basic L = 0 states and their radial excitations. But, first, letus overlook the situation with the observed baryons – some of them need comments. The masses of the well-established states (3 or 4 stars in the Particle Data Group classification[22]) are taken as a mean value over the interval given by PDG, with errors covering thisinterval. But the states established not so definitely require special discussion.We have introduced two S states in the region of 1900 and 2200 MeV, which areclassified by PDG as S (2090). Indeed, the observation of a state with mass 2180 ±
80 MeVby Cutkosky [23] can be hardly compatible with observations [24, 25, 26] of an S state withthe mass around 1900 MeV.The same procedure has been applied to the states D around 2000 MeV. Here the firststate is located in the region of 1880 MeV and was observed in the analyses [23, 24, 27].This state is also well compatible with the analysis of photoproduction reactions [28]. Thesecond state is located in the region 2040 MeV: its mass has been obtained as an averagevalue over the results of [23, 25, 29].The P (1880) state has been observed by Manley [24] as well as in the analyses ofthe photoproduction data with open strangeness [28, 30]: we consider this state as wellestablished. Thus, for the state P (2100) we have taken the mass as an average value overall the measurements quoted by PDG: [23, 25, 26, 29, 31, 32].We also consider D (2070) as an established state. It has been observed in the η photo-production data [33], although we understand that a confirmation of this state by other data14s needed. Furthermore, we have taken for D (2200) the average value, using [23, 29, 32]analyses which give compatible results.As to the ∆ sector, we see that the − state observed in the analysis of the GWU group[34] with mass 2233 MeV and quoted as D (1930) can be hardly compatible with otherobservations, which give the results in the region of 1930 MeV. Moreover, the GWU result iscompatible with the analysis of Manley [24], which is quoted by PDG as D (2350), thoughit gives the mass 2171 ±
18 MeV. Thus, we introduce the ∆( − ) state with the mass 2210MeV and the error which covers both these results. Then, the mass of the D (2350) stateis taken as an average value over the results of [23, 25, 29].We also consider the D (1940) state, which has one star by PDG classification, as anestablished one. It is seen very clearly in the analysis of the γp → π ηp data [35, 36].One of the most interesting observations concerns ∆(
52 + ) states. The analyses of Manley[24] and Vrana [25] give a state in the region 1740 MeV with compatible widths. However,this state was confirmed neither by πN elastic nor by photoproduction data. This state islisted by PDG as F (2000) together with the observation [23] of a state in the elastic πN scattering at 2200 MeV. Here we consider these results as a possible indication to two states:one at 1740 MeV and another at 2200 MeV. We consider N J P =
12 + and ∆ J P =
32 + states in two variants:( ) M D = M D (see Fig. 3a), the SU (6) symmetry being imposed, and( ) M D = M D (see Fig. 3b) with the broken SU (6) symmetry constraints. SU (6) symmetry for the nucleon N
12 + (940) , isobar ∆
32 + (1238) and theirradial excitations
In this way we assume M D = M D (see Fig. 3a) and suppose the SU (6) symmetry for thelowest baryons with L = 0. It gives us two ground states, the nucleon N
12 + (940) and theisobar ∆
32 + (1238) as well as their radial excitations, see section 2: L = 0 S = , N (
12 + ) S = , ∆(
32 + ) n = 1 938 ± ± n = 2 1440 ±
40 1635 ± n = 3 1710 ± ∼ n = 4 1900 ± ∼ δ n M ( N
12 + ), isof the order of 1 . ± .
