Diquark approach to calculating the mass and stability of multiquark states
DDepartment of Physics, erdowsi University of Mashhad,
A. R. Haghpayma
F Mashhad, Iran
Do other multiquark hadrons exist 4q , 6q ,7q , ... , Nq, ?is there an upper limit for N? study of these issues will deepen our understandingof the low- energy sector of QCD.Thus we needand concepts concepts such .??
Can low - energy QCD describe the underlying dynamical forces between quarks and gluons in multiquark states and generate their mass and width correctly? ? It is very difficult to calculate the whole hadron spectrum from first principles in QCD, under such a circumstance various models which are QCD - based or incorporate some important properties of QCD were proposed to explain the hadron spectrum and other low-energy properties. For more details, see Ref in which the papers are explained further to some extent.
In the high temperature case, it is evident that strong interacting matter exists in the form of a quark - gluon plasma, but at low temperatures where the baryon density is high, bosonic states, diquarks and even longer quark clusters may play an important role, at this region of energy the diquarks form a Bose condensate and also there is a possibility of a pentaquark.
At 2 Gev/200 MeV hadronic systems can undergo a phase transition from the hadronic to a deconfined state of quarks and gluons, quark - gluon plasma (QGP) where quarks behave like free
Thus diquarks may play an important role in hadronic physics particularly near the phase transitions ( chiral, deconfinement points ).
In fact the notion of diquark is as old as the quark model itself, for example Gell - Mann mentioned the possibility of diquarks in his of a constituent diquark and quark by Ida and Kobayashi and Lichtenberg and Tassie up to now many articles have been written on this subject. [4] [5] [6,7]
The microscopic origin of diquarks is not completely clear and its connection with the fundamental theory (QCD) not fully understood, apart from the cold asymptotic high dense baryon case, in which the quarks form a fermi surface and perturbative gluon interactions support the existence of a diquark BCS state known as colour superconductor.
There is hyperfine interaction (HF) between quarks in a diquark which indeed is induced by instantons, and besides of these interactions there is confinement interaction between them. The hyperfine energy lies between the deconfinment and the asymptotic freedom regions. lattice QCD (LQCD) determinations of baryon charge distributions do not support the concept of substantial u - d scalar diquark clustering as an appropriate description of the internal structure of the baryons. Furthermore,
Diquarks may play an important role in hadronic physics phase transitions ( chiral, deconfinementbaryon charge diquark clustering as an appropriate description of the internalthus vector diquarks are more favourable. chiral limit diquark between quarks in a vector diquark we calculated the mass of two( pentaquark and H dibaryon ), also by using simultaneously calculated their decay all multiquarkin the mass region 1480 Mev
Although the diquark approach in our model is extendable to all possible multiquark states, we consider the pentaquark and H Θ + dibaryon as two examples of multiquark states and use the diquark approach for calculating their mass and decay width simultaneously. original paper on quarks. Effective field theories are not just models, they represent very general principles such as analyticity, unitarity, cluster decompositionof quantum field theory and the symmetries of the systems. For example chiral perturbation theory ( PT ) describes the low - energy behaviour of QCD within the framework of an effective theory(at least in the meson sector), χ With a conventional PT if we consider group theoretical clustering between quarks and hyperfine QCD interactions between them, we χ χ [15] have an correlated PT between quarks or correlated perturbative chiral quark model CP QM. In fact this theory describes correctly the physics of QCD between the confinement scale and the chiral symmetry breaking scale. [14]
In other case we are working with non - correlated theories.
If we discuss about an correlated P QM with explicit symmetry on configuration, the hyperfin interactions ( FS , CS ) would be considered χ on the n-quark subsystems( diquarks,...), and a special form of confining The paper suggest that the baryon is a composed state of two vector diquarks and a single antiquark, the spatially wave function of these diquarks has a p - wave and a s - wave in angular momentum in the first and second versions of our model respectively. + ∗ fermi gas. From the description of baryons as composed χ respectively, QCD itself does not exclude the existence of the nonconventional − − [1] Θ + The limit on the width is very low Θ MeV << Γ B Γ ≃ for conventional baryon resonances and in quark model this picture is less clear. [16] the overlap of a displaced scalar diquark source and the positive parity ground state for pentaquark is 0.03, indicating that vector diquarks maybe [9] more favourable.Both the LQCD ad QCD sum rule results reject existence of a low-lying positive parity state for . this paper calculate the mass Θ + [10] Θ + [3] [8] Abstract.
