πN and ππN Couplings of the Δ(1232) and its Chiral Partners
aa r X i v : . [ h e p - ph ] A p r πN and ππN Couplings of the ∆(1232) and its Chiral Partners
K. Nagata , ∗ A. Hosaka , and V. Dmitraˇsinovi´c Department of Physics, Chung-Yuan Christian University,Chung-Li 320, Taiwan Research Center for Nuclear Physics, Osaka University, Mihogaoka 10-1, Osaka 567-0047, Japan Vinˇca Institute of Nuclear Sciences, lab 010, P.O.Box 522, 11001 Beograd, Serbia (Dated: November 17, 2018)We investigate the interactions and chiral properties of the four spin- -baryons : N − ( D ), N + ( P ), ∆ + ( P ) and ∆ − ( D ) together with the nucleon. We construct the SU (2) R × SU (2) L invariant interactions between the spin- and - baryons with the aid of a new, specially developedspin and isospin projection technique for these baryon fields, where the chiral invariant interactionscontain one- and two-pion couplings. We obtain simple relations for the coupling constants of theone- and two-pion spin − transitions terms. The relation for the one-pion interactions reasonablyagrees with the experiments, which suggests that these spin- baryons are chiral partners. PACS numbers: 13.30.Eg, 14.65.Bt, 12.39.Fe
Chiral symmetry SU (2) R × SU (2) L is a key propertyof the strong interaction. When the spontaneous break-down SU (2) R × SU (2) L → SU (2) V occur, the brokensymmetry plays a dynamical role in various scatteringprocesses involving its Nambu-Goldstone bosons, i.e. thepions. Hadrons are then classified by isospin multipletsof the residual symmetry. If chiral symmetry is restoredat high temperature or density, hadrons should form de-generate multiplets as classified by the full chiral grouprepresentations ( I R , I L ), where I R ( I L ) is isospin for the SU (2) R ( SU (2) L ) group. Even in the broken world wemay expect particles are expressed by one of chiral mul-tiplets or by their simple superpositions [1]. A familiarexample is for the chiral mesons ( σ, ~π ) and for the vec-tor mesons ( ~ρ, ~a ). With regards to the baryons, the roleof chiral symmetry in the classification has not been ex-plored much, as one does not know which hadrons wouldform which chiral multiplets as degenerate partners.The linear realization of the chiral symmetry offers twomerits. First, hadrons in the same chiral multiplet butwith different isospins are related by the larger symme-try SU (2) R × SU (2) L than SU (2) V . This can help re-ducing the number of the free parameters. Second itis easy to investigate the property changes towards thechiral restoration as functions of the chiral condensate.Having these advantages, the purpose of this paper is toinvestigate the properties of baryons with respecting thechiral symmetry SU (2) R × SU (2) L , especially we focuson baryon resonances.It is particularly interesting that the recent stud-ies [2, 3, 4, 5, 6, 7] show that the ∆ + P (1232) and N − D (1520) resonances are qualitatively reproduced inthe quenched lattice QCD, which validates to some ex-tent the empirical assumption that the baryons are dom-inated by their 3 q Fock components. Recently, we clar-ified the relation between the chiral multiplets and thequark structures [8]. For instance, interpolating fields ∗ Electronic address: [email protected],[email protected] used in Ref. [2, 3, 4, 5, 6, 7] belong to the chiral multiplet(1 , ) ⊕ ( ,
1) [8]. This is our starting assumption wherea set of spin- baryons form the chiral multiplet as chiralpartners. We extend this idea to include two other four-star resonances, the N + P (1720) and ∆ − D (1700), follow-ing Jido et. al. [9], where the four spin- baryons form acertain set of chiral multiplets, so-called Quartet scheme.Ref. [9] mostly discussed the interactions between thesechiral multiplets with the same spins, but not with thedifferent spins especially the nucleon. Inclusion of thenucleon enables us to testify such a framework in com-parison with the experimental data for not only massesbut also other quantities related to the dynamical pro-cesses such as resonance decays and scatterings.In this Letter, we construct an effective Lagrangianfor four types of four-star resonances, ∆(1232), N (1520) N (1720) and ∆(1700) together with the nucleon and in-vestigate the structures of the one- and two-pion cou-pling strengths. We derive a relation among the one-pioncoupling constants of the four baryon resonances, whichagree well with the experimental data. We also considerthe property changes of the one-pion couplings towardsthe chiral restoration.We begin with the nucleon’s chiral multiplet for whichthere are two possibilities ( , ⊕ (0 , ) and (1 , ) ⊕ ( , , ) ⊕ ( ,
