The role of the qqqq q ¯ components in the electromagnetic transition γ ∗ N→ N ∗ (1535)
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Role of the qqqq ¯ q components in the electromagnetic transition γ ∗ N → N ∗ (1535) C. S. An , and B. S. Zou , Institute of High Energy Physics, CAS, P.O.Box 918(4), Beijing 100049, China Helsinki Institute of Physics, POB 64, 00014 University of Helsinki, Finland Theoretical Physics Center for Science Facilities, CAS, P.O.Box 918(4), Beijing 100049, ChinaReceived: date / Revised version: date
Abstract.
The helicity amplitudes A p / and S p / for the electromagnetic transition γ ∗ N → N ∗ (1535) arecalculated in the quark model that is extended to include the lowest lying qqqq ¯ q components in additionto the qqq component. It is found that with admixture of 5-quark components with a proportion of 20%in the nucleon and 25-65% in the N ∗ (1535) resonance the calculated helicity amplitude A p / decreasesat the photon point, Q = 0 to the empirical range. The qqqq ¯ q components contain s ¯ s pairs, which isconsistent with the substantial width for Nη decay of the N ∗ (1535). The best description of the momentumdependence of the empirical helicity amplitudes is obtained by assuming that the qqqq ¯ q components aremore compact than the qqq component. However, this version of extended quark model still does not leadto a satisfactory simultaneous description of both A p / and S p / although with significant improvement. PACS.
The structure of the nucleon resonances with spin-parity1 / − has continued to be somewhat enigmatic. On theone hand the number of observed resonances coincideswith that predicted by the constituent quark model witha monotonic confining interaction [1,2], while on the otherhand several of these resonances can be dynamically gen-erated in chiral meson-nucleon models [3,4]. The questionthen is whether a 3 quark or a 4 quark + 1 antiquarkdescription is the more appropriate one. The N ∗ (1535)resonance is particularly interesting in this regard, as ithas a sizeable N η decay branch, even though its energyis very close to the threshold of that decay. The electro-magnetic transition form factors of this resonance may infact contain crucial information to settle this issue. It hasrecently been shown that the chiral dynamical approachcan provide a description of both the A p / and the S p / transition form factors [5], while a simultaneous descrip-tion of both these form factors has proven elusive in thebasic qqq constituent quark model [6].To bridge the gap between the constituent quark modeland the chiral meson-nucleon resonance models, it is nat-ural to extend the former to include explicit qqqq ¯ q com-ponents in addition to its basic qqq structure. In the caseof the ∆ (1232) and the N ∗ (1440) resonances it has beenshown that already a modest admixture of qqqq ¯ q compo-nents can remove the main under-prediction of the de- cay widths that is typical of the qqq quark model [7,8,9].Moreover, it has recently been shown that the coupling of N ∗ (1535) N φ may be significant [10], which is consistentwith the previous indications of the notable N ∗ (1535) KΛ coupling [11]. These suggest that there are large s ¯ s com-ponents in the N ∗ (1535) resonance. With this suggestion,we can give a natural explanation of the mass orderingof the N ∗ (1440)1 / + , N ∗ (1535)1 / − and Λ ∗ (1405)1 / − resonances [11].Here we calculate the helicity transition amplitudes A p / and S p / for the electromagnetic transition γ ∗ p → N ∗ (1535), by including contributions from both qqq com-ponents and those qqqq ¯ q components with configurationsexpected to have the lowest energy in the proton and N ∗ (1535). These amplitudes are not well described in theconventional qqq constituent quark model [12,13]. The re-sults show that the calculated transition amplitudes canbe improved by taking inclusion of the qqqq ¯ q components.We take the flavor-spin configurations of the four-quarksubsystem in the qqqq ¯ q components in the proton to be[4] F S [22] F [22] S , which is likely to have the lowest energy[14]. In the case of the negative parity N ∗ (1535) reso-nance the corresponding most likely lowest energy config-uration is [31] F S [211] F [22] S [14]. The flavor configuration[211] F in the latter requires that the qqqq ¯ q component canonly be qqqs ¯ s . This implies a large hidden strangeness inthe N ∗ (1535) resonance and strong couplings to N φ and KΛ states, which is consistent with the results in Refs. C. S. An, B. S. Zou: Role of the qqqq ¯ q components in the electromagnetic transition γ ∗ N → N ∗ (1535) [10] and [11]. This feature is also in line with the mech-anism of dynamical resonance formation within the cou-pled channel approach based on chiral SU (3) with the KΣ quasi-bound state explanation for the N ∗ (1535) [3,4,5,15,16]. For completeness we also consider the contributions of d ¯ d or u ¯ u components in the N ∗ (1535) and consequently,which appear in the next-to-next-to-lowest-energy config-uration [31] F S [22] F [31] S .Some of the empirical results on the strangeness mag-netic form factors may be described, at least qualitatively,by uuds ¯ s configurations in the proton, where the ¯ s anti-quark is in the S − state [17,18]. Since configurations withthe antiquark in the S − state cannot be represented bylong range pion or kaon loop