15 GeV . This value is close to that observed in meson sector [17, 37]: M [ N
12 + (1440)] − M [ N
12 + (940)] ≃ M [ N
12 + (1710)] − M [ N
12 + (1440)] ≡ δ n M ( N
12 + ) ≃ . . (47)The state with n = 4 cannot be unambiguously determined. Namely, in the region of 1880MeV a resonance structure is seen, which may be either nucleon radial excitation ( n = 4)15r ( S = 3 / , L = 2 , J P = 1 / + ) state. Also it is possible that in the region ∼ M plane and is notobserved yet.One can see that the mass-squared splitting of the ∆
32 + isobars, δ n M (∆
32 + ), coincideswith that of a nucleon, δ n M ( N
12 + ), with a good accuracy: δ n M (∆
32 + ) = 1 . ± . . (48)Let us emphasize that two states, ∆
32 + (1600) and ∆
32 + (1920), are considered here as radialexcitations of ∆
32 + (1232) with n = 2 and n = 3. However, the resonances ∆
32 + (1600) and∆
32 + (1920) can be reliably classified as S = 1 / , L = 2 and S = 3 / , L = 2 states, with n = 1 (see Section 4). Actually, it means that around ∼ ∼ SU (6) symmetry, M D = M D Here, we consider alternative scheme supposing diquarks D and D to have different masses,thus being different effective particles – arguments in favor of different effective masses of D and D may be found in [5, 21].In the scheme with two different diquarks, D and D , we have two basic nucleons withcorresponding sets of radial excitations.The first nucleonic set corresponds to the qD states, the second one describes the qD states: L = 0 S = , N (
12 + ) S = , N (
12 + ) S = , ∆(
32 + ) n = 1 938 ± ±
40 1232 ± n = 2 1440 ±
40 1710 ±
30 1635 ± n = 3 1710 ±
30 2100 ± ∼ n = 4 2100 ± ∼ M ∼ D and D .The ( L = 0) set of isobar states coincides with that defined in the ( M D = M D ) scheme. L > as ( qD , qD ) states Considering excited states, we analyze several variants, assuming M D = M D for the L ≥ SU (6) constraints for L = 0 ones.In Fig. 4 we demonstrate the setting of baryons on the ( J P , M ) planes. We see reason-ably good description of data, although the scheme requires some additional states as wellas double pole structures in many cases.The deciphering of baryon setting shown in Fig. 4 is given in (50), (51), (52), (53) – themass values (in MeV units) are taken from [22, 38, 39, 40].16 ) - (J D L=1 L=3 L=5 J P M ) + (J D L=0
L=2 L=4 J P M ) - N(J
L=1 L=3 L=5 J P M ) + N(J
L=0
L=2 L=4 J P M Figure 4: ( J P , M ) planes for baryons at M D = M D with the SU (6) constraints for L = 0states. Solid and dashed lines are the trajectories for the states with S = 1 / S = 3 / n = 1) with S = 1 /
2; stars and rombs: ground states ( n = 1) with S = 3 /
2; circles: radially exited states ( n >
1) with S = 1 / , / J P tra-jectories have δJ ± = 2 ± and M J +2) ± − M J ± ≃ . Clear examples give us ∆(
32 + )trajectory (the states ∆
32 + (1231), ∆
72 + (1895), ∆
112 + (2400), ∆
132 + (2920) ) and N ( − ) trajec-tory (the states N − (1670), N − (2250), N − (2270)). At the same time, in Fig. 4 we seethe lines with δJ ± = 1 ± and M J +2) ± − M J ± ≃ : actually, such a line represents twooverlapping trajectories.For better presentation of the model, let us re-draw the ( J, M ) planes keeping the basic( n = 1) states only – they are shown in Fig. 5.For L = 0 we see two basic states: N (1 / + ) and ∆(3 / + ).In the I = 3 / L = 1 and six states for each L at L > L = 1 , I = 1 /
