USITP-96-01 January 4, 1996 A possible description of the quantum 6 numbers in a hadronic string model 9 9 1 V.A. Kudryavtsev 1 n Petersburg Nuclear Physics Institute a Gatchina, 188350 St. Petersburg, Russia J 4 G. Weidl2 1 Institute of Theoretical Physics v University of Stockholm, Box 6730 8 0 S-113 85 Stockholm, Sweden 0 1 0 ABSTRACT: We consider a critical composite superconformal 6 string model to desribe hadronic interactions. We present a new 9 / approach of introducing hadronic quantum numbers in the scat- h tering amplitudes. The physical states carry the quantum num- t - p bers and form a common system of eigenfunctions of the opera- e tors in this string model. We give explicit constructions of the h : quantum number operators. v i X r a 1e-mail: [email protected] 2e-mail:[email protected] 1 1 Introduction In 1967 the Regge hadronic resonances have been discovered. The experi- mental data gave evidence of a typical relation between the spins and the I square of the masses M of these strongly interacting particles [1]. It turned out, that the resonances form the so-called Regge trajectories (M2) = α +α′M2 n . (1) 0 I − For n = 0,1,2,... one finds a family of parallel linear daughter trajectories. The constants α and α′ denote the intercept and the Regge slope. The 0 leading ρ-meson trajectory corresponds to n = 0. So far QCD has not been able to explain this phenomenon. On the other hand in the 1970’s it was observed, that in string theory the mass shell condition (L a) phys >= 0 0 − | generatesamassspectrumwithparalleltrajectories. Theinterceptadepends on the cancellation condition of the conformal anomaly in the respective theory. Thus hadronic strings became a natural candidate for describing the Regge spectrum phenomena. However such efforts failed because of the necessary appearance of massless states of spin one and two, which do not correspond to physical hadronic resonances. While these massless states gave rise to treat string theory as a fun- damental theory of all interactions including gravity at Planck energy scale, the discrepance in hadronic string theory at typical strong interaction energy scales of E 1 GeV remained unresolved for a long time. In [7], [10] one ∼ of the authors suggested a new critical composite superconformal hadronic string. It consists of the Neveu-Schwarz (NS) superstring and a fermionic superconformal string [5]. The NS field components are associated with the space-time degrees of freedom, while the fermionic sector carries the internal degrees of freedom, namely the hadronic quantum numbers. In accordance with the conformalanomaly cancelation in this model we have to impose new gauge constraints on the physical states. This allows to eliminate the prob- lematic massless states from the string mass spectrum, which now becomes compatible with the Regge resonance spectrum [7]. In this paper we present a possible explicit construction of the opera- tors, corresponding to the hadronic quantum numbers spin, isospin, electric 2 charge, hyper-charge, baryon charge, strangeness, charm, beauty and top. We give the structure of the hadronic wave functions. Insection2wereviewthemainfeaturesofthecompositestringmodeland the structure of its superconformal generator. We discuss the cancellation of the anomaly achieved in D = 10 or D = 4 space-time dimensions and D′ internalfermionicdegreesoffreedom. Thenadditional D′ or D′ 3conditions 6 6 − are needed respectively for the anomaly-free solution. In section 3 we give all gauge conditions eliminating ghosts in the physical states of the composite string model and the spectral equations for the quantum numbers. 