fluctuations, this motivates toa systematic extension of the qqq quark model to includethe qqqq ¯ q configurations explicitly. On the other hand, theoverall descriptions of the baryon magnetic moments maybe improved by taking the qqqq ¯ q components contribu-tions into account [19,20].This present manuscript is organized in the followingway. The wave functions of the proton and N ∗ (1535) aregiven in section 2. The helicity amplitudes A p / and S p / for γ ∗ p → N ∗ (1535) are calculated in section 3. Finallysection 4 contains a concluding discussion. N ∗ (1535) The internal nucleon wave functions that include qqqq ¯ q components in addition to the conventional qqq compo-nents for the proton and N ∗ (1535) may be written in thefollowing form: | P, s z i = A ( P )3 q | qqq i + A ( P )5 q X i A i | qqqq i ¯ q i i , | N ∗ (1535) , s z i = A ( N ∗ )3 q | qqq i + A ( N ∗ )5 q (1) × X i A i | qqqq i ¯ q i i . Here A ( P )3 q and A ( P )5 q are the amplitudes for the 3-quark and 5-quark components in the nucleon, respec-tively; A ( N ∗ )3 q and A ( N ∗ )5 q are the corresponding factorsfor N ∗ (1535). The sum over i runs over all the possible qqqq i ¯ q i components, and the factors A i denotes the cor-responding coefficient for the qqqq i ¯ q i component. We givethe explicit forms of the qqq and qqqq ¯ q components in theproton and N ∗ (1535) in the next two subsections. qqq components Here we take the wave functions for the qqq components inproton and N ∗ (1535) to be the conventional ones, whichcan be expressed in the following way in the harmonicoscillator quark model: | P, s z i q = 1 √ | , t z i + | , s z i + + | , t z i − | , s z i − ) × φ ( κ ) φ ( κ ) , | N ∗ (1535) , s z i q = 12 X ms C ,s z m, ,s { φ m ( κ ) φ ( κ )[ | , t z i + | , s i + −| , t z i − | , s i − ] − { φ ( κ ) φ m ( κ )[ | , t z i + ×| , s i − + | , t z i − | , s i + ] } . (2)Here | , s ( z ) i ± and | , t z i ± , with t z being the z-componentof the isospin, are spin and isospin wave functions of mixedsymmetry [21] S and [21] F , respectively, in which ’+’ de-notes a state that is symmetric and ’-’ denotes a state thatis anti-symmetric under exchange of the spin or isospin ofthe first two quarks. The momenta κ i are defined by thethree quarks momenta as κ = 1 √ p − p ) , κ = 1 √ p + p − p ) . (3)The harmonic oscillator wave functions are φ ( κ ; ω ) = 1( ω π ) / exp {− κ ω } , (4) φ , ± ( κ ; ω ) = ∓√ κ ± ω φ ( κ ; ω ) , (5) φ ( κ ; ω ) = √ κ ω φ ( κ ; ω ) . (6)Here κ ± ≡ √ ( κ x ± iκ y ), and κ ≡ κ z . The subscriptsdenote the quantum numbers ( lm ) of the oscillator wavefunctions. qqqq ¯ q components If the hyperfine interaction between the quarks depends onspin and flavor [2], the qqqq subsystems of the qqqq ¯ q com-ponents with the mixed spatial symmetry [31] X are ex-pected to be the configurations with the lowest energy, andtherefore most likely to form appreciable components ofthe baryons with positive parity. Consequently the flavor-spin state of the qqqq subsystem is most likely totallysymmetric: [4] F S . Moreover in the case of the proton,the flavor-spin configuration of the four quark subsystem[4]
F S [22] F [22] S , with one quark in its first orbitally excitedstate, and the anti-quark in its ground state, is likely tohave the lowest energy [14], consequently, the qqqq i ¯ q i com-ponents and the corresponding amplitudes in the protonmay be [19]: r | [ uudd ] [22] F ¯ d i + r | [ uuds ] [22] F ¯ s i . (7)The explicit form of the wave functions for these two q ¯ q components may be expressed as [18,19,9]: | p, s z i q = 1 √ X a,b X m,s C s z m, s C [1 ][31] a [211] a [211] C ( a )[31] X,m ( a ) . S. An, B. S. Zou: Role of the qqqq ¯ q components in the electromagnetic transition γ ∗ N → N ∗ (1535) 3 × [22] F ( b )[22] S ( b ) ¯ χ s ψ ( κ i ) . (8)Here the flavor-spin, color and orbital wave functions ofthe qqqq subsystem have been denoted by the Young pat-terns, respectively. The sum over a runs over the three con-figurations of the [31] X and [211] C representations of S permutation group, and the sum over b runs over the twoconfigurations of the [22] F and [22] S representations of S permutation group, respectively. The factors C [1 ][31] a [211] a denotes the S Clebsch-Gordan coefficients.The explicit forms of the flavor-spin configurations havebeen given in Ref. [18], and the explicit color-space partof the wave function (8) may be expressed in the form [9]: ψ C ( { κ i } ) = 1 √ { [211] C ϕ m ( κ ) ϕ ( κ ) ϕ ( κ ) − [211] C × ϕ ( κ ) ϕ m ( κ ) ϕ ( κ ) + [211] C × ϕ ( κ ) ϕ ( κ ) ϕ m ( κ ) } ϕ ( κ ) . (9)Here the two additional Jacobi momenta κ and κ are defined as κ = 1 √
12 ( p + p + p − p ) , (10) κ = 1 √
