Electroproduction of the Roper resonance on the proton: the role of the three-quark core and the molecular N-sigma component
Igor T. Obukhovsky, Amand Faessler, Dimitry K. Fedorov, Thomas Gutsche, Valery E. Lyubovitskij
aa r X i v : . [ h e p - ph ] A p r Electroproduction of the Roper resonance on the proton:the role of the three-quark core and the molecular
N σ component
Igor T. Obukhovsky , Amand Faessler , Dimitry K. Fedorov , Thomas Gutsche , Valery E. Lyubovitskij ∗ Institute of Nuclear Physics, Moscow State University,119991 Moscow, Russia Institut f¨ur Theoretische Physik, Universit¨at T¨ubingen,Kepler Center for Astro and Particle Physics,Auf der Morgenstelle 14, D–72076 T¨ubingen, Germany (Dated: November 9, 2018)The Roper resonance is considered as a mixed state of a three-quark core configuration and ahadron molecular component N + σ . Based on this ansatz we study electroproduction of the Roperresonance. The strong and electromagnetic couplings induced by the quark core are calculated inthe P model. The contribution of the vector meson cloud to the electromagnetic transition is givenin the framework of the VMD model. Results are compared with the recent JLab electroproductiondata. PACS numbers: 12.39.Ki, 13.40.Gp, 14.20.Gk, 13.40.Hq, 14.20.GkKeywords: Roper resonance, quark model, hadron molecules, strong and electromagnetic form factors
I. INTRODUCTION.
The structure issue of the lowest lying nucleon reso-nance N (1440) with J P =
12 + (the Roper resonance P or simply R ) has been a long standing problem of hadronphysics. One of the indication that the inner structure ofthe Roper is possibly more complicated than the struc-ture of the other lightest baryons was obtained some timeago in the framework of the constituent quark model(CQM). It was found (see, e.g. [1]) that the observedmass of the Roper resonance is much too low and thedecay width is too large when compared to the predictedvalues of the CQM.The simplest description of the Roper consists of thethree-quark (3 q ) configuration sp [3] X , i.e. the first (2 S )radial excitation of the nucleon ground state s [3] X , butit fails to explain either the large decay width Γ R ≃
300 MeV or the branching ratios for the πN (55-75%)and σN (5-20%) decay channels [2, 3]. Evaluation ofthese values in the framework of the CQM is often basedon the elementary emission model (EEM) with single-particle quark-meson (or quark-gamma) couplings qqπ , qqσ , qqγ , etc.. The calculation of decay widths (or ofthe electroproduction cross section at small virtuality ofthe photon with Q ≃
0) results in anomalous small val-ues. These underestimates can especially be traced tothe strict requirement of orthogonality for the ground(0 S ) and excited state (2 S ) radial wave functions of the N - and R states belonging to the quark configurations ∗ On leave of absence from Department of Physics, Tomsk StateUniversity, 634050 Tomsk, Russia with the same spin-isospin ( S = 1 / T = 1 /
2) and sym-metry ([3] ST [3] X ) quantum numbers. To overcome thisdiscrepancy it is suggested that either the Roper is notan ordinary 3 q state or the ”true” transition operatorshave a more complicated form than the single-particleoperators used in calculations.Quark models with Goldstone boson interactions [4]can explain why the mass of the Roper resonance isshifted to the observed value including the correct levelordering. But these models still fail to get the strong de-cay widths and electromagnetic couplings under control.On the experimental side there has been noticeableprogress in the experimental study of the Roper reso-nance in the last decade. The Roper resonance has beenstudied in π [5] and ππ [6] electroproduction processeson the proton with the polarized electron beam at theJLab (CLAS Collaboration) followed by a combined anal-ysis of pion- and photo-induced reactions made by CB-ELSA and the A2-TAPS Collaborations [3]. These recentdata present new possibilities for the study of the lightestbaryon resonances.Several models for the description of the Roper res-onance electroexcitation were proposed during the lastthree decades [7–9, 11–17] (see the review [18] for a de-tailed discussion). Now model predictions can be com-pared with the new high-quality photo- and electropro-duction data [3, 5, 6], and updated versions [19–21] of themost realistic models give a good description of the dataat intermediate values of 1.5 . Q . . However,in the ”soft” region, i.e. at low values of Q (0 ≤ Q .