1) and ( , ⊕ (0 , ). We choose the formercase as is commonly used to describe the ∆(1232), as wellas in Ref. [9]. Now we define two types of the diquarks; aLorenz vector iso-scalar diquark V µ ( I ( J ) P = 0(1) − )andaxial-vector iso-vector diquark A µi (1(1) + ), V µ = ˜ qγ µ q, (1a) A µi = ˜ qγ µ γ τ i q, (1b)where ˜ q = q T C ( iτ ) γ is a transposed quark field. Thesediquarks form the chiral multiplet ( , ), just like the σ and ~π mesons, which is a key ingredient in our construc-tion of the chiral invariant interactions. We will comeback to this point later. There is one possible operatorfor I ( J ) = ( ), ∆ µi = A jν Γ µν / P ij / q, (2a)and two for I ( J ) = ( ), N µV = V ν Γ µν / γ q, (2b) N µA = A iν Γ µν / τ i q. (2c)Here the isospin- P ij / and the isospin- projection op-erator P ij / , satisfy δ ij = P ij / + P ij / [8]. Similarly thespin- projection operator Γ µν / , with the spin pro-jection operator Γ µν / , satisfy the completeness relation g µν = Γ µν / + Γ µν / . Note that with the spin- projec-tion operator the baryon fields still contain four fictitiousspin- components. However, the chiral transformationproperties does not depend on the choice of the spin pro-jection operators, the local or non-local type [8]. Ourstrategy is firstly use the local projection operators in theconstruction of the Lagrangian, later eliminate the spin- components in the calculations of the physical quantities,the one-pion decays in the present context.Taking into account the normalization and Pauli-principle, which is implemented by the Fierz transfor-mation, we define the baryon fields as∆ µi = ∆ µi , (3a) N µ = √ N µV N µA √ , (3b)where we separate the coefficients to show explicitlythe normalized baryon fields ∆ µi / , N µV / N µA / √ N µV and N µA results fromthe chiral transformations of V µ and A µi , and the mixingangle between N µV and N µA are determined by the Fierztransformation [8]. The chiral transformation propertiesare straightforwardly given by δ ~a N µ = 53 i a · τ γ N µ + 4 √ iγ a · ∆ µ , (4a) δ ~a ∆ µi = 4 √ iγ a j P ij / N µ − iτ i γ a · ∆ µ + i a · τ γ ∆ µi , (4b)implying that a set of N µ and ∆ µi form the chiral mul-tiplet (1 , ) ⊕ ( , V µ , A µi ) and( σ, π i ) has been used. Since such composite operators arereducible under both the chiral and the spin and isospintransformations, we perform the decomposition into irre-ducible parts containing only spin and isospin-projectedbaryons.As an illustration let us consider the vector and axial-vector diquarks ( V µ , A µi ). As explained, they belong tothe chiral multiplet ( , ) similar to ( σ, ~π ). Therefore the V µ + A µ combination is a chiral scalar, which immediatelyleads to the chiral invariant term ¯ q ( V µ + A µ ) U q , where U = σ + iγ τ · π . The direct products of a quark anddiquarks V µ q and A µi q contain several kinds of baryonswith I ( J ) = ( ), ( ), ( ) [14]. The decompositioninto irreducible spin and isospin parts is carried out byusing the completeness relations of both the spin andisospin projection operators. The resulting interactionLagrangian is given by L πBB = g (cid:18) ¯∆ i µ U ∆ µi −
34 ¯ N µ U N µ + 112 ¯ N µ τ i U τ i N µ + √
36 ¯ N µ τ i U ∆ µi ! . (5)Note that the relative weights for N µ and ∆ µi are un-ambiguously fixed by Eq. (4) without dependence on anyfree parameters [15].Now, following Ref. [9], we introduce a new set of spin- baryons ( N µ , ∆ µi ) that have the SU (2) A transforma-tion properties opposite in sign to those of ( N µ , ∆ µi ),so called mirror baryons [9, 10, 11, 12]. They may ap-pear due to some more complicated structures such asnon-local nature of three-quark states and multiquarkcomponents. Here we assume the existence of the mirrorbaryons without considering their internal structures indetail. The diagonal interactions for the mirror baryonsare easily obtained: L πBB = g (cid:18) ¯∆ i µ U † ∆ µi −
34 ¯ N µ U † N µ + 112 ¯ N µ τ i U † τ i N µ + √