1 A.R. Haghpayma.: Diquark Approach to Calculating ...
Diquark approach to calculating the mass and stability of multiquark states ∗ Here, pentaquark states are studied in a correlated perturbative chiral quark model (CP QM) Hamiltonian.Any pentaquark state can be formally decomposed in combinations of simpler coulour-singlet clusters, the energyof the state is computed with the state matrix element of the Hamiltonian. χ Θ It is well known that the multiquarks cannot be a simple n-quark states in ground state, because they would freely recombine and decay in baryons and mesons, with a very broad decay width. potential. [2] for the mass and width of which are in good agreement Γ Θ + MeV ≃ M 1540 MeV and ≃ Θ + with experimental limits. and decay width of two interesting multiquark states by using of vector diquarks. for a recent review see e.g. , and for some examples . [13] [12][11] Θ + ***e - mail: [email protected] he [ 2 ] flavour symmetry of each diquark leads to [ 22 ] flavour symmetry for q and the [ 2 ] spin symmetry of each diquark leads to [ 22 ] and [31] spin symmetry for q in the first version and second version of the model respectively. The colour symmetry of each diquark is [ 11 ] and for the first version of the model we assume [ 2 ] for one of the diquark pairs, this leads to [ 211] colour symmetry for q .The orbital symmetry of each diquark is [ 2 ] and for the first version we assume [ 11 ] for one of the diquark pairs, this leads to [ 31 ] and [ 4 ] orbital symmetry for q in the first and second version of the model respectively. f f − ss s ccc 4 oo oo As the result of these considerations we would have for q contribution of quarks ,( [ 1111 ] and [ 4 ] ) and ( [ 211 ] ,[ 31 ] ) for the first and second versions ; and [ 1111 ] for q is the same for the two versions and lead to a totally antisymmetric wave function for q due to pauli principle. o c f s f s o c o cf s The flavour symmetry contribution for q configuration comes from the decomposition formula for q q in which we have two 6 the second one is used in J W model and the first one is used in our model due to vector diquark contribution of it. with (C = 0 and C =/ 0 ) also the tensor notations for 8 in our model comes from Eq with C = 0 =/ 5 ,C =/ 0 . − f , f = = f = i = = − (3)(4) (2) Inserting 10 tensor notation into symmetry lagrangian leads to SU ( 3 ) symmetry interactions which experimental − f L - f By introducing the quark and antiquark operators and by taking direct product of two diquarks and one antiquark the state can be represented by SU ( 3 ) tensors. Θ + We denote a quark with and antiquark with in which 1,2,3 denote u , d , s and impose normalization relations as and in notation we consider symmetric in upper and lower q i q i = ( q i ,q j ) = δ ij , ( q i ,q j ) = δ ij , ( q i ,q j ) = 0( p q ) , T b ,...,b q a ,...,a p We introduce and ,then for a quark and an antiquark we would havequark 3 :antiquark 3 : where is the levi-civita tensor and we havefor q we haveand for q q we haveThe tensor notation of all of these multiplets can be constructed as S jk √ q j q k q k q j ) = A jk = √ q j q k q k q j ) + − T i T i − ǫ ijk A jk , = ijk = ( T i ,T j ) = 4 δ ij . ( S jk ,S lm ) = δ jl δ km + δ jm δ kl , ( T i ,T j ) = 4 δ ij . ( S jk T i ) = , − ⊗ ⊗ ⊗ ⊕ ⊕ ( ) == )( ⊗ ⊕ ( ) ( ) ⊗ ⊕ ( ) ⊗ ⊕ ( ) ⊗ ⊕ ( ) ⊗⊕ ⊕⊕ = ( ) ( ⊕ ) ⊕ ( ⊕ ) ⊕ ( ) ⊗ ⊗ ⊗ ⊗ [ ] = ( ) [ ⊕ ⊕ ] ⊕ ⊕ ][ ⊕ ⊕ ][ ⊕ ⊕ [ ] ⊗ = ( ) ⊗ ⊕ ( ) ⊗ ⊕ ( ) ⊗ ⊕ ( ) ⊗ = ( ) ⊕ ⊕ ⊕ ( ) ⊕ ( ) ⊕ ⊕ ( ) ⊕ ⊕ T i T i (1)(2) For example the tensor notations for 10 and 8 are The nomenclature for pentaquark states based on hypercharge is as follow −