1- 1.5 GeV ), the data differ qualitatively from the theo-retical predictions: the experimental helicity amplitude A / changes sign at Q ≈ and it is large andnegative at the photon point Q = 0. Theoretical predic-tions for A / are large and positive at Q ≈ and quickly go to a small negative (or zero) value at thephoton point.For pion electroproduction in the resonance region W ≃ m R the behavior of the transverse helicity ampli-tude A / near the photon point Q & Q CLAS pπ + π − data [25] wasobtained in Ref. [6] on the basis of JLab-Moscow (JM)model [26, 27] with taking into account the π ∆ channelalong with additional contact terms and the direct 2 π production. The contribution of the meson (pion) cloudto the Roper resonance mass was recently calculated inRefs. [28, 29].As a result, there are essentially two comprehensivetheoretical approaches to the Roper electroproduction onthe market. One of them (the coupled channel model ofthe meson cloud [6, 22–24]) is only successful in the softregion 0 ≤ Q . and the other one (the light front(LF) three-quark model [13, 19] or the covariant quarkspectator model [20]) is compatible with data in the hardregion 1.5 . Q . .Universal, but more phenomenological approaches whi-ch pretend to cover both regions of Q were also sug-gested (see, e.g. Refs. [16] and [21]). In Ref. [16] a 3 q + ¯ qq approach was suggested using the P model [7] and vec-tor meson dominance (VMD) in combination with theEEM. In Ref. [21] a generalization of the Cloudy BagModel (CBM) [30] was used for the case of the open in-elastic channels π ∆ and σN in combination with a phe-nomenological strong background interaction.In such combined approaches two types of electromag-netic transition operators are used, the operator designedfor the soft Q region and one for hard values of Q . How-ever, in the transition amplitude they are summed for anyvalue of Q . For example, in the generalized P + EEMapproach [16] the transition operator includes the sum oftwo vertices, schematically sketched in Figs. 1a and b. V= ρ,ω, ...γ γ e v f f v2s0s0s 0s0s2s Ψ v N e f v Q < Q > ~ χχ ~ V= ρ,ω, ... p p p p p ** R Rp N(0s ) a b FIG. 1: Diagrams of (a) “soft” (non-local) and (b) “hard”(local) coupling of vector mesons to the nucleon quark core.
In the present paper we follow a more physical con-cept (see, e.g., Ref. [31] where the constituent quark and parton approaches to the γqq vertex are discussedin the context of the nucleon electromagnetic form fac-tors). We can consider that the diagram in Fig. 1a rep-resents the unknown large-distance physics described bya phenomenological model (the P model in our case),which is adjusted to low-energy data (i.e. meson-nucleoncoupling constants πN N , ρN N , magnetic moments anddecay widths). In the hard Q -region these contributionsbecome less important and an adequate description of theelectromagnetic transition will be given by the diagramin Fig. 1b. In this case the unknown short-range physicsis encoded by adjusted parameters of a parton model.In the region of moderate values of Q (1.5 . Q . ) we can consider the constituent quarks as par-tons and corresponding unknown short-range physics canbe included in a few constituent quark parameters (suchas quark form factors given by the intermediate vectormesons in the VMD model and scale parameters of quarkconfigurations in the baryons). In this case it is not nec-essary to sum the contributions of the two diagrams inFig. 