36 ¯ N µ τ i U † ∆ µi ! . (6)In addition, the following mass terms are allowed L BB = − m (cid:16) ¯∆ i µ ∆ µi + ¯ N µ N µ (cid:17) . (7)In contrast to Eqs. (5) and (6), the mixings between N µ (∆ µi ) and N µ (∆ µi ) occur only after the mass di-agonalization when the so-called mirror mass m is fi-nite [9, 11, 12]. Combining Eqs. (5) ∼ (7), the quartetscheme of Jido et. al. [9] is exactly reproduced.Next, we include the nucleon and its couplings withthe spin − baryons, which is new in this work. Simi-larly to the above discussion, ( V µ , ~A µ ) and ( σ, ~π ) form achiral scalar σV µ + i π · A µ . Hence we find two chirallyinvariant interactions: (1) ¯ NU [( ∂ µ σ ) V µ + i ( ∂ µ π ) · A µ ] q ,and (2) ¯ N ( ∂ µ U )( σV µ + i π · A µ ) q . With the irreducibledecomposition, we obtain L πNB = g Λ h ¯ N U ( i∂ µ π i )∆ µi + √
32 ¯
N U ( γ ∂ µ σ + i ∂ µ π · τ ) N µ , (8) L πNB = g Λ h ¯ N ( ∂ µ U )( iπ i )∆ µi + √
32 ¯ N ( ∂ µ U )( γ σ + i π · τ ) N µ , (9)where the dimensional parameter Λ is introduced tokeep the coupling constants dimensionless. We neglectthe higher-order terms for the nucleon ¯ NU /∂U † N and¯ N ( ∂ µ U ) U † γ µ N . For the mirror baryons, we obtain thesingle-meson coupling L πNB = g Λ (cid:0) ¯ N ( i∂ µ π i )∆ i µ − √
32 ¯
N ∂ µ ( γ σ − i τ · π ) N µ ! . (10)Again, we neglect the nucleon term ¯ N/∂U † N m , where N m is another nucleon field having the mirror proper-ties. Note that the interactions Eqs. (8) and (9) involvetwo mesons, while Eq. (10) contains only the single me-son couplings. TABLE I: Masses (second column) and Coupling constants(third column). For masses, we follow Jido et. al [9]. Theexperimental values are taken from the PDG tables [13]. Theexperiments determine only the absolute values of the cou-pling constants, the positive values are our assumption.States Masses [MeV] g πNB / Λ [MeV − ] Γ B → πN [MeV]Theo (Exp) Theo (Exp)∆ µi + ( P ) 1320 (1232) 15 (16) 118∆ µi − ( D ) 1770 (1700) 9.2 (9.5) 45 N µ − ( D ) 1430 (1520) 9.4 (8.6) 69 N µ + ( P ) 1660 (1720) 2.4 (2.4) 30 m = 1550 g = g = 2 . g / Λ = 17 g f π / Λ = 4 . g f π / Λ = 8 . Having constructed the Lagrangian with the nucleonand spin − baryons, let us determine the parameters g , and m , following Ref. [9]. The results are shown in Ta-ble I [16]. After the diagonalization of the mass term andthe corresponding parity (re)definition, the mass eigen-states are obtained as: for the ∆s, ∆ µi + = (∆ µi +∆ µi ) / √ µi − = γ ( − ∆ µi + ∆ µi ) / √
2, and for the N ∗ s, N µ − = γ ( − N µ + N µ ) / √ N µ + = ( N µ + N µ ) / √
2, where thesubscripts ± denote the parity [17]. After the spontaneous breaking SU (2) R × SU (2) L → SU (2) V , the one-pion interactions in Eqs. (8)-(10) arereduced to L πNB = g πN ∆ + Λ ¯ N ( i∂ µ π i )∆ µi + + g πN ∆ − Λ ¯ N ( iγ ∂ µ π i )∆ µi − + g πNN ∗− Λ ¯ N ( iγ ∂ µ π · τ ) N µ − + g πNN ∗ + Λ ¯ N ( i∂ µ π · τ ) N µ + , (11a)where the coupling constants are given by g πN ∆ ± = 1 √
2Λ ( g Λ ± g f π ) , (11b) g πNN ∗± = √ g + 3 g ) f π ∓ g Λ) . (11c)The three coupling constants g , , are determined fromthe one-pion decay widths of the resonances as shown inTable I. We obtain quantitatively reasonable results forall the four coupling constants Eqs. (11). Eliminating g , , from Eqs. (11), we obtain a new relation:( g πN ∆ + + g πN ∆ − ) = 2 √ g πNN ∗− − g πNN ∗ + ) , (12)which satisfies the experimental data with a numericalerror of less than 10%. Considering the simplicity of thepresent description, this is an encouraging result suggest-ing that the spin − baryons are the candidates of thechiral partners.One of interesting properties of the present model isthe two-pion contact terms, which are an inevitable con-sequence of the chiral invariance. They involve only the g and g , while g , which is a leading contribution in theone-pion couplings, does not contribute to the two-pioncouplings. The two-pion decay of ∆(1232) is thereforesuppressed by the smallness of the coupling constants ascompared to the one-pion decay. On top of this, thederivative coupling causes an additional suppression ofthe two-pion decay rate, due to the small final state pionmomentum. Hence we can expect strong suppression ofthe two-pion decay of ∆(1232). Explicitly the two-pioncontact interactions are given by L πNB = 1 √ ¯ NA iµ ∆ µi + − √ ¯ N A iµ γ ∆ µi − + √ ¯ N B µ γ N µ − − √ ¯ N B µ N µ + , (13a)with A iµ = g ( iγ π · τ )( i∂ µ π i ) + g ( iγ ∂ µ π · τ )( iπ i ) , (13b) B µ = g ( iγ π · τ )( i∂ µ π · τ ) + g ( iγ ∂ µ π · τ )( i π · τ ) . (13c)Hence we obtain a relation between the 2- π contactterms: | g πN ∆ + | = | g πN ∆ − | = 2 √ | g πNN ∗ + | = 2 √ | g πNN ∗− | . (14) f p [MeV]
93D +D -N *-N *+ g pNB / L [ M e V - ] FIG. 1: The illustration of f π dependence of the one-pioncoupling constants g πNB / Λ in the case (1).