20 ( p + p + p + p − p ) . (11)and the harmonic oscillator wave functions are ϕ ( κ ; ω ) = 1( ω π ) / exp {− κ ω } , (12) ϕ , ± ( κ ; ω ) = ∓√ κ ± ω ϕ ( κ ; ω ) , (13) ϕ ( κ ; ω ) = √ κ ω ϕ ( κ ; ω ) . (14)In the case of the wave functions for the qqqq ¯ q compo-nents in the S resonance N ∗ (1535), the parity of whichis negative, are definitely not same as that of the pro-ton. Negative parity demands that all of the quarks andthe anti-quark be in their ground state, or states withhigher even angular momentum. If the flavor- and spin-dependent hyperfine interaction between the quarks areemployed, for the qqqq subsystem, the flavor-spin config-uration [31] F S [211] F [22] S is expected to have the lowestenergy, and the orbital state should be completely sym-metric state [4] X [14]. Consequently, the wave function ofthe qqqq ¯ q component in N ∗ (1535) may be expressed as | N ∗ (1535) , s z i q = X abc C [1 ][31] a [211] a C [31] a [211] b [22] c [4] X [211] F ( b ) × [22] S ( c )[211] C ( a ) ¯ χ s z ϕ ( { κ i } ) . (15)Note that the total spin of the four quark subsystemwith symmetry [22] S is S = 0. The flavor configuration[211] F implies that the qqqq ¯ q components can only be uuds ¯ s . Consequently there are appreciable s ¯ s componentsin N ∗ (1535), which is consistent with the strong couplingof N ∗ (1535) KΛ [11] and N ∗ (1535) N φ [10]. From this as-sumption it follows that the proportion of the s ¯ s in the N ∗ (1535) resonance might equal P q . In the Roper reso-nance, the probability of the s ¯ s component would then beonly P q / P q of the qqqq ¯ q components in N ∗ (1535) and N ∗ (1440) the larger propor-tion of the strange component may then make N ∗ (1535)heavier than the Roper resonance as desired. This hasbeen a puzzle in the conventional qqq constituent quarkmodel with flavor independent hyperfine interactions.In addition, there may be smaller d ¯ d or u ¯ u compo-nents in N ∗ (1535). The next-to-lowest-energy qqqq ¯ q con-figuration is however in this scheme [4] X [31] F S [211] F [31] S [14], which also rules out the u ¯ u and d ¯ d components.The contributions to the electromagnetic transition γ ∗ p → N ∗ (1535) of this configuration may be replaced by thatof certain proportion of the lowest energy qqqq ¯ q compo-nent. Consequently, one has to consider the configura-tion with qqqq subsystem of the orbital-flavor-spin sym-metry [4] X [31] F S [22] F [31] S , which is the lowest energyconfiguration allowing uudd ¯ d component with spin-parity J P = 1 / − [14]. For this configuration, the qqqq i ¯ q i com-ponents and the amplitudes in the N ∗ (1535) may be [19]: r | [ uudd ] [22] F ¯ d i + r | [ uuds ] [22] F ¯ s i . (16)The explicit form of the wave function for these two q ¯ q components may be expressed as: | N ∗ (1535) , s z i q ′ = X abc C [1 ][31] a [211] a C [31] a [211] b [22] c [4] X [22] F ( b ) × [31] S ( c )[211] C ( a ) ¯ χ s z ϕ ( { κ i } ) . (17)Actually, we can also obtain the probabilities of the qqqq ¯ q components from a recent manuscript [21]. The van-ishing or small axial charge of N ∗ (1535) requires thatthere should be a cancelation between the contributionsof the qqq and qqqq ¯ q components. Consequently, the mostobvious five-quarks components should be the ones in whichthe flavor-spin configurations of the four-quarks subsys-tem are [31] F S [211] F [22] S , [31] F S [211] F [31] S or [31] F S [31] F [31] S .And the large coefficients A n obtained from the latter twoconfigurations indicate that the probabilities of these twocomponents should be smaller than the first one. γ ∗ N → N ∗ (1535) The calculation of the helicity amplitudes A p / and S p / for the electromagnetic transition γ ∗ p → N ∗ (1535) thattakes into account the contributions of the qqqq ¯ q compo-nents discussed above falls into three parts: (1) the contri-butions of the direct transition γ ∗ qqq → qqq , i. e. the con-tributions of the qqq components in proton and N ∗ (1535),(2) the contributions of the lowest energy qqqq ¯ q compo-nent in N ∗ (1535), and (3) the contributions of the next-to-next-to-lowest-energy qqqq ¯ q component in N ∗ (1535). Andeach of the latter two parts contains contributions fromtwo processes: the direct transition γ ∗ qqqq ¯ q → qqqq ¯ q , i.e. C. S. An, B. S. Zou: Role of the qqqq ¯ q components in the electromagnetic transition γ ∗ N → N ∗ (1535) the diagonal transitions, and the annihilation transition γ ∗ qqq → qqqq ¯ q (or γ ∗ qqqq ¯ q → qqq ), i.e. the off-diagonaltransitions.In the non-relativistic approximation the elastic andannihilation current operators are: h p ′ | j | p i elas = 1 , h p ′ | j | p i anni = 12 m [ σ · ( p ′ + p )] , h p ′ | j | p i elas = p ′ + p m + i m ( σ × q ) , h p ′ | j | p i anni = σ , (18)respectively. Here we have set q = p ′ − p . qqq components to thehelicity amplitude A P / and S P / For point-like quarks the electromagnetic transition oper-ator for elastic transitions between states with n q quarksis ˆ T A = − n q X i =1 e i m i [ √ σ i + k γ + ( p ′ i + + p i + )]ˆ T S = n q X i =1 e i . (19)Here ˆ T A and ˆ T S are the corresponding operators for thetransverse and longitudinal helicity amplitudes, which areobtained by coupling the current operators (18) to thetransverse ( ǫ +1 = − √ (ˆ x + i ˆ y ) and ǫ = 0) and longitu-dinal ( ǫ = 0 and ǫ = 1) photon [22], respectively. And e i and m i denote the electric charge and constituent massof the quark which absorbs the photon, respectively. p i + and p ′ i + are defined as the initial and final momenta of thequark that absorbs the photon: p ( ′ ) i + ≡ √ p ( ′ ) ix + ip ( ′ ) iy ) , (20)and ˆ σ i + is defined asˆ σ i + ≡