10 : − T ijk = c √ (cid:0) S ij q k + S jk q i + S ki q j (cid:1) + c √ (cid:0) T ij q k + T jk q i + T ki q j (cid:1) ,P ij = c √ (cid:18) T j q i − δ ij T m q m (cid:19) + c √ (cid:18) Q j q i − δ ij Q m q m (cid:19) + c √ (cid:18) e Q j q i − δ ij e Q m q m + c √ ǫ jab S ia q b + c √ ǫ jab T ia q b + c √ T ijk q k + c √ e T ijk q k + c √ S ijk q k . e Q m = 1 √ T l S ml ,T ij = 1 √ ǫ iab ǫ jcd S ac S bd ,S ijk = √ ǫ ilm ( S jl S km + S kl S jm ) . e T ijk = T i S jk − √ (cid:0) δ ij δ mk + δ ik δ mj (cid:1) e Q m . e Q m = 1 √ T l S ml ,Y = 2 Θ ,Y = 1 N, ∆ , Y = 0 Σ , Λ ,Y = − , Y = − Y = − XT ijk = S jk T i − √ (cid:0) δ ij δ mk + δ ik δ mj (cid:1) Q m (3)(4) which in the notation we have Now the SU ( 3 ) symmetry lagrangian would be T b ,...,b q a ,...,a p Y = p − q + p − q p − q ) ,I = 1( p − q ) − p − q ) .Q = I + Y/ + p + p + p = p and q + q + q = q f L - = g - ǫ ilm T ijk B jl M km + (H.c.) (5) i and f a tensor which is completely indices and traceless on every pair of indices. in which . .. ,, i . evidence of them would be explored and the discovery of them will Thus the tensor notations for 10 in our model comes from Eq . − − Considering flavour configuration of q in Eq(1), one can see that there are 15 and 15 multiplets which coms from 6 6 normal products and this leads to two 45 multiplets for flavour configurations of q q in Eq(2) in which there is octet, decuplet , 27 plet and 35 plet. ⊗ − The SU ( 3 ) configurations of these multiplets are [ 321 ] , [ 411 ][ 42 ] and [ 51 ] and the SU ( 6 ) configurations of them are f f s [ 33111] , [ 321111] , [ 41111] and [ 51111]
560 70 56 700
The tensor notations of 15 and 15 are and respectively and by multiplying to tensor notation of q ( ) we lead to tensor notations of ( 8 , 10 , 27 , 35 ) plets. T jklm = √ S jk S lm + S lk S jm + S jm S kl + S lj S km + S km S jl + S lm S jk ) , S ijk = 1 √ ǫ ilm ( S jl S km + S kl S jm ) . S ijk . T jklm T i − (6) In fact there is noting in quark model to prevent us for constructing such multiplets in which they have vector diquarks and the discovery of them will be evidence supporting the vector diquark approach.
Briefly the spin - flavour - color and parity of our model for the first version and second one are as follows We have considered [ 4 ] and [ 31 ] for the flavour - spin configurations of q in the first and second versions of our model respectively, this leads to [ 51111] and [ 42111] for the flavour - spin configurations of q q , but if one assume the angular momentum 1 for the four quarks in q there are several allowed SU ( 6 ) representations for q q which are ÿ[ 51111 ] , [ 42111 ] , [33111 ] and [ 32211 ] based on [ 4 ] , [ 31 ] , [ 22 ] and [ 211 ] SU ( 6 ) representations for q respectively. (cid:12)(cid:12)(cid:12)
QQQQ ℓ =1 , c , f ¯ q j = , c , f E J Π , c , ( f ⊕ f ) (cid:27)(cid:26) ¯ ⊕ )=( (cid:12)(cid:12)(cid:12) QQQQ ℓ =0 , c , f ¯ q j = , c , f E J Π =( ⊕ ) , c , ( f ⊕ f ) (cid:27)(cid:26) − −
126 210 f s f s
700 1134 − ℓ = s s −
700 1134 560 540 f s f s f s f s (7)(8) and by using of diquark ideas in the chiral limit diquark correlations in the relativistic region and imposing HF interactions between quarks in a diquark,fig.1,we led to introducing a conventional containing Kinetic energy, binding and hyperfine interactions h T ( q ¯ q ) V ( q ¯ q ) V ( q ¯ q ) H ( q ¯ q )= + + bin cs V ( q i q j ) = a ~λ i · ~λ j ( ~r i − ~r j ) bin V ( q i q j ) = − bbbbbbbbbb ~λ i · ~λ j ~S i .