In contrast to the ∆(1232) case, it is expected that thetwo-pion contact term leads to larger contributions forother baryon resonances, because of the larger final statepion momenta. In particular, the two-pion coupling con-stants of N ∗ + (∆ − ) has the same magnitude as comparedwith that of N ∗− (∆ + ), while the one-pion coupling con-stant is suppressed by the negative sign in Eqs. (11).This qualitatively explains the observed feature of thetwo-pion decay enhancement in the decays of N (1720)and ∆(1700). Due to the lack of other resonances, suchas ρ -meson and N (1440), we do not consider this pointin details.As an application of our results to one of the recent in-terests, we consider a situation towards the chiral restora-tion at high temperature or density. We briefly considerthe property changes of the one-pion coupling constantsin two cases: (1) the scale parameter Λ is constant and(2) Λ = f π . In the case (1), as f π decreases, the threecoupling constants g πN ∆ + and g πNN ∗± decrease, whilethe remaining one g πN ∆ − increases. At f π = 0, we ob-tain the simple relation g πN ∆ + = g πN ∆ − = √ g πNN ∗− = − √ g πNN ∗ + , which is shown in Fig. 1. In the case (2), all the coupling constants simply increase proportional to f − π . Eq. (12) does not depend on the value of Λ, henceit always holds in both cases.In summary, we have investigated the chiral propertiesof four spin- baryon resonances together with the nu-cleon. We have constructed the effective interactions forthe spin − transition terms with the aid of the spinand isospin projection formalism for the baryon fieldscomprised of three quark fields. Of course, we can provethe chiral invariance of the derived interactions directlyfrom the chiral transformation laws, but the results aregeneral from the group-theoretical point of view. Withinthe J = sector, the projection formalism reproducesthe Quartet scheme proposed by Jido et al. [9]. In ad-dition, we derived the minimal chiral invariant one- andtwo- meson couplings with spin − baryons. We foundthat the one-pion couplings describing the spin − tran-sitions are constrained by the chiral symmetry via the Eq.(12), which quantitatively agrees with the experiment.Considering the simplicity of our assumptions on the ef-fective Lagrangian, it is an remarkable result suggest-ing that these baryons are chiral partners. In addition,we obtain chiral two-pion couplings, whose strengths areentirely determined by the one-pion coupling constants.This enable us to predict two-pion decays of the reso-nances that can be tested in experiments. In this Letterwe employed a new projection technique to derive theeffective chiral interaction Lagrangians between baryonsof different spin and isospin.We thank Prof. D. Jido for fruitful discussions. K.Nand V.D thank Prof. H. Toki for hospitality in the stayat RCNP. K.N is supported by National Science Council(NSC) of Republic of China under grants No. NSC96-2119-M-002-001. [1] S. Weinberg, Phys. Rev. Lett. , 1177 (1990).[2] D. B. Leinweber, T. Draper, and R. M. Woloshyn, Phys.Rev. D46 , 3067 (1992).[3] J. M. Zanotti et al. (CSSM Lattice), Phys. Rev.
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G33 ,1 (2006).[14] I ( J ) = ( ) state is forbidden by the Pauli-principle [8]when it is described as a local three quark state.[15] Eq. (5) predicts a mass relation m (∆ µi ) : m ( N µ ) = 2 : 1.There are no candidates for this sort of N ∗ and ∆ in theobserved spectrum [13].[16] In Fig.1 of Jido et. al. the state labeled as 1770 and 1660ought to be turned upside down.[17] The slight difference from Ref. [9] is caused by our defini-tion that all the basis ∆ , and N ,2