2 The critical composite superconformal string model Inorder to describe the physical hadronicstates withtheir quantum numbers and masses we have to consider a string model in the four dimensional space- time. Such models can be obtained by compactification of the critical ten dimensional superstring. One way of compactification to four dimensions is the fermionization of six dimensions [2]-[3], namely, by introducing free world sheet fermions ν(σ,τ) carrying all internal quantum numbers of the string [2], [5], [11]. A generalization and concrete realization of this approach has been achieved in the critical composite superconformal string model [10], [12]. This model unifies the superconformal structures of the Neveu-Schwarz operator G(NS) [9] and of the fermion operator G(f) [5] by introducing the r r composite superconformal operator G = G(NS) +G(f). (2) r r r The operator G(f) is constructed in such a way, that G is a singlet in all r r quantum numbers, and G(NS),G(f) = 0. { r r } The specific superconformal algebras of G(NS) and G(f) are closed separately r r (see the Appendix). The canonical superconformal operator G(NS) reads [8],[9] r 1 G(NS) = Hµ(∂ X )e+irτdτ. r 2πi I τ µ 3 We use the notations a(i) X(i) = x(i) +ip(i)lnz + nµzn, z = e−iτi µ 0µ µ i in i i X n6=0 [a(i),a(i) ] = ng δ , g = 1, g = 1, nµ mν − µν n,−m 00 ii − for the i-th string space-time coordinate of zero conformal weight, and H(i)(z ) = b(i)zr, b(i),b(i) = g δ µ i rµ i { rµ sν} − µν r,−s for its superpartner in the Neveu-Schwarz string model. The NS-states are given by the Fock space of the products of the creation operators a† b† 0 > . Yn,µmY,ν{ nµ}{ mν}| As usual the operatorG of the composite string shall be of the conformal r weight I = 3/2.Therefore thefermionoperatorG(f) isofthesameconformal c r weight. It can be constructed as a three-linear combination [2] of the fermion fields ν of I = 1/2 and their currents JνA of I = 1, i.e. [5] A c c G(f)(ν) = ν ν ν ε + JνAν . (3) r A B C ABC A X X A,B,C A By construction G(f) is a singlet in the quantum numbers. The fermion r operator generates all internal quantum numbers, what we show in detail in the next section. We turn now to the description of the structure of the fields ν and their A currents JνA, entering in (3). Let ψ ψ , µ = 0,1,2,3, j = 1,2 be α µj ≡ Majorana spinor with I = 1/2. Its eight components are Lorentz spinors in c µ and isospinors in j simultaneously. The respective currents JνA JνA = ψ˜ T ψ , ψ˜ = T ψ = γ τ ψ , α αβ β α 0 α 0 2 α are of conformal weight one. These currents are non-zero, if the respective 8 8 matrices T T are antisymmetric. The matrices T can be choosen 0 αβ αβ × of the form γ , γ γ τ , τ , γ τ , [γ ,γ ]. µ 5 µ i i 5 i µ ν 4 Thus we obtain the 28 components for the vector, axial vector, scalar, pseu- doscalar and tensor currents JνA = ψ˜ T ψ = JV,JA,JS,JP,JT . α αβ β { µ µi i i µν} They generate a Kac-Moody algebra, see [4]. With these currents we asso- ciate the respective 28 fermion components ν := ψ ;φ ,ρ ,θ ,η ,ξ , µ = 0,1,2,3, i = 1,2,3. (4) A α µ µi i i µν { } These fields satisfy the standard anticommutation relations. We consider the physical quarks as compositions of the elementary field components (4). We see, that the eight spinor components ψ generate 28 vector, axial α vector, scalar, pseudoscalar and tensor fermionic components. Thus G(f) is r a scalar in D′=36 fermionic components. It is well known, that the Neveu-Schwarz string is critical in the space- time dimension D = 10. This comes from the condition of cancellation of the conformal anomaly, or equivalently the nilpotency of the BRST-charge Ω2 = 0 [3], [4] 3 3 D 26+11 = (D 10) = 0. (5) 2 − 2 − We point out, that the new composite string remains critical for D=10. This is due to the ghost contributions 3 from the introduced fermionic fields and their currents [10]. Indeed, in the composite model the condition (5) turns into 3 1 3 1 D 26+11+( D′ 3N) = (D 10)+( D′ 3N) = 0, (6) 2 − 2 − 2 − 2 − where 3N denotes the ghost contribution. In the dimension D = 10 the − condition (6) is satisfied, if we choose N = D′/6 [10]. On the other hand, compactifying the string to the space-time dimension D = 4,wecan imposeN = D′ 3 gaugeconstraints. These gaugeconstraints 6 − on the fermionic sector are used to cancel the superfluous (in compairison 3The ghost contribution is given by c = 2ǫ(6I (I 1)+1), where ǫ = 1 is the g c c − − ghost statistic for integer I and ǫ= 1 is the ghost statistic for half integer I . Hence N c c − fermions with I = 1/2 contribute c = N and N corresponding currents with I = 1 c g c − contribute c = 2N. g − 5 with the Regge spectrum) states, and to shift the vacuum to G 0 > . −1/2 | This shift ensures the absence of a tachyon in the spectrum. The amplitude of the hadron