~S j bcs Hamiltonian for λ S i i a b . . (9)(10)(11) q q i . j ( ~r i − ~r j ) ~r i ~r j where and are iteraction constantes also and are colour and spinmatrices respectively. binding We have neglected confinement and flavour - spin interactions and considered V as a noncontact interaction.Now in the q q rest frame, we define the internal variables,fig.2 Thus we would have for the Kinetic energy h cs − q q q q ¯ q √ √ q ¯ q rest frame. hbinding ~r = − µ m + 12 ~ + 1 √ ~r = − µ m +12 ~ − √ ~r = − µ m − ~ + 1 √ ~r = − µ m − ~ − √ ~r ¯ q = µm ¯ q R R R R R R R R R R R R R ~~~ ~~~~~~ R R R R T ( q ¯ q ) = ~ ∇ µ + ~ ∇ m + ~ ∇ m + ~ ∇ m R R R R µ is the q , ¯ q reduced mass, m ¯ q m m + m ¯ q . (47) (12) The orbital wave function is ψ m = N [ Y m (ˆ) e − / ][ Y (ˆ) e − / ][ Y (ˆ) e − / ][ Y (ˆ) e − / ] a a R R R R R β β γ R R R R we have . h i R γ h i = 52 R a h i = h i = β R R γ h ~ ∇ i = 2 . R h ~ ∇ i = 5 . R a h ~ ∇ i = h ~ ∇ i = 3 β R R V ( q ) = 53 a ( + ) + 23 V (¯ q ) = 13 a ( + ) + 13 + 43 V ( q ) i = 23 (cid:0) ~S .~S ~S .~S (cid:1) + 16 (cid:0) ~S .~S ~S .~S ~S .~S ~S .~S V (¯ q ) = 13 ~S ( q ) .~S (¯ q ) R R a R q-q ih ih − q-q R R a R a R h cs b + b + + + (cid:1) q-q h i cs − q-q b (13)(14) binbin Now we consider this solution for the second version of our model which = =0, after calculating the mass of a vector diquark from colour-spin interaction M 520Mev and assuming m 450 MeV and T 50MeV the resulted mass of pentaquark would be 1540 MeV. ud ≃ s ≃ ≃ Θ + ℓ = between quarks ǫ ijk − The constituent quark model have not been yet derived from QCD, therefore it is useful to consider the Effective Hamiltonian approach pentaquark where and are the 3 - quark baryon octet and meson octet respectively and is the universal coupling constant. B jl M km g - = Θ + (cid:0) pK − nK + (cid:1) −√ + ..... support the vector diquark approach. A.R. Haghpayma.: Diquark Approach to Calculating ... 2
FIG.1 :Diquark configuration in coordinate space.FIG.2: [13] e suppose that the small Decay widths of is due to tunneling of one of the quarks between the two vector diquarks. Thus in the decayprocess K N a (d) quark tunnels from a diquark (ud) to the other diquark to form a nucleon (udd) and an off-shell (u) quark which is annihilated by the anti- strange quark. Θ + Θ + → + Since the diquark masses ( e.x, scalar or vector, .... ) are smaller than the constituents, they are stable against decay near mass shell, in such a configuration, the diquarks are nearby and "tunneling "of one of the quarks between the two diquarks may take place.
The decay width of this process is Γ Θ + ≃ . e − S g g A πf K | ψ (0) | . (15) Which we have used WKB approximation for the tunneling amplitude and .The is the 1S wave function of quark - diquark at the origin and can be written as E = ( m u + m d − ∆ ψ (0) ψ (0) = 2 a / √ π, (16) ) M ud According to our model the Kinetic energy this leads to 148 and then = < > from the quark model and and 520 MeV in our model.Inserting this values into Eq we findWhich is unusually narrow and camparable with the experimental T ≃ = 3 am ≃
50 MeV a g A = 0 . E = ( m u + m d ) − M ud ∆ M ud ≃ Γ Θ + ≃ MeV < (15) r = h i = 52 R a √ ≃ = g , (17) limit 1 MeV. Γ Θ + (18) ≃ We have calculated the pentaquark decay width for a range of its mass, 1500 Mev Θ + < M Θ + < M Θ + T MeV r Γ Θ + MeV pentauark decay width for a range of its mass. is the
MeV r distance between two vector (ud) diquarks. Table.1: Θ + Now the paper extend the diquark model which previously formulated to describe exotic baryon , to dibaryons . Θ + One of the great open problems of intermediate energy physics is the question of existence or nonexistence of dibaryons. Early theoretical models based on SU(3) and SU(6) symmetries and on Regge theory suggest that dibaryons should exist. There is QCD-based models predict dibaryons with strangeness S = 0, -1, and -2. The invariant masses range between 2 and
3 GeV . The masses and widths of the expected models predict dibaryons and none forbids them. Until now, about 30 years after the first predictions of the S = -2 H-dibaryon by Jaffe this question is still open.