12 (ˆ σ ix + i ˆ σ iy ) , (21)which raises the spin of the quark which absorbs the pho-ton. Here we consider a right-handed virtual-photon inthe operator ˆ T A , and the momentum of photon has beenset to be: k = (0 , , k γ ) in the center-of-mass frame of thefinal resonance N ∗ (1535). It is related to the initial andfinal momentum of the quark which absorbs the photonas k γ = p ′ i − p i , and the magnitude of four-momentumtransfer Q = √− k as: k γ = Q + ( M ∗ − M N − Q ) M ∗ , (22)where M ∗ and M N denote the mass of N ∗ (1535) resonanceand proton, respectively. With the diagonal transition operators (19), the he-licity amplitudes of the electromagnetic transition γ ∗ p → N ∗ (1535), which include only the contributions from the qqq components in the proton and the N ∗ (1535), may beexpressed as A p / = 1 p K γ h N ∗ (1535) , , | ˆ T A | p, , − i ,S p / = 1 p K γ h N ∗ (1535) , , | ˆ T S | p, , i . (23)Here K γ is the real-photon three-momentum in the centerof mass frame of N ∗ (1535).With the wave functions (2) and the operators (19),we can obtain the helicity amplitudes A p / and S p / inthe following form: A p (3 q )1 / = A ( p ) q A ( N ∗ )3 q p K γ e m ( k γ ω + 2 ω {− k γ ω } S p (3 q )1 / = − A ( p ) q A ( N ∗ )3 q p K γ e √ k γ ω exp {− k γ ω } . (24)Here e denotes the electric charge of the proton. Themodel parameters are the constituent masses of the lightquarks m , the oscillator parameter ω , and the qqq com-ponents amplitudes A q . For the qqq model, we set theconstituent quark mass to be m = 340 MeV. The oscilla-tor parameter ω may be determined by the nucleon ra-dius as ω = √ h r i , which yield the value 246 MeV, whilethe empirical value for this parameter falls in the range110 −
410 MeV [9,24]. We give the numerical results asfunctions of Q by setting ω = 340 MeV, ω = 246 MeVand ω = 200 MeV, respectively, which are shown in figure1 and 2 for the transverse and longitudinal helicity am-plitudes, respectively, compared to the experimental dataextracted from Ref. [25,26,27]. As in this case there areno qqqq ¯ q components in the proton and N ∗ (1535) the am-plitudes are A ( p )3 q = A ( N ∗ )3 q = 1.As shown in figure 1, none of the three curves candescribe the experimental data satisfactorily. At the pho-ton point, Q = 0, the calculated helicity amplitudes are A p / = 0 . / √ GeV and A p / = 0 . / √ GeV with theparameter ω = 340 MeV and ω = 246 MeV, respec-tively, both of which are larger than the experimentalvalue A p / = 0 . ± . / √ GeV [25]. For the curvewhich is obtained by setting the parameter ω = 200 MeV,the calculated amplitude decribes the data better at thephoton point, but it falls too fast in comparison with thedata when Q increases. Similarly in the case of the longi-tudinal helicity amplitude S p / , none of the three curvescan fit the data well, as that all of them are too small nearthe photon point. . S. An, B. S. Zou: Role of the qqqq ¯ q components in the electromagnetic transition γ ∗ N → N ∗ (1535) 5 qqqq ¯ q components of N ∗ (1535) to the helicity amplitude A P / and S P / The contributions of the qqqq ¯ q components contain thedirect transition matrix elements γ ∗ qqqq ¯ q → qqqq ¯ q , andthe annihilation transition elements γ ∗ qqq → qqqq ¯ q and γ ∗ qqqq ¯ q → qqq , i. e. the diagonal and non-diagonal tran-sitions.The contributions from the diagonal transition ele-ments of the qqqq ¯ q components are obtained as matrixelements of the operator (19) (with n q = 5) between the5 q wave functions (8) and (15), and between the wavefunctions (8) and (17). At the first step, we consider thecontributions of the lowest energy qqqq ¯ q components in N ∗ (1535).The lowest energy configuration in the qqqq ¯ q compo-nents in N ∗ (1535) is the one in which the flavor-spin con-figuration of the four quark subsystem has the mixed sym-metry [31] F S [211] F [22] S , as mentioned in section 2. Forthe spin symmetry [22] S ( S = 0), both in the protonand N ∗ (1535), the matrix elements of the operator ˆ σ i + of these four quarks vanish. On the other hand, by the or-thogonality of the four-quark orbital states [31] X (for theproton) and [4] X (for the N ∗ (1535)), the matrix elementof the operator ˆ T A of the anti-quark does not contributeto the transition γ ∗ qqqq ¯ q → qqqq ¯ q . Finally the contri-butions to A p (5 q )1 / only come from the matrix element ofthe second term of the operator ˆ T A (19) between the fourquark states. In the case of the helicity amplitude S p (5 q )1 / ,the matrix element of the operator should vanish for theorthogonality of the different qqqq ¯ q states in proton and N ∗ (1535).Explicit calculation yields: A p (5 q )1 / = A ( p ) s ¯ s A ( N ∗ ) s ¯ s p K γ ω
24 ( e m − e m s ) exp {− k γ ω } S p (5 q )1 / = 0 . (25)Here we have neglected the transition between the uudd ¯ d component in the proton and the uuds ¯ s component in N ∗ (1535), which should be very tiny.Consider the the contributions of the non-diagonal tran-sition elements, i. e. the transitions γ ∗ qqq → qqqq ¯ q and γ ∗ qqqq ¯ q → qqq . By equation (18), the operator for thesetransitions may be expressed asˆ T Aanni = − X i =1 √ e i ˆ σ i + , ˆ T Sanni = X i =1 e i m i σ · ( p i + p ) . (26)Here ˆ T Aanni and ˆ T Sanni denote the transition operatorsfor A p ( anni )1 / and S p ( anni )1 / , respectively, and e i and m i arethe electric charge and the constituent mass of the annihi-lating quark, and ˆ σ i + raises the spin of the corresponding quark. p i is the momentum of the annihilating quark and p the momentum of the antiquark.First, we calculate the matrix elements for the tran-sitions γ ∗ qqq → qqqq ¯ q . These involve calculations of theoverlap between the γ ∗ qqq wave function and that for the qqqq ¯ q component in N ∗ (1535).In the case of the lowest energy