1. Instead it would be desirable to use some mech-anism for a smooth transition from one regime to theother.In our opinion such a mechanism can be described ingeneral by a smooth transition from a typical hadronradius b V ≈ b V = 0 corresponding to thequark-parton picture sketched in Fig. 1b. Here we use theapproximation b V ( Q ) = b V (0) e − Q /χ , where χ ≃ − Q where the parton model phenomenology in deep inelastic ep scattering sets in.Another important issue related to the Roper reso-nance is a possible combined structure of this state whichimplies a virtual hadron-hadron component (e.g. σN or/and π ∆) [32] in addition to the radially excited three-quark structure. Here we consider an admixture of thehadronic molecular state N + σ in an effective descriptionof such a component. We also consider to what degreesuch a combined structure for the Roper is compatiblewith the new high-quality data of JLab. II. COMPOSITE STRUCTURE OF THE ROPERRESONANCE
We consider the Roper resonance ( R ) as a superposi-tion of the radially excited three-quark configuration 3 q ∗ and the hadron molecule component N + σ as: | R i = cos θ | q ∗ i + sin θ | N + σ i , (1)where θ is the mixing angle between the 3 q ∗ and thehadronic component: cos θ and sin θ represent theprobabilities to find a 3 q ∗ and hadronic configuration,respectively. The parameter θ is adjusted to optimizethe description of data on the Roper resonance electro-production. The limiting case of cos θ = 1 corresponds tothe pure 3 q ∗ interpretation of R , while the value cos θ = 0corresponds to the situation, where R is a pure loosebound state of N + σ (analogous to the deuteron – boundstate of proton and neutron). Note, in a first step wesimplify the model by reducing it to two independent(decoupled) systems, R = 3 q ∗ and R = N + σ , anddo not consider the full coupled channel problem. More-over, we consider the dynamics of the R component inthe framework of the nonrelativistic P model (see, e.g.Refs. [7, 33]), while the dynamics of the R componentis considered in the framework of the hadronic molecu-lar approach [34] which is manifestly Lorentz invariant.In future we intend to improve the description of the R component by applying a relativistic quark model.First we briefly outline the basic notions of the P mo-del. The effective interaction term of the P model [33,35] is set up as H eff q = g q Z d x ¯ ψ q ψ q , g q = 2 m q γ , (2)where γ is dimensionless constant. It can be consideredas a static variant of the coupling γ ′ ¯ q ( x ) q ( x ) S ( x ) wherethe external field S ( x ) represents some scalar combina-tion of gluon fields in the hadron. At low energy, wherethe dynamics is ruled by nonperturbative QCD, we passto an effective description in terms of constituent quarks ψ q ( x ) and substitute a constant for the field S ( x ).Apart from some drawbacks [see, e.g. Eqs. (B1) and(B3) in Appendix], the P model [33, 36, 37] is a goodphenomenological method for the evaluation of hadrontransitions [38–41] on the basis of the quark model start-ing from Eq. (2) with a single strength parameter γ . Theinteraction term (2) gives rise to Feynman amplitudes forthe ¯ qq -pair creation (annihilation)(2 π ) δ (3) ( p + p ) i M eff fi = h q, p , µ |h ¯ q, p , µ | i Z d x L eff q ( x ) | i , (3)which are used here for the calculation of meson-baryoncouplings. The quark is labelled by its 3-momentum p and spin projection µ (for simplicity the isospin pro-jection t and the color are omitted), similarly for theantiquark. For the numbering of the quarks see Fig. 1(or Fig. 8 in Appendix B).The corresponding non-relativistic interaction term V eff q is defined as T eff fi = nr h q, p , µ | nr h ¯ q, p , µ | V eff q | i . = 12 m q M eff fi , (4)where a noncovariant normalization nr h p , µ | p ′ , µ ′ i nr = (2 π ) δ (3) ( p − p ′ ) δ µ,µ ′ (5)is implied instead of the covariant