interactions are constructed by the opera- tor formalism. The quantum numbers spin, isospin, electric charge, hyper- charge, baryon charge, strangeness, charm, beauty and top and the masses of the hadronic states are included in the structure of the hadronic wave func- tion which enters in the vertex operator structure, and hence in the structure of the N hadrons amplitude A . N The factorization of the amplitude gives the wave function of the cor- responding hadronic state. This wave function contains a certain number of components, corresponding to its experimental mass. These components carry all hadron quantum numbers and will reproduce explicitly the quan- tum numbers in the hadronic amplitude structure. This has been shown in [6] where the superconformal string amplitudes for π mesons interactions have been constructed and in [11], [12] where the general N hadrons interac- tion amplitudes are obtained as multiparticle generalization of the Lovelace- Shapiro amplitude and are treated as composite superconformal strings. Duality, crossing and cyclic symmetry for these superconformal compos- ite string amplitudes hold together with the description of hadron quantum numbers. 3 The Quantum Numbers In this section we give a possible explicit construction of the quantum num- bers of hadron wave functions. This wave functions shall satisfy a set of gauge constraints L phys >= 0, n > 0, G phys >= 0, r > 0, (7) n r | | which eliminate ghosts in the physical states. Here L = L(NS)+L(f) denotes n n n the Virasoro operator, the fermion part of which is explicitly given in the Appendix. As usual we impose the mass shell condition (L 1/2) phys >= 0, (8) 0 − | where the intercept 1/2 is fixed by the nilpotency of the BRST charge. 6 In addition to this we have to require new gauge constraints D′ ν(l) phys >= 0, Jνr(l) phys >= 0, l = 1,...,N = 3, (9) r | | 6 − eliminating theghostscontributions fromtheaddedfermionic fieldsandtheir currents. Such new constraints are in agreement with the anomaly cancella- tion condition (6). They are responsible for the reduction of the composite string spectrum to the Regge resonance spectrum. The gauge constraints (7)-(9) build a supermultiplet L ,G ,Jνr(l),ν(l) with the conformal weights n r r I = 2, 3,1, 1 respectively. c 2 2 We search for a common system of eigenfunctions for (7)-(9) and the spectral equations ˆ phys >= q phys > (10) m m Q | | ontheremainingD′ N fermioncomponents. Theoperator ˆ runsoverthe m − Q quantumnumber operatorsforthespin, isospin, electriccharge, hypercharge, baryon charge, strangeness, charm, beauty and top. An appropriate choice of the quantum number operators are those zero components of the corresponding currents ( νA) = TνA := G(f),(ν ) , (11) J 0 { r A −r } which are Number operators, i.e. count the component carrying the spin, isospin, baryon charge and so on. The operators (11) automatically commute with the Virasoro operators L(f) and G(f) n r [L(f),TνA] = 0, [G(f),TνA] = 0, n r what ensures the existence of a common system of eigenfunctions. Since G(f) is a quantum number singlet, the charge TνA carries the same r quantum numbers as the field ν . Therefore the isospin operator shall be A defined as Tη := G(f),(η ) . (12) i { r i −r} It generates the algebra [Tη,Tη] = ε Tη and has eigenvalues i j ijk k (Tˆη)2 phys >= Tη(Tη +1) phys > . i | i i | Analogously the Lorentz spin operator is defined by J (k Tξµν) := G(f),(k ξ ) (13) ≡ µ { r µ µν −r} 7 with the momentum k . The specific realization of the spin and isospin op- µ erators is given in the Appendix. The baryon charge B = 1 is carried by hadron states with half-integer Lorentz spin. Among the fields in (4) only ψ is a Lorentz spinor. Therefore, if a wave function contains an even number of ψ components, it is a meson, otherwise it is a baryon. Thus the baryon charge is defined by 1 B = (1 ( 1)NB) (14) 2 − − ˜ where N = ψ ψ is the number of spinor ψ-fields. B l −l l Among thPe currents (11) one can not find other operators with Number structure. On the other hand all 36 components (4) were used in the specific realization of (12)-(14). Therefore with them we can only construct meson andbaryonstatescontaininguanddquarks. Todefinethequantumnumbers