In 1977 a bound six-quark state ( uuddss ),the H -dibaryon,was predicted in a bag-model calcu lation by Jaffe.This state is the lowest SU(3) flavor singlet state with spin zero, strangeness -2 and J P = 0 + where ~ ∇ m R ≃ We calculated the masses of vector diquarks using colour - spin interactions If we take over the results and consider a T = 450 Mev Kinetic energy as the binding energy for the H dibaryon the mass of it would be equal to the sum of the and N masses. = Ξ as HF intraction between quarks in a diquark. We suppose that Decay widths of is due to tunneling of one of the quarks between the two vector diquarks.In the decay process N a (d) quark tunnels from a diquark ud to the other diquark to form a nucleon (udd) and an off-shell (u) quark which forms with the other diquark . → H H Ξ Ξ According to our model the Kinetic energy this leads to 444 and then = < > from the quark model and where 520 MeV in our model.Inserting this values into Eq we find T ≃ = 34 am ≃ a g A = 0 . .75 E = ( m u + m d ) − M ud ∆ M ud ≃ Γ
45 MeV. < (15) r = h i = 52 R a √ ≃ = g , ≃ H (21) (22) we have ~ ∇ m R ≃ Now the paper consider another six-quark state uu-dd-cc ( ) composed of three vector diquarks.
The decay width of the process would be N → H X ++U cc H cc Γ
52 MeV. ≃ H cc (23) Another method for calculating the mass and stability of the pentaquark is based on our diquark - diquark - antiquark approach of our model in which we have used a long - range nonperturbative binding energy for the confinement interaction between two diquarks ( QQ ) ,fig.6, in the where is the binding streanth.In the model the two light quarks compose a bound diquark system in the antitriplet colour state with HF ( CS , FS ) interactions between quark pairs; now we suppose each diquark as a localized colour source in which the light quarks moves; interaction forces between diquarks ( Q s ) then lead to the formation of a two - particle bound state of a QQ system, the scale of which is determined by the quantity 1 / m which is smaller than the QCD scale 1 / ( is 200 400 MeV ).We assume the light antiquark influence on the diquark ( Q ) dynamics is small due to universal nature of diquarks.
Thus we use a shrodinger - like equation for the QQ system as Θ + V ( ) r , r i j = a r i r j −| | a Q Λ Λ
QCD QCD `~ ~ .. ~ ~ (24) , binding Q Q r ¯ q r i r j −| | r = ~ ~ (25) whereand E is the total binding energy of the system, this leads to if we consider for the wave function of the system. The expectation value of potential isNow for the second version of our model we have 520Mev and m = 450 MeV, thus if we consider E = 50 MeV, the resulted mass of pentaquark would be 1540 Mev .Finding ( ) from Eq and inserting it intoThis result is in agreement with our previous one Eq and leads to 1.30 MeV width for pentaquark using of tunneling approach. ψ ( r ) = ψ ( r ) H EH = VT + ( r ) + N e r − ( r )= ψ a a a = < > ud M ≃ s ≃ ≃ Θ + a Θ + ≃ Γ < > = r a (17) E = / m = a V > = 3 / 4 . < > = 0.009 r ≃ leads to for the relative distance of diquarks in the QQ system. (26)(27)(28)(29)(30)(31)(28) Assuming a (CP QM) effective field theory, we suggested that the At the first we constructed the total OCFS symmetry of q q contri baryon is a bound state of two vector diquarks and a single antiquark, the spatially wave function of these diquarks has a P-wave anda S -wave in angular momentum in the first and second version of ourmodel respectively.bution of quarks. Θ + ¯ χ [16,17] the width as narrow as a few MeV. Furthermore, suchchoices of the wave function, which are expected to givea small width, seem quite artificial. In this work, we point MeV -1 Γ(Θ + → K + n ) ≃ ∼ MeV,