configurations, theflavor-spin configuration has the mixed symmetry[31] F S [211] F [22] S , the explicit form of which is shown inappendix A. For the case of the color overlap, the onlycontribution comes from the color symmetry configurationof the qqqq ¯ q component in N ∗ (1535) which is denoted by[211] C . The matrix elements between the color singlet ofthe qqq component in the proton and the other two colorconfigurations ([211] C and [211] C ) vanish.Explicit calculation leads to the result: A p (3 q → q )1 / = − A ( p )3 A ( N ∗ ) s ¯ s p K γ √ eC exp {− k γ ω } ,S p (3 q → q )1 / = A ( p )3 A ( N ∗ ) s ¯ s p K γ e m s k γ C × exp {− k γ ω } . (27)Here the factor C denotes the orbital overlap factor: h ϕ ( κ ) ϕ ( κ ) | ϕ ( κ ) ϕ ( κ ) i = ( 2 ω ω ω + ω ) . (28)Here we have obtained the helicity amplitudes A p / and S p / for the electromagnetic transition γ ∗ p → N ∗ (1535),which contains the contributions both of the qqq and thelowest energy qqqq ¯ q components in the proton and N ∗ (1535).The results are shown in figure 3 and 4, respectively. Herewe have taken the probability of the qqqq ¯ q componentsin the proton as the tentative value P q = 20%, and in N ∗ (1535) P q = 45%. Taking the qqqq ¯ q components intoaccount, the constituent quarks masses should be a bitsmaller than the ones we employ in the previous section.To reproduce the mass for the nucleon when the five-quark components have been included, we take the values m u = m d = m = 290 MeV, and m s = 430 MeV. The oscil-lator parameters are ω and ω . The latter one is treatedas a free parameter in this manuscript. Note that the valuefor the ω may be different from that for ω . In our ex-tended quark model with each baryon as a mixture of thethree-quark and five-quark components, the two compo-nents represent two different states of the baryon. For the qqqq ¯ q state, there are more color sources than the qqq state, and may make the effective phenomenological con-finement potential stronger. This is consistent with otherempirical evidence favoring larger value of the ω [7,8,9,19]. An intuitive picture for our extended quark model islike this: the qqq state has weaker potential; when quarksexpand, a q ¯ q pair is pulled out and results in a qqqq ¯ q statewith stronger potential; the stronger potential leads to amore compact state which then makes the ¯ q to annihi-late with a quark easily and transits to the qqq state; this C. S. An, B. S. Zou: Role of the qqqq ¯ q components in the electromagnetic transition γ ∗ N → N ∗ (1535) leads to constantly transitions between these two states.Here presented are the results by setting ω = 340 MeVand ω = 600 MeV, ω = 246 and ω = 600 MeV, and ω = 340 MeV and ω = 1000 MeV, respectively. Asshown in figure 3, the results describe the experimentaldata for A p / well when the oscillator parameters are giventhe values ω = 340 MeV and ω = 600 MeV, both at thephoton point and larger Q . While in figure 4, the mag-nitude of the values for S p / are however larger than theexperimental value, even though the momentum depen-dence appears reasonable. Intriguingly when Q increasesto about 1 . GeV the calculated values change sign. Sohere we should conclude that this model can work whenthe Q is less than 1 . GeV , if Q is larger than this value,maybe we should consider the relativistic effect.There are another important parameter in our model,the phase factors δ between the qqq and qqqq ¯ q componentsof the N ∗ (1535) resonance, which has been taken to be +1.But in principle, this factor could be an arbitrary complexone exp { iφ } . As we know, the helicity amplitude A p / isreal, so here we may choose δ to be ±
1. And as shown infigure 3, the non-diagonal transition contributes a minusvalue to the A p / . If we assume δ to be −
1, the numericalvalue for A p / may be about 0 . / √ GeV, the result isnot so good. Consequently, the best choice for us may be δ = +1.Note that the diagonal contributions of the lowest en-ergy qqqq ¯ q component in N ∗ (1535) to S p / is 0. Actually,the contributions of the diagonal transition to A p / arealso very small, which is less than 0 . / √ GeV. But thenon-diagonal contributions are significant, and negativefor A p / and positive for S p / , which are shown in figure 3and 4 by the dash dot curves, with the parameter values ω = 340 MeV and ω = 600 MeV. For instance, at thephoton point, Q = 0, with the parameter values ω = 340MeV and ω = 600 MeV, the contribution to A p / is about − . / √ GeV, which can decrease the helicity amplitude A p / to describe the data. qqqq ¯ q components in N ∗ (1535) to the helicity amplitude A P / and S P / The next-to-next-to-lowest-energy configuration of the qqqq ¯ q components in N ∗ (1535) is the one in which theflavor-spin configuration of the four quark subsystem hasthe mixed symmetry [31] F S [22] F [31] S , as mentioned insection 2. For the same reason as that in section 3.2,the matrix element of the operator ˆ T A between the anti-quark states vanishes, and the diagonal contributions to S p / is 0. The difference is, there are two qqqq ¯ q compo-nents in N ∗ (1535) with the four quark flavor-spin config-uration [31] F S [22] F [31] S . Here we neglect the transitions γ ∗ uudd ¯ d → uuds ¯ s and γ ∗ uuds ¯ s → uudd ¯ d . Calculation leads to the result: A p (5 q ′ )1 / = − A ( p ) d ¯ d A ( N ∗ ) d ¯ d p K γ k γ ω e m exp {− k γ ω } − A ( p ) s ¯ s A ′ ( N ∗ ) s ¯ s p K γ k γ ω ( e m − e m s ) × exp {− k γ ω } ,S p q ′ / = 0 . (29)In the case of the annihilation transitions straightfor-ward calculation leads to the result that the flavor-spinoverlap factors for the transition γ ∗ qqq → qqqq ¯ q C F S van-ish, both for A p / and S p / . Therefore this transition doesnot contribute to the process γ ∗ p → N ∗ (1535).The next step is to consider the contributions of thetransitions γ ∗ qqqq ¯ q → qqq . The flavor-spin configurationfor the four-quarks subsystem of the qqqq ¯ q componentsin the proton is [4] F S [22] F [22] S , for which the explicitform has been given in Ref. [18]. After some calculation,it emerges that the flavor-spin overlap factors are also 0.Consequently, this process also yields no contribution tothe transition γ ∗ p → N ∗ (1535).Finally, we find that the the next-to-next-to-lowest-energy qqqq ¯ q components in N ∗ (1535) does not contributeto the helicity amplitude S p / for the electromagnetic tran-sition γ ∗ p → N ∗ (1535). The contributions to A p / onlycomes from the diagonal transition elements. As in thecase of the contributions of the lowest energy qqqq ¯ q com-ponents, this contribution is also very small. And as thethe non-diagonal contribution to A p / is 0, this 5-quarkcomponent does not contribute significantly to the lastresults. The result, which contains all the contributionsfor A p / that we have considered is shown in figure 5. Forcomparison, we have considered several different probabil-ities of the qqqq ¯ q components in N ∗ (1535), as is discussedin details in section 3.5. A n / at the photon point As we know, the ratio of the helicity amplitudes at photonpoint for proton and neutron is not a trivial issue. So wecalculate the A n / in this section. For isospin symmetry,the wave functions for neutron and n ∗ (1535) can be ob-tained by the wave functions we have given for the protonand p ∗ (1535). Consequently, we can calculate the matrixelements directly. After some calculations, we can get thefollowing results: A n (3 q )1 / = − A ( p ) q A ( N ∗ )3 q p K γ e m ( k γ ω + 2 ω {− k γ ω } A n (5 q )1 / = − A ( p ) s ¯ s A ( N ∗ ) s ¯ s p K γ ω
24 ( e m + e m s ) exp {− k γ ω } A n (3 q → q )1 / = − A ( p )3 A ( N ∗ ) s ¯ s p K γ √ eC exp {− k γ ω } . (30) . S. An, B. S. Zou: Role of the qqqq ¯ q components in the electromagnetic transition γ ∗ N → N ∗ (1535) 7 As we can see in equation (30), the non-diagonal contri-butions to the helicity amplitude A n / is same as the oneto A p / . Note that here we have only considered the con-tributions of the lowest energy five-quark components in n ∗ (1535), for that the probability of the next-to-next-to-lowest-energy qqqq ¯ q components should be some smaller,and on the other side, we can see in the section 3.3, thenon-diagonal contributions between the three-quark com-ponent and this five-quark configuration is 0, so it mayonly give a very small correction to our result, so we canneglect it here.If we set ω = 340 MeV, and the constituent quarkmasses are taken to be the same value as that in section3.1, then we can get that A n / = − . / √ GeV, which ismuch larger than the data A n / = − . ± . / √ GeV[25]. It indicates that we should consider the contribu-tions of the five-quark components. When the contribu-tions of the qqqq ¯ q components are taken into account,with the same parameter employed in section 3.2 (Thatfor the best fit), we can get that the helicity amplitude A n / = − . / √ GeV, which falls in the range of thedata. And the ratio A n / /A p / is then − .
82, which alsofalls well in the range of the data 0 . ± .
15. Note thatthe non-diagonal contributions to A n / is same as thatto A p / , and as we have obtained in section 3.2 that thiscontribution should be the major one of the qqqq ¯ q compo-nents, so we can conclude that the ratio A n / /A p / is notsensitive with the free parameter ω . qqqq ¯ q components in N ∗ (1535)Here we have calculated the helicity amplitude A p / byconsidering all the contributions of the qqq and low lying qqqq ¯ q components in the proton and N ∗ (1535). The lastresult is shown in figure 5 as a function of Q . Here all thelines are obtained by taking ω = 340 MeV and ω = 600MeV. The solid line is obtained by taking the probabilityof the lowest energy qqqq ¯ q components in N ∗ (1535) to bethe totally P q , and the dash dot line 0 . P q MeV, i.e.the probability of the next-to-next-to-lowest-energy qqqq ¯ q components is 0 . P q , and both the two lines are obtainedby setting P q = 55%. The dot line is obtained by setting P q = 35%, and the dash line P q = 75%, and both ofthe two lines are obtained by taking the probability of thelowest energy qqqq ¯ q component to be the totally P q .As shown in figure 5, the best description of A p / isgiven by the curve obtained by taking the probability ofthe lowest energy qqqq ¯ q components in N ∗ (1535) to be P q = 45%. This indicates that this electromagnetic tran-sition does not favor any large probability of the next-to-next-to-lowest energy qqqq ¯ q components in N ∗ (1535),although its contribution is tiny and the result is not verysensitive to it. The main result is that if the probability ofthe lowest energy 5 q components falls in the range 25-65%the calculated helicity amplitudes may fall within the datarange 90 ± / √ GeV at the photon point.