one of Eq. (3).Substitution of the non-relativistic reduction of the ef-fective interaction (2) into Eqs. (3) and (4) leads to the expression V eff q . = g q m q ( − − µ − t h − µ | σ · ( p − p ) | µ i× h − t | t i (2 π ) δ (3) ( p + p ) , (6)which is the nonrelativistic analogue of the ¯ qq pair cre-ation (annihilation) operator.The description of the hadronic N + σ component ofthe Roper resonance is based on the compositeness con-dition [42, 43]. This condition implies that the renor-malization constant of the hadron wave function is setequal to zero or that the hadron exists as a bound stateof its constituents only. In the case of mixed states (asin the present situation where the Roper is a superposi-tion of the 3 q ∗ and N + σ components) the amplitude forthe N + σ component is defined by the parameter sin θ .The compositeness condition was originally applied tothe study of the deuteron as a bound state of proton andneutron [42]. Then it was extensively used in low–energyhadron phenomenology as the master equation for thetreatment of mesons and baryons as bound states of lightand heavy constituent quarks (see e.g. Refs. [43, 44]).By constructing a phenomenological Lagrangian includ-ing the couplings of the bound state to its constituentsand of the constituents to other particles in the possi-ble decay channels we calculated hadronic-loop diagramsdescribing different decays of the molecular states (seedetails in [34]). R σ R N
FIG. 2: The Nσ loop diagram contributing to the Roper massoperator. In the present case the R → N + σ coupling is fixedfrom the compositeness condition Z R = 1 − Σ ′ Nσ ( p ) | p = m R = 0 , (7)where Σ Nσ ( p ) is the mass operator of the N σ bound state(Fig. 2), calculated with the use of the phenomenologicalLagrangian L str R ( x ) = g R σ N ¯ R ( x ) Z dy Φ R ( y ) × N ( x + w σN y ) σ ( x − w Nσ y ) + H . c . , (8)where w ij = m i / ( m i + m j ). Here Φ R ( y ) is the cor-relation function describing the distribution of N σ in-side R , which depends on the Jacobi coordinate y . ItsFourier transform used in the calculations has the formof a “modified” Gaussian, i.e. the Gaussian multipliedby a polynomial. In Euclidean space it may be writtenas ˜Φ R ( − k E ) = (cid:18) − λ k E Λ M (cid:19) exp (cid:18) − k E Λ M (cid:19) , (9)where k E is the Euclidean momentum. This present akind of generalization of the nonrelativistic quark modelwave function to the 4-dimensional case. But the rela-tivistic parameters λ and Λ M should differ from the cor-responding nonrelativistic ones. Here Λ M is the molecu-lar size parameter and λ is a free parameter which shouldbe fixed by the orthogonality condition, i.e. h N | R i = 0. III. ROPER ELECTROPRODUCTION
The diagrams which contribute to the Roper resonanceelectroproduction are shown in Fig. 1 (contribution of the3 q ∗ component) and Fig. 3 (contribution of the hadronic N σ component). In the following we discuss the separatecontributions of the structure components of the Roperresonance. + +
N R γ * γ * γ * N R N R σσσ
NNN + γ * σ NN R + N σ N R γ * cba d e FIG. 3: Nσ hadronic-loop diagrams contributing to the Roperelectroproduction: the triangle diagram (a), the bubble dia-grams (b) and (c), the pole diagrams (d) and (e). A. Contribution of the q ∗ component The contribution of the 3 q ∗ component to the hadroniccurrent of the Roper electroproduction is generally givenas J µ = h R | j µq | N i . = h R, p ′ , S ′ z , T ′ z | j µq | N, p , S z , T z i . (10)The current j µq is derived by starting from the vectormeson absorption amplitudes described in the P model T q ( λ ) V + N → R = 3 nr h R, , S ′ z , T ′ z | V effq | N, − q , S z , T z