Mev Γ Θ + r MeV -1 FIG.3:The calculated width of the negative- parity pentaquark is Θ + shown for the decay K N Θ + → + K N Θ + → + = = We consider H dibaryon, fig.5, which composed of uuddss, three vector ud - ud - ss diquarks, each of which is a colour antitriplet and is symmetric in flavour and spin and orbital space, this leads to a six - quark state which is colour singlet, we ignore the pauli principle for quarks in different diquarks in the limit that diquarks are pointlike, but two quarks in each diquark satisfy this principle. T ( ) = ~ ∇ m R H Now by using of diquark ideas in the chiral limit diquark correlations in the relativistic region and imposing HF interactions between quarks in a diquark, we led to introducing a conventional Hamiltonian h h T ( ) V ( ) H ( + cs H ) = H H `~ binding R ( ud) Q ( ud) Q with (19) The orbital wave function is ψ m = N [ Y m (ˆ) e − / ] a a R RR (20)
We consider the direct product space of the J π = baryon octet with itself in terms of irreducible representations of SU ( ⊗ = ⊕ ⊕ ⊕ ⊕ ⊕ . The hypercharge (Y) ranges from +2 (NN,nucleon-nucleon) to _-2 ( Ξ Ξ ) N for these dibaryon states. The Y = 2 member include the deuteron ofthe 10 - multiplet. The Y = 1 states contain and and the Y= 0 members contain , , and dibaryons. There is several N Λ Σ Λ Λ N Λ Σ ΣΣ Ξ doubly-strange dibaryon states in the SU ( ff directly decay to and may appear as resonances in the and systems .The lowest - order processes in reaction resulted Λ Λ Λ Λ N Ξ K − d → K Λ Λ to H - dibaryon production is shown in fig.4. - p n FIG.4: The lowest-order processes in reaction. K − d → K Λ Λ f [18] Q ss FIG.6 : pentaquark Θ + two vector diquarks which composed of and one antiquark. FIG.5 :H dibaryon three vector diquarks.which composed of For the (1540), particle production is not KKkinematically possible and the motion is non-relativistic + Θ v /c ( = 0.08 for the nucleon and 0.30 for the kaon ).Therefore it should be possible to use the methods of non-relativistic theory based on schroinger equation. .. [19,21] [19,20] [18,22,23,24,25,26,27,28,29,30] − MeV -1 MeV -1 MeV -1
4 Dibaryon states 5 Conclusions
3 A.R. Haghpayma.: Diquark Approach to Calculating ...
Table1 and fig.3. form
Λ Λ H KK Ξ y extendeding our diquark model to dibaryons the paper calculated some estimations of the mass and width of the H dibaryon. QQ system in our model, we calculated the mass and the width of the pentaquark. Our results in this method, for example the average distance between diquarks, are in agreement with our previous conclusions.In an another method by using of a shrodinger - like equation for the.. Θ + Our theoretical results on the mass and width of and H are in agreement with many experimental limits and one can use the vector diquark approach for simultaneously calculatig the mass and width of other multiquark Θ + A. R. Haghpayma, hep-ph/0606214 v1 20 Jun (2006), hep-ph/0606270 v1 26 Jun (2006).
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Int. J. Mod. Phys. E ,995 (2005), hep-ph/0407305 v2 25 Nov (2005). v1 14 Mar (2006)
Eq(2) for pentaquarks in which they have vector diquarks.
Then by using of diquark ideas in the chiral limit diquark correlations in the relativistic region and imposing HF interactions between quarks in a diquark, we led to introducing a conventional Hamiltonian and by h considering its solution for the second version of our model we led to pentaquark Θ + mass.According to these considerations the paper calculated the average distance between two diquarks which leads to a reasonable width for the due to tunneling model. The observed width Θ + Although the 2005 JLab experiment found no pentaquarks in the mass region 1480 MeV m 1700 MeV, our paper predicts that Θ + << for m 1700 MeV its width 53 MeV, and there is a possibility Θ + > Γ Θ + > Γ Θ + ≃ of is unexpectedly narrow, considering that the decay of requires Θ + Θ + no pair creation and goes through a fall-apart process. Attempts have been [31] made to explain the width based on the quark model and QCD sum rule.However most of them suggest the symmetry properties of the orbital - colourspin - flavour wave function. Furthermore, such choises of the wave function seem quite artificial. [32] [33-35][36] J. Bardeen ",Phys. Rev. 108 (5), 1175 (1957). D. Ebert ,
Phys.Rev.C
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