The helicity amplitudes A p / and S p / for the electromag-netic transition γ ∗ p → N ∗ (1535) were calculated, and thepossible role of the qqqq ¯ q was investigated. The resultsindicate that the extension of the qqq quark model to in-clude qqqq ¯ q components can bring about a much betterdescription of the empirical results. The results indicatethat the admixtures of qqqq ¯ q components in the N ∗ (1535)resonance might be in the range 25 − qqqq ¯ q components of the proton was as-sumed to be [31] X [4] F S [22] F [22] S , a configuration whichis expected to have the lowest energy, and therefore mostlikely to form appreciable components of the proton. In thecase of the resonance N ∗ (1535), the lowest energy config-uration is [4] X [31] F S [211] F [22] S , which requires that the5 q component of N ∗ (1535) should only be s ¯ s component.This means that there may be large strangeness compo-nents in N ∗ (1535) resonance, which is consistent with thestrong couplings of N ∗ (1535) N φ and N ∗ (1535) KΛ , pre-dicted in Refs. [10] and [11]. This is also in line with the KΣ quasibound state explanation for N ∗ (1535) by themechanism of dynamical resonance formation within thecoupled channel approach based on chiral SU (3) [3,4,5,15]. In addition we considered the contributions of the uudd ¯ d and uudu ¯ u components in the N ∗ (1535), whichhave the orbital-flavor-spin configuration [4] X [31] F S [22] F [31] S .The results show that this electromagnetic transition doesnot favor large qqqq ¯ q components with the next-to-next-to-lowest-energy.The suggested large probability s ¯ s components in N ∗ (1535),may be naturally consistent with the mass ordering ofthe resonances N ∗ (1440) and N ∗ (1535). For the Roperresonance, the largest 5 q component is uudd ¯ d [9], whilethat for N ∗ (1535) is uuds ¯ s component, which may makeit heavier than the roper resonance. If we neglect higherenergy configurations of the qqqq ¯ q components in thesetwo resonances, the s ¯ s components in the Roper shouldbe P q / N ∗ (1535), it is P q , whichmay make it heavier than the Roper resonance. We are grateful to Professor D. O. Riska for helpful dis-cussions and English editing of the draft, and Dr Q. B.Li for helpful discussions. This work is partly supportedby the National Natural Science Foundation of China un-der grants Nos. 10435080, 10521003, and by the ChineseAcademy of Sciences under project No. KJCX3-SYW-N2.
C. S. An, B. S. Zou: Role of the qqqq ¯ q components in the electromagnetic transition γ ∗ N → N ∗ (1535) A The explicit form of the flavor-spinconfiguration [31]
F S [211] F [22] S The explicit decomposition of the flavor-spin configuration[31]
F S [211] F [22] S may be expressed as [23] | [31] F S i = 12 {√ | [211] i F | [22] i S − | [211] i F | [22] i S + | [211] i F | [22] i S } , (31) | [31] F S i = 12 {√ | [211] i F | [22] i S + | [211] i F | [22] i S + | [211] i F | [22] i S } , (32) | [31] F S i = 1 √ {−| [211] i F | [22] i S + | [211] i F | [22] i S } , (33)and the explicit forms of the flavor symmetry [211] F andspin symmetry [22] S are | [211] i F = 14 { | uuds i − | uusd i − | duus i − | udus i−| sudu i − | usdu i + | suud i + | dusu i + | usud i + | udsu i} , (34) | [211] i F = 1 √ { | udus i − | duus i + 3 | suud i − | usud i +2 | dsuu i − | sduu i − | sudu i + | usdu i + | dusu i − | udsu i} , (35) | [211] i F = 1 √ {| sudu i + | udsu i + | dsuu i − | usdu i−| dusu i − | sduu i} , (36) | [22] i S = 1 √ { | ↑↑↓↓i + 2 | ↓↓↑↑i − | ↓↑↑↓i−| ↑↓↑↓i − | ↓↑↓↑i − | ↑↓↓↑i} , (37) | [22] i S = 12 {| ↑↓↑↓i + | ↓↑↓↑i − | ↓↑↑↓i − | ↑↓↓↑i} . (38) B The explicit form of the flavor-spinconfiguration [31]
F S [22] F [31] S The explicit decomposition of the flavor-spin configuration[31]
F S [22] F [31] S may be expressed as [23] | [31] F S i = 1 √ {| [22] i F | [31] i S + | [22] i F | [31] i S } , (39) | [31] F S i = 12 {√ | [22] i F | [31] i S + | [22] i F | [31] i S −| [22] i F | [31] i S } , (40) | [31] F S i = 12 {√ | [22] i F | [31] i S − | [22] i F | [31] i S −| [22] i F | [31] i S } , (41)the explicit forms of the flavor symmetry [22] F has beenshown in Ref. [18], and spin symmetry [31] S may be (for -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0020406080100120140160180 =340MeV =246MeV =200MeV A p 1 / ( - G e V - / ) Q (GeV ) Fig. 1.
The helicity amplitude A p / for γ ∗ p → N ∗ (1535) inthe qqq model. Here the solid line is obtained by taking ω =340 MeV, and the dash and dot lines are obtained by taking ω = 246 MeV and ω = 200 MeV, respectively. The datapoint at Q = 0 (square) is from Ref. [25], the other pointsare taken from Ref. [26] (triangles), [27] (open circles) and [28](filled circles). S z = +1) | [31] i S = 1 √ { | ↑↑↑↓i − | ↑↑↓↑i − | ↑↓↑↑i−| ↓↑↑↑i} , (42) | [31] i S = 1 √ { | ↑↑↓↑i − | ↑↓↑↑i − | ↓↑↑↑i} , (43) | [31] i S = 1 √ {| ↑↓↑↑i − | ↓↑↑↑i} , (44)(for S z = 0) | [31] i S = 1 √ {| ↑↑↓↓i + | ↓↑↑↓i + | ↑↓↑↓i − | ↓↑↓↑i−| ↑↓↓↑i − | ↓↓↑↑i} , (45) | [31] i S = 1 √ { | ↑↑↓↓i − | ↓↓↑↑i + | ↑↓↓↑i − | ↓↑↑↓i + | ↓↑↓↑i − | ↑↓↑↓i} , (46) | [31] i S = − {| ↓↑↑↓i − | ↑↓↓↑i + | ↓↑↓↑i − | ↑↓↑↓i} (47) References
1. S. Capstick and N. Isgur, Phys. Rev.
D34 (1986)28092. L.Ya. Glozman and D. O. Riska, Phys. Rept. (1996)2633. N. Kaiser, P. Siegel and W. Weise, Phys. Lett.