i×| V, q , λ V , t V i nr (11)[see Appendix B for details] and use of the vector me-son dominance (VMD) mechanism in the photon-quarkcoupling: e ǫ ( λ ) µ J µ = e X V = ρ,ω M q ( λ ) V + N → R g V NN M V Q + M V . (12)The vector meson-nucleon coupling constant g V NN is cal-culated in the P model [see Appendix B] and we use M q ( λ ) V + N → R = 2 m M √ M V T q ( λ ) V + N → R by taking into accounta noncovariant normalization (5) in Eq. (11). M V is thevector meson mass approximated as M V = M ρ ≈ M ω ; p , S z , T z ( p ′ , S ′ z , T ′ z ) and q , λ ρ , t ρ are the 3-momentum,spin and isospin projections of the nucleon (the Roper)and of the vector meson, respectively. For conveniencewe choose the photon momentum as q µ = ( q , , , | q | ).After substitution of the quark substructure (see Ap-pendix A 2) for | N i , | R i and | V i into Eqs. (10) – (12)and with a simple algebra (here we use m q = m N / J µ ǫ ( λ ) µ = √ (cid:20) n ( y o ) n ( y ) (cid:21) / e − ζ ( y ) q b / M V Q + M V × (cid:26) h τ z i T δ S ′ z ,S z (cid:20)(cid:18) ǫ ( λ )0 + q · ǫ ( λ ) m N n ( y ) (cid:19) × p ( y, q ) + p ( y ) q · ǫ ( λ ) m N n ( y ) (cid:21) − h τ z i T h i [ σ × q ] · ǫ ( λ ) m N i S p ( y, q ) (cid:27) . (13)We use the notation h· · · i S = h S ′ z | · · · | Sz i , h· · · i T = h T ′ z | · · · | T z i for the spin and isospin matrix elements, respectively, ǫ ( λ ) µ = { ǫ ( λ )0 , ǫ ( λ ) } is the photon polarization vector and n, ζ, p , are polynomials in y = b V /b and q : n ( y ) = 1 + 23 y , ζ ( y ) = 1+ y n , p ( y ) = 43 1+ y n ,p ( y, q ) = 23 y n − (cid:18) y n (cid:19) q b . (14)The transverse ( λ = ±
1) and longitudinal ( λ = 0) helic-ity amplitudes for electroproduction of the Roper reso-nance on the proton ( T z = 1 /
2) are defined by the matrixelements (13) for λ = +1 and 0 respectively [11, 13, 20] A / = r παq R h R, , + | j µq ǫ (+) µ | N, − q , − i ,S / = r παq R h R, , + | j µq ǫ (0) µ | N, − q , + i | q | Q (15)where α = 1 /
137 is the fine-structure constant. We in-troduce q R = m R − m N m R (16)for the threshold value of the photon 3-momentum forRoper electroproduction.In the rest frame of the Roper resonance (the c.m.frame of γ ∗ N collision) the absolute value of the trans-fered three-momentum q in Eqs. (15) is defined by q = Q + (cid:18) Q + m N − m R m R (cid:19) . (17)Note that in the region of 0.5 . Q . the c.m.frame is very close to the Breit frame, i.e. Q ≈ q , whichis very convenient for comparison of our q -dependentresults with the relativistic Q -dependent ones (substi-tution of q → Q does not really change our results ifone considers the region 0.5 . Q . ).We have several remarks regarding current conserva-tion connected to the gauge symmetry of theory. Thecurrent conservation condition q µ J µ = 0 for the matrixelements (10) is not automatically satisfied for the VMDamplitudes. To provide q µ J µ = 0 for a transition cur-rent in the VMD amplitudes one needs the conservationof the neutral vector meson currents ∂ µ J Vµ = 0 [47]. Inour model these currents J Vµ are expressed via the am-plitudes (11) for which the relation J = | q | q J is notexactly satisfied.One can try to construct the electromagnetic currentusing the transverse projector g µν ⊥ = g µν + q µ q ν Q . Thisprojector does not change the A / amplitude, but couldlead to some corrections for the components J and J .However, the expression for S / in Eq. (15) is invariantwith respect to such corrections because of the contrac-tion of the current matrix elements with the longitudinalpolarization vector ǫ (0) µ = (cid:8) | q | Q , , , q Q (cid:9) . (18)Some problem appears in a small region near the photonpoint with Q = 0 where the last factor for S / in ex-pression (15) shows singular behavior. In this region weuse the following trick. We start from the exact equality J = J at Q = 0 (19)which follows from the Ward identity at the photon pointwhere q = | q | . Note that at Q = 0 we really have J ≈ J if we use realistic parameters of the constituentquark model (CQM) for the wave functions of baryonsand mesons. Thus it is not difficult to transform the ap-proximate equality J ≈ J to the exact one of Eq. (19)by slightly varying one of the free parameters of the CQM(e.g. the radius b R of the quark core of the Roper res-onance which is not strictly fixed otherwise). The con-straint (19) imposed on the parameters of the quark wavefunctions in the P amplitudes only stabilizes the behav-ior of S / near the photon point Q . anddoes not give pronounced effects for S / in the remainingregion for Q , where | q | Q ≈ A / = − r παq R √ µ p h σ + i (cid:20) y R n ( y ) N ( y, y R ) (cid:21) / M V Q + M V × e − ˜ ζ ( y,y R ) q b | q | m N P ( y, y R , q ) (20) and S / = − r παq R √ (cid:20) y R n ( y ) N ( y, y R ) (cid:21) / M V Q + M V e − ˜ ζ ( y,y R ) q b × q Q (cid:26)(cid:2) q ( y R − )2 m N N ( y, y R ) (cid:3) P ( y, y R , q )+ q m N N ( y, y R ) P ( y, y R ) (cid:27) (21)where y ≡ y ( Q ) = y exp( − Q /χ ). We also take intoaccount a possible difference of the R -resonance radius b R and the one of the nucleon, which is b , by introducingthe ratio y R = b R /b which does not depend on Q . Asa result the polynomials (14) become y R -dependent onesnow denoted as N, P , , ˜ ζ [see Eqs. (B17) – (B18) in theAppendix] and only for y R = 1 they are identical with n, p , , ζ : n ( y ) = N ( y, y R = 1) , p ( y, q ) = P ( y, y R = 1 , q ) ,p ( y ) = P ( y, y R = 1) , ζ ( y ) = ˜ ζ ( y, y R = 1) . (22) Q (GeV /c ) G p M / µ p FIG. 4: Normalized magnetic form factor of the proton G pM /µ p in the modified P model with a Q -dependent vectormeson radius and the VMD approach to the qqγ interaction(dashed line). Here we use the same set of parameters as inthe N → R vertex of Fig. 6a. For comparison, the dipoleapproximation is also shown (solid line). The vector meson contribution to the amplitude athigh Q should also contain contributions of vectormesons of higher mass M V & M ρ . Following thework [31] we use the approximation M V Q + M V = x M ρ Q + M ρ + (1 − x ) 4 M ρ Q + 4 M ρ ,x = 0 . , (23)which we have checked in the description of the nucleonmagnetic form factor.Note the matrix element (10) for the diagonal transi-tion N + γ ∗ → N has the same form as Eq. (13) exclud-ing the algebraic factor √ and the polynomial p whichshould be changed to 1. In the static limit | q | , q → m q = m N / e = e I + τ z , ˆ µ = µ N I + 5 τ z , µ N = e m N . (24)The values of µ p and µ n are reproduced with an accuracyof about 10%. Moreover, at low and moderate values of Q this amplitude describes the nucleon magnetic formfactor G M with a reasonable accuracy (see Fig. 4). Suchan accuracy is sufficient (at least in the region Q . ) for the present calculation of the Roperelectroproduction amplitudes.For the non-diagonal process N + γ ∗ → R the matrixelement (13) defines ‘the transition magnetic moment‘ inthe limit | q | , q → q R (i.e. at the photon point):ˆ µ N → R = − e m N ( I + 5 τ z )2 √ exp [ − ζ ( y ) q R b / × (cid:20) y / y / − (cid:18) y y / (cid:19) q R b (cid:21) . (25)The quantity ˆ µ N → R gives the value (apart from a kine-matical factor h σ + i p q R /