2) states we have five basic states with J P = − , − , J P =17 ) - (J D L=1 L=3 L=5 J P M ) + (J D L=0
L=2 L=4 J P M ) - N(J
L=1 L=3 L=5 J P M ) + N(J
L=0
L=2 L=4 J P M Figure 5: (
J, M ) planes for baryons at M D = M D within the SU (6) constraints for wavefunctions of the L = 0 states – the basic states are shown only (notations are as in Fig. 4). − , − , − , J P = − , − , while for the states with L > J P = ( L + ) P , ( L − ) P , J P = ( L + ) P , ( L + ) P , ( L − ) P , ( L − ) P , J P = ( L + ) P , ( L − ) P .The states belonging to the same trajectory have δJ ± = 2 ± and M J +2) ± − M J ± ≃ . One pair of states with S = and I = is formed by the D diquark – we cannotassert definitely what pair is built by D , heavier or lighter one.18 .1 The setting of N ( J + ) states at M D = M D and L ≥ In this sector the trajectories of Fig. 4 give us the following states at L ≥ L = 2 S = N (
32 + ) N (
52 + ) n = 1 (1720 ±
30) (1685 ± n = 2 ∼ ( ∗ ) (2000 ± n = 3 ∼ ( ∗ ) ∼ ( ∗ ) n = 4 ∼ ( ∗ ) ∼ ( ∗ ) S = N (
32 + ) N (
52 + ) n = 1 ∼ ( ∗ ) ∼ n = 2 ∼ ( ∗ ) ∼ ( ∗ ) n = 3 ∼ ( ∗ ) ∼ ( ∗ ) n = 4 ∼ ∼ ( ∗ ) S = N (
12 + ) N (
32 + ) N (
52 + ) N (
72 + ) n = 1 (1880 ±
40) (1915 ± ∼ ± n = 2 ∼ ∼ ∼ ∼ n = 3 ∼ ∼ ∼ ( ∗ ) ∼ ( ∗ ) n = 4 ∼ ∼ ∼ ( ∗ ) ∼ ( ∗ ) L = 4 S = N (
72 + ) N (
92 + ) n = 1 ∼ ± n = 2 ∼ ∼ ( ∗ ) n = 3 ∼ ( ∗ ) ∼ ( ∗ ) n = 4 ∼ ( ∗ ) ∼ ( ∗ ) S = N (
72 + ) N (
92 + ) n = 1 ∼ ∼ ( ∗ ) n = 2 ∼ ( ∗ ) ∼ ( ∗ ) n = 3 ∼ ( ∗ ) ∼ ( ∗ ) S = N (
52 + ) N (
72 + ) N (
92 + ) N (
112 + ) n = 1 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ∼ n = 2 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ∼ n = 3 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ( ∗ ) ∼ ( ∗ ) L = 6 S = N (
112 + ) N (
132 + ) n = 1 ∼ ± n = 2 ∼ ∼ S = N (
112 + ) N (
132 + ) n = 1 ∼ ∼ n = 2 ∼ ∼ S = N (
92 + ) N (
112 + ) N (
132 + ) N (
152 + ) n = 1 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ∼ n = 2 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ∼ ( ∗ ) means that in this mass region we should have two poles.We see reasonable agreement of our predictions with data. N ( J − ) states at M D = M D In the N ( J − ) sector the lightest states have L = 1, and we see that these states are inagreement with model predictions. But let us stress that the scheme requires a series of19adial excitation states at J P =
12 + ,
32 + ,
52 + at M ≃ L = 1 S = N ( − ) N ( − ) n = 1 (1530 ±
30) (1524 ± n = 2 (1905 ±
60) (1875 ± n = 3 (2180 ± ( ∗ ) (2160 ± ( ∗ ) n = 4 ∼ ( ∗ ) ∼ ( ∗ ) S = N ( − ) N ( − ) n = 1 ∼ ∼ n = 2 ∼ ( ∗ ) ∼ ( ∗ ) n = 3 ∼ ( ∗ ) ∼ ( ∗ ) n = 4 ∼ ( ∗ ) ∼ ( ∗ ) S = N ( − ) N ( − ) N ( − ) n = 1 (1705 ±
30) (1740 ±
20) (1670 ± n = 2 ∼ ∼ ∼ n = 3 ∼ ∼ ( ∗ ) ∼ ( ∗ ) n = 4 ∼ ∼ ( ∗ ) ∼ ( ∗ ) L = 3 S = N ( − ) N ( − ) n = 1 (2150 ±