strangeness sˆ, charm cˆ, beauty ˆb and top tˆindependently, we have to extend the set (4) of 36 fermion components with an analogous partner-set of 36 field components (Lorentz scalars and isoscalars) ω := χ ; σ ,σ , α = 1...8, i,j,k = 1...7. (15) A α i jk { } The fermion fields (15) show the following anticommutation properties (χ˜ ) ,(χ ) = δ δ , χ˜ = χ γ τ , α r β s αβ r,−s α α 0 2 { } (σ ) ,(σ ) = δ δ , i r j s ij r,−s { } (σ ) ,(σ ) = (δ δ δ δ )δ , σ = σ . ij r kl s ik jl il jk r,−s ij ji { } − − From the 8 component spinor χ we can define 28 components of the Kac- α Moody currents JωA = J := χ˜Γ χ; J := χ˜[Γ ,Γ ]χ . (16) i i jk i j n o which are Lorentz scalars and isoscalars. Here Γ denote the 8 8 Clifford i × matrices, obeying the standard anticommutation relations Γ ,Γ = 2δ . j k jk { } The currents J ,J carry the same quantum numbers as the fields σ ,σ . i jk i jk This guaranties the scalarity of the superconformal operator G(f)(ω) = ω ω ω ǫ + JωAω . (17) r A B C ABC A X X ABC A 8 Composing the superconformal operators (3) and (17) G(f) = G(f)(ν)+G(f)(ω) (18) r r r we describe the full set of hadron quantum numbers 4. From (12) the third isospin component in the ν -space is given by A Tη := G(f),(η ) . 3 { r 3 −r} Byconstructionσ ω isthepartner-componenttoη .Thereforewedefine 12 A 3 ∈ the isospin projection in ω by A T := G(f),(σ ) . (19) 12 { r 12 −r} The full projection of the isospin of the hadron state corresponds to the sum Tˆ phys >= (Tη +T ) phys > . (20) 3| 3 12 | In order to describe the flavour quantum numbers strangeness, charm, beauty and top we have to construct four commutating operators in ω with A structure similar to the zero component of the currents JωA. We will find them using the properties of the spinor representation of the O(6) group. The 8 component spinors in the compactified D = 6 space transforms under O(6). The generators of this group are given via the 8 8 Γ matrices by i × [Γ ,Γ ] = 4iM , i,j = 1...6. It holds [M ,M ] = 0, if and only if i = k,l i j ij ij kl 6 and j = k,l. Thus among the generators one can choose not more then three 6 commutating with each other M , e.g. M ,M ,M . Since the Clifford ij 12 34 56 matrices Γ transforms under O(6) as vectors i [M ,Γ ] = i(δ Γ δ Γ ), i,j,k = 1...6, ij k ik j jk i − neither of the matrices Γ ,...,Γ commutes with all of the three generators 1 6 M ,M ,M . On the other hand, the matrix Γ := 6 Γ commutes with 12 34 56 7 i=1 i all generators M . Thus Γ is the fourth independentQoperator. ij 7 4G(f) isaquantumnumbersingletinD′ =72,whichcanbereducedtoD=10according r ′ to (6) by imposing N =D /6=12 conditions on the physical states, i.e. 6 on each of the sets (4) and (15). Then for the compactification of the composite string to D=4 are used ′ N =D /6 3=9 constraints. − 9 According to thiswe introduce first three independent generatorsT ,T , 12 34 T . We choose T according to (19), since it defines the s,c,b,t-flavour part 56 12 of the isospin projection. The remaining two generators are given by T := G(f),(σ ) (21) 34 { r 34 −r} T := G(f),(σ ) . (22) 56 { r 56 −r} The fourth charge operator has the form T := G(f),(σ ) . (23) 7 { r 7 −r} It commutes with (19), (21), (22) and with L ,G . Hence there exists a n r common system of eigenfunctions for all these operators. TheoperatorsT ,T ,T ,T cannotdirectlybeidentified asflavor quan- 12 34 56 7 tum number operators. In fact one has to consider the following four linear combinations, which define strangeness, charm, beauty and top 1 1 sˆ:= (T +T ) (T +T ), (24) 34 56 12 7 −2 − 2 1 1 cˆ:= (T +T )+ (T +T ), (25) 34 56 12 7 −2 2 1 1 ˆ b := (T +T )+ (T +T ), (26) 12 34 56 7 −2 2 1 1 tˆ:= + (T +T ) (T +T ). (27) 12 56 34 7 2 − 2 Thus thephysical quarks aredescribed assuperpositions ofelementary string fields. Hence the quarks are composite objects in this model. If we substitute (20) and (24)-(27) into the expression for the electric charge Q = T +(sˆ+ˆb cˆ tˆ+B)/2 3 − − with the baryon charge B, we obtain Q = Tη +B/2. (28) 3 This can be taken as a definition, which is independent of the quark flavors contained in the hadronic state. Analogously the expression for the hyper- charge in the quark model Y = sˆ+ˆb cˆ tˆ+B − − 10