B362 (1995)234. D. Jido et al., Nucl. Phys.
A725 (2003) 1815. D. Jido, M. D¨oring and E. Oset, Phys. Rev.
C77 (2008)065207.. S. An, B. S. Zou: Role of the qqqq ¯ q components in the electromagnetic transition γ ∗ N → N ∗ (1535) 9 eV eV eV S / ( - G e V / ) Q (GeV ) Fig. 2.
The helicity amplitude S p / for γ ∗ p → N ∗ (1535) inthe qqq model. The data points are extracted from Ref. [26]. =340MeV, =600MeV =246MeV, =600MeV =340MeV, =1000MeV A p 1 / ( - G e V - / ) Q (GeV ) Fig. 3.
The helicity amplitude A p / for γ ∗ p → N ∗ (1535) con-tains the contributions both of the qqq and lowest energy qqqq ¯ q components in the proton and N ∗ (1535). Here the solid line isobtained by taking ω = 340 MeV and ω = 600 MeV, thedash line ω = 246 MeV and ω = 600 MeV, and the dot line ω = 340 MeV and ω = 1000 MeV, respectively. And thedash dot line is the absolute value of the contributions of the5-quark component with ω = 340 and ω = 600 MeV. Datapoint as in figure 1.6. S. Capstick, B. D. Keister and D. Morel, J. Phys. Conf.Ser. (2007) 0120167. Q. B. Li and D. O. Riska, Nucl. Phys. A766 (2006) 1728. Q. B. Li and D. O. Riska, Phys. Rev.
C73 (2006) 0352019. Q. B. Li and D. O. Riska, Phys. Rev.
C74 (2006) 01520210. J. J. Xie, B. S. Zou and H. C. Chiang, Phys. Rev.
C77 (2008) 01520611. B. C. Liu and B. S. Zou , Phys. Rev. Lett. (2006) 04200212. S. Capstick and W. Roberts, Pro. Par. Nucl. Phys. (2000) S24113. V. D. Burkert, Pro. Par. Nucl. Phys. (2005) 10814. C. Helminen and D. O. Riska, Nucl. Phys. A699 (2002)62415. N. Kaiser, T. Waas and W. Weise, Nucl. Phys.
A612 (1997) 297 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0-50-40-30-20-1001020 =340 Mev, eV =246 Mev, eV =340 Mev, eV S / ( - G e V - / ) Q (GeV ) Fig. 4.
The helicity amplitude S p / for γ ∗ p → N ∗ (1535) con-tains the contributions both of the qqq and lowest energy qqqq ¯ q components in the proton and N ∗ (1535). The dash dot line isthe emperical value of the contributions of the 5-quark com-ponent with ω = 340 and ω = 600 MeV. Data point as infigure 2. -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0020406080100120140160180 A p 1 / ( - G e V - / ) Q (GeV ) Fig. 5.
The total helicity amplitude A p / for γ ∗ p → N ∗ (1535).Here all the lines are obtained by taking ω = 340 MeV and ω = 600 MeV. The solid line is obtained by taking the prob-ability of the lowest energy qqqq ¯ q components in N ∗ (1535) tobe the totally P q , and the dash dot line 0 . P q MeV, i. e. theprobability of the next-to-next-to-lowest energy qqqq ¯ q compo-nents is 0 . P q , and both the two lines are obtained by setting P q = 0 .
55. The dot line is obtained by setting P q = 0 . P q = 0 .
75, and both of the two lines areobtained by taking the probability of the lowest energy qqqq ¯ q component to be the totally P q . Data point as in figure 1.16. M. D¨oring, E. Oset and B.S. Zou, Phys. Rev. C78 (2008)02520717. B. S. Zou and D. O. Riska, Phys. Rev. Lett. (2005)07200118. C. S. An, D. O. Riska and B. S. Zou, Phys. Rev. C73 (2006) 03520719. C. S. An, Q. B. Li, D. O. Riska and B. S. Zou, Phys. Rev.
C74 (2006) 055205, Erratum-ibid.
C79 (2007) 0699010 C. S. An, B. S. Zou: Role of the qqqq ¯ q components in the electromagnetic transition γ ∗ N → N ∗ (1535)20. C. S. An, Nucl. Phys. A797 (2007) 131 , Erratum-ibid.
A801 (2008) 8221. C. S. An and D. O. Riska, Eur. Phys. J.
A37 (2008) 26322. B. Juli´a-D´ıaz, T.-S. H. Lee, T. Sato and L. C. Smith, Phys.Rev.
C75 (2007) 01520523. J. Q. Chen,
Group Representation Theory for Physi-cists,2nd edition (World Scientific, Singapore, 1989)24. R. Koniuk and N. Isgur, Phys. Rev.
D21 (1980) 1868,Erratum-ibid.
D23 (1981) 81825. Particle Group Data, J. Phys.
G33 (2006) 126. I. G. Aznauryan et al., Phys. Rev.