80) (2170 ± n = 2 ∼ ( ∗ ) ∼ ( ∗ ) n = 3 ∼ ( ∗ ) ∼ ( ∗ ) n = 4 ∼ ( ∗ ) ∼ ( ∗ ) S = N ( − ) N ( − ) n = 1 ∼ ( ∗ ) ∼ ( ∗ ) n = 2 ∼ ( ∗ ) ∼ ( ∗ ) n = 3 ∼ ( ∗ ) ∼ ( ∗ ) S = N ( − ) N ( − ) N ( − ) N ( − ) n = 1 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ± n = 2 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ∼ n = 3 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ( ∗ ) ∼ ( ∗ ) L = 5 S = N ( − ) N ( − ) n = 1 ∼ ∼ n = 2 ∼ ( ∗ ) ∼ ( ∗ ) n = 3 ∼ ( ∗ ) ∼ ( ∗ ) S = N ( − ) N ( − ) n = 1 ∼ ( ∗ ) ∼ ( ∗ ) n = 2 ∼ ( ∗ ) ∼ ( ∗ ) S = N ( − ) N ( − ) N ( − ) N ( − ) n = 1 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ∼ n = 2 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ∼ .3 The setting of ∆( J + ) states at M D = M D and L ≥ Considering the equation (52), we should remember that the ( L = 0) states are excludedfrom the suggested classification – they are given in section 3. L = 2 S = ∆(
32 + ) ∆(
52 + ) n = 1 ∼ ∼ n = 2 ∼ ∼ n = 3 ∼ ∼ n = 4 ∼ ∼ S = ∆(
12 + ) ∆(
32 + ) ∆(
52 + ) ∆(
72 + ) n = 1 (1895 ±
25) (1990 ± ( ∗ ) (1890 ±
25) (1895 ± n = 2 ∼ ∼ ( ∗ ) ∼ ∼ n = 3 ∼ ∼ ( ∗ ) ∼ ( ∗ ) ∼ ( ∗ ) n = 4 ∼ ∼ ( ∗ ) ∼ ( ∗ ) ∼ ( ∗ ) L = 4 S = ∆(
72 + ) ∆(
92 + ) n = 1 ∼ ∼ n = 2 ∼ ∼ n = 3 ∼ ∼ n = 4 ∼ ∼ S = ∆(
52 + ) ∆(
72 + ) ∆(
92 + ) ∆(
112 + ) n = 1 ∼ ( ∗ ) (2390 ± ( ∗ ) (2400 ±
50) (2420 ± n = 2 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ∼ n = 3 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ( ∗ ) ∼ ( ∗ ) n = 4 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ( ∗ ) ∼ ( ∗ ) L = 6 S = ∆(
112 + ) ∆(
132 + ) n = 1 ∼ ∼ n = 2 ∼ ∼ n = 3 ∼ ∼ n = 4 ∼ ∼ S = ∆(
92 + ) ∆(
112 + ) ∆(
132 + ) ∆(
152 + ) n = 1 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ± n = 2 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ∼ n = 3 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ∼ n = 4 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ∼ ∆( J − ) states at M D = M D In the ∆( J − ) sector we face a problem with the − state observed in the analysis [34] withmass 2233 MeV and quoted as D (1930) [22]. However, it can be hardly compatible withother observations, which are in the region of 1930 MeV. In addition, the result of [34]is compatible with the analysis of Manley [24], quoted by PDG as D (2350). Thus, weintroduce the ∆( − ) state with the mass 2210 MeV and with the error which covers theresults of [24, 34]. Then, the mass of the D (2350) state is taken as an average value overthe results of [23, 25, 29]. 21 = 1 S = ∆( − ) ∆( − ) n = 1 (1650 ±
25) (1640 ± n = 2 (1900 ±
50) (1990 ± n = 3 (2150 ± ∼ n = 4 ∼ ∼ S = ∆( − ) ∆( − ) ∆( − ) n = 1 ∼ ∼ ± n = 2 ∼ ∼ ∼ n = 3 ∼ ∼ ( ∗ ) ∼ ( ∗ ) n = 4 ∼ ∼ ( ∗ ) ∼ ( ∗ ) L = 3 S = ∆( − ) ∆( − ) n = 1 ∼ ± n = 2 ∼ ∼ n = 3 ∼ ∼ S = ∆( − ) ∆( − ) ∆( − ) ∆( − ) n = 1 ∼ ( ∗ ) (2350 ± ( ∗ ) ∼ ∼ n = 2 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ∼ n = 3 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ( ∗ ) ∼ ( ∗ ) L = 5 S = ∆( − ) ∆( − ) n = 1 (2633 ± ∼ n = 2 ∼ ∼ n = 3 ∼ ∼ S = ∆( − ) ∆( − ) ∆( − ) ∆( − ) n = 1 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ± n = 2 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ∼ n = 3 ∼ ( ∗ ) ∼ ( ∗ ) ∼ ∼ In equations (50), (51), (52), (53), the overlapping resonances are denoted by the symbol ( ∗ )– the model predicts a lot of such states. Decay processes lead to a mixing of the overlappingstates (owing to the transition baryon (1) → hadrons → baryon (2)). It results in a specificphenomenon, that is, when several resonances overlap, one of them accumulates the widthsof neighbouring resonances