2) of the transverse helicity am-plitude A / at the photon point. The first term in thesquare brackets of the r.h.s. of Eq. (25) Z V = 2 y / y / , y = b V /b (26)(or the first term of the polynomial p in Eq. (14)) ispresent because of the nonlocality of the V qq interactiondefined by Eq. (11). There the operator V effq leads toan insertion of the inner ¯ qq wave function of the vectormeson into the V qq vertex.The size of the nonlocal region is defined by the spa-tial scale of the meson wave function. For a point-likevector meson ( b V = 0) the value of Z V reduces to zero,and the matrix element for the transition N + γ ∗ T → R reduces to the matrix element of the elementary-emissionmodel (EEM) with a local V qq vertex. The EEM ma-trix element vanishes in the limit | q | →
0, as it shouldbecause of the orthogonality of the spatial parts of thewave functions of N and R .Such behavior of the A / amplitude near the photonpoint Q = 0 is characteristic of all the models whichstart from local γqq or V qq vertices at high Q and con-tinue to use such interaction in the ‘soft‘ region of small Q . /b V , where the e.-m. interaction is modified bythe inner structure of vector mesons as ¯ qq bound states.As a result, in models with a local operator for the γqq (or V qq ) interaction (see, e.g. the relativistic models [10–13, 20]) the transverse helicity amplitude A / vanishesin the limit Q → V qq interac-tion in the description of Roper electroproduction near the photon point was first noted by the authors of the P model [7]. In Ref. [16] this nonlocal P interactionwas used for the calculation of the helicity amplitudes onthe basis of a dynamical quark model of baryons. Unfor-tunately, the authors of [16] have only used a trivial sumof P and EEM interaction terms (a ‘generalized EEM‘).With this ansatz they describe both the low- and high- Q amplitudes with a common mechanism, and the samequark dynamics was used for both the nucleon and theRoper resonance.Now it becomes evident that intermediate meson-baryon states (‘hadron loops‘) can play a considerablerole in the quark dynamics of excited baryons, and suchmeson-baryon states should be taken into account (see,e.g., Ref. [48, 49]). Since the resonance pole of theRoper [2] 1365 - i95 MeV is rather close to the N + σ threshold the intermediate N + σ configuration will playa more important role in the inner dynamics of the Roperas compared for example to the case of the nucleon.In our opinion, a first step in the study of the non-trivial inner structure of the Roper resonance could bean evaluation on the basis of the recent CLAS data [5],where a nonvanishing probability for a possible N + σ component of the Roper is compatible with the data.. B. Contribution of the hadronic N + σ component The hadronic
N σ loop diagrams contributing to theRoper electroexcitation are shown in Fig. 3. The
RN σ vertex is defined by the nonlocal Lagrangian L R ofEq. (8). For the N N σ vertex we use a similar nonlo-cal Lagrangian with the correlation function Φ N ( y ) L N = g NN σ σ ( x ) Z dy Φ N ( y ) × ¯ N ( x + y/ N ( x − y/ , (27)where g NNσ is the
N N σ coupling constant, ˜Φ N ( − k E ) =exp (cid:16) − k E Λ N (cid:17) is the Fourier transform of Φ N ( y ) in Eu-clidean space with Λ N = 0.7 – 1 GeV.The electromagnetic interaction Lagrangian containstwo pieces L emint = L em(1)int + L em(2)int (28)which are generated after the inclusion of photons. Thefirst term L em(1)int is standard and is obtained by minimalsubstitution in the free Lagrangian of the proton andcharged Roper resonance: ∂ µ B → ( ∂ µ − ie B A µ ) B , (29)where B stands for p, R + and e B is the electric charge ofthe field B . The interaction Lagrangian L em(1)int reads L em(1)int ( x ) = e B ¯ B ( x ) A B ( x ) . (30)The second electromagnetic interaction term L em(2)int isgenerated when the nonlocal Lagrangians (8) and (27)are gauged. The gauging proceeds in a way suggestedand extensively used in Refs. [44–46]. In order to guaran-tee local U (1) gauge invariance of the strong interactionLagrangian one multiplies each charged field in (8) and(27) with a gauge field exponential e − ie B I ( y,x,P ) . Theexponent contains the term I ( y, x, P ) = y Z x dz µ A µ ( z ) , (31)where P is the path of integration from x to y . Then weobtain L str+em(2) R ( x ) = g R σ N ¯ R ( x ) Z dy Φ R ( y ) × n ( x + w σN y ) σ ( x − w Nσ y )+ g R σ N ¯ R + ( x ) Z dy Φ R ( y ) × e − ie p I ( x + w σN y,x,P ) p ( x + w σN y ) × σ ( x − w Nσ y ) + H . c . (32)and L str+em(2) N ( x ) = g NN σ σ ( x ) Z dy Φ N ( y ) × (cid:18) ¯ n ( x + y n ( x − y p ( x + y e − ie p I ( x − y ,x + y ,P ) p ( x − y (cid:19) . An expansion of the gauge exponential up to terms linearin A µ leads to L em(2)int .The full Lagrangian consistently generates the requiredmatrix element of the electroexcitation amplitude whichis linked to coming the hadronic molecular component ofthe Roper. Because of gauge invariance the electromag-netic vertex function Λ µ ( p, p ′ ) is orthogonal to the photonmomentum q µ Λ µ ( p, p ′ ) = 0. As a result, the vertex func-tion Λ µ ( p, p ′ ) is given by the sum of the gauge-invariantpieces of the triangle (∆), the bubble (bub) and the pole(pol) diagrams, while the non gauge-invariant parts ofthese diagrams cancel in the sum:Λ µ ( p, p ′ ) = Λ ⊥ µ, ∆ ( p, p ′ ) + Λ ⊥ µ, bub ( p, p ′ )+ Λ ⊥ µ, pol ( p, p ′ ) . (34)The contribution of each diagram can be split into agauge invariant piece and a reminder term, which is notgauge invariant, by introducing the decomposition γ µ = γ ⊥ µ + q µ qq , p i µ = p ⊥ i µ + q µ p i qq , (35)with γ ⊥ µ q µ = 0, p ⊥ i µ q µ = 0, where p i is p or p ′ . Thevertex function Λ ⊥ µ ( p , p ) can then be expressed in termsof γ ⊥ µ and p ⊥ i µ . In the case of the triangle diagram of Fig. 3a we in-clude the q -dependence of the photon-nucleon verticesin correspondence with data. Taking into account thenucleon structure the e p ¯ pγ µ ⊥ p vertex is modified as¯ N (cid:18) γ µ ⊥ F N ( q ) + i σ µν q ν m N F N ( q ) (cid:19) N, (36)where F N ( q ) and F N ( q ) are the Dirac and Pauli formfactors, which are normalized as F N (0) = e N (nucleonelectric charge) and F N (0) = κ N (nucleon anomalousmagnetic moment). The form factors F N , are expressedthrough the electric and magnetic Sachs form factors G NE , G NM of the nucleon as F N = ( G NE + τ G NM ) / (1 + τ ), F N =( G NM − G NE ) / (1 + τ ), τ = − q / m N . For the Sachs formfactors we use the Kelly parametrization [50]: G ( τ ) ∝ P nk =1 a k τ k P n +2 k =0 b k τ k . (37)Two additional contributing diagrams to the electro-production of the Roper resonance are shown in Fig. 5.The amplitudes of the σγ ∗ V ( V = ρ , ω ) transition arewritten in the form e g σγV M V ( g µν q · k − k µ q ν ) , (38)where k is the vector meson momentum. The values forthe coupling constants g σγV are estimated in the branchratio model [51] with g σγρ ≃ . g σγω ≃ . + N R γ N * σρ N N R ω σγ * a b FIG. 5: σγ ∗ V ( V = ρ , ω ) processes in the electroexcitationof the Nσ bound state. The contributions of the amplitudes of Fig. 5 were es-timated using the local limit for the
N N ρ and
N N ω ver-tices. We found a very small contribution compared tothe diagrams of Fig. 3. Both σγV diagrams are explicitlytransverse under contraction with the photon momentum q µ .Finally, the helicity amplitudes for the electromagneticexcitation are defined like in (15) A / = r παq R (cid:28) R, (cid:12)(cid:12)(cid:12)(cid:12) J + (cid:12)(cid:12)(cid:12)(cid:12) N, − (cid:29) ξS / = r παq R