and transforms into the broad state, see [17, 41] and referencestherein.This phenomenon had been observed in [42, 43] for meson scalar–isoscalar states andapplied to the interpretation of the broad state f (1200 − q ¯ q systematics. Indeed, being among q ¯ q resonances, the exoticstate creates a group of overlapping resonances. The exotic state, after accumulating the”excess” of widths, turns into a broad one. This broad resonance should be accompaniedby narrow states which are the descendants of states from which the widths have beenborrowed. In this way, the existence of a broad resonance accompanied by narrow ones maybe a signature of exotics. This possibility, in context of searching for exotic states, wasdiscussed in [46]. 22n the considered case of quark–diquarks baryons (equations (50), (51), (52), (53)), wedeal mainly with two overlapping states: it means that we should observe one comparativelynarrow resonance together with the second one which is comparatively broad. Experimentalobservation of the corresponding two-pole singularities in partial amplitudes looks as ratherintricate problem. M D = M D on ( n, M ) planes Equations (50), (51), (52), (53) allow us to present the setting of baryons ( M D = M D ) on( n, M ) planes – they are shown in Figs. 6 and 7.We have three trajectories on the ( n, M ) plot for N (
12 + ) with the starting states shownin Fig. 5 in the L = 0 group. The low trajectory is filled in by the states of equation (49).In the plot for N (
32 + ) we also have three trajectories, with the starting L = 2 states, seeFig. 5. Likewise, all other ( n, M ) plots are constructed: the starting states are those shownin Fig. 5.We have a lot of predicted radial excitation states, though not many observed ones – thematter is that in case of overlapping resonances the broad state is concealed under narrowone. As is well known, the mixing states repulse from one another. The mixing of overlappingstates, due to the transition into real hadrons baryon (1) → real hadrons → baryon (2) alsoleads to the repulsing of resonance poles in the complex- M plane: one is moving to real M axis (i.e. reducing the width), another is diving deaper into complex- M plane (i.e. increasingthe width) – for more detail see [17], sections 3.4.2 and 3.4.3. To separate two overlappingpoles, one needs to carry out the measurements of decays into different channels – differentresonances have, as a rule, different partial widths, so the total width of the ”two-poleresonance” should depend on the reaction type.Radially excited states are seen in (see Figs. 6 and 7), namely, N (
12 + ) sector (four states on the lowest trajectory), N (
52 + ) sector (two states on the lowest trajectory), N ( − ) sector (three states on the lowest trajectory), N ( − ) sector (three states on the lowest trajectory),∆(
32 + ) sector (three states on the lowest trajectory),∆( − ) sector (three states on the lowest trajectory),∆( − ) sector (two states on the lowest trajectory),∆( − ) sector (two states on the lowest trajectory).However, in Figs. 6 and 7 we do not mark specially the resonances which are ”shadowed”by observed ones. The slopes of all trajectories in Figs. 6 and 7 coincide with each otherand with those of meson trajectories (see [17, 37]).23 N( ) + M n N( ) + M n N( ) + M n N( ) + M n N( ) - M n N( ) - M n N( ) - M n N( ) - M n Figure 6: ( n, M ) planes for N ( J ± ) states, M D = M D .24 ( ) D + M n ( ) D + M n ( ) D + M n ( ) D + M n ( ) D - M n ( ) D - M n ( ) D - M n ( ) D - M n Figure 7: ( n, M ) planes for ∆( J ± ) states, M D = M D .25 A variant with M D = M D and overlapping qD ( S =1 / and qD ( S = 1 / states Here we consider the case with further decrease of states which can be easily seen. Wesuppose that M D = M D and the states qD ( S = 1 /
2) and qD ( S = 1 /
2) overlap. So, fora naive observer, who does not perform an analysis of double pole structure, the number ofstates with S = 1 / J, M ) plots as they look like for ”naive observers”, while Fig. 9demonstrates the ( J, M ) plots for ground states ( n = 1) only. ) - (J D L=1 L=3 L=5 J P M ) + (J D L=0
L=2 L=4 J P M ) - N(J
L=1 L=3 L=5 J P M ) + N(J
L=0
L=2 L=4 J P M Figure 8: Baryon setting on ( J P , M ) planes in the model with overlapping qD ( S = 1 / qD ( S = 1 /
2) states (notations are as in Fig. 4).26 ) - (J D L=1 L=3 L=5 J P M ) + (J D L=0
L=2 L=4 J P M ) - N(J
L=1 L=3 L=5 J P M ) + N(J
L=0
L=2 L=4 J P M Figure 9: Setting of basic baryons ( n = 1) on ( J P , M ) planes in the model with overlapping qD ( S = 1 /
2) and qD ( S = 1 /
2) states (notations are as in Fig). 4.Figures 8, 9 present us more compact scheme than that given in Figs. 4, 5.
We have systematized all baryon states in the framework of the hypothesis of their quark–diquark structure. We cannot say whether such a systematization is unambiguous, so wediscuss possible versions of setting baryons upon multiplets. To carry out a more definitesystematization, additional efforts are needed in both experiment and phenomenologicalcomprehension of data.Concerning the experiment, it is necessary:(i) To investigate in details the ∆ spectrum in the region around 1700 MeV. Here one shouldsearch for the D and/or F states. The double pole structures should be searched for,first, in the regions N
12 + (1400) and ∆
32 + (1600).27ii) To increase the interval of available energies in order to get a possibility to investigateresonances up to the masses 3.0–3.5 GeV.(iii) To measure various types of reactions in order to analyze them simultaneously.As to the phenomenology and theory, it is necessary to continue the K -matrix analysis,the first results of which were obtained in [35, 47], in order to cover a larger mass intervaland the most possible number of reactions. One should take into account the expectedoverlapping of resonances. Namely, the standard procedure should be elaborated for singlingout the amplitude poles in the complex- M plane in case when one pole is under another.We thank L.G. Dakhno and E. Klempt for helpful discussions. This paper was supportedby the grants RFBR No 07-02-01196-a and RSGSS-3628.2008.2. References [1] V.V. Anisovich, Pis’ma ZhETF , 439 (1965) [JETP Lett. , 272 (1965)].[2] M. Ida and R. Kobayashi, Progr. Theor. Phys. , 846 (1966).[3] D.B Lichtenberg and L.J. Tassie, Phys. Rev. , 1601 (1967).[4] S. Ono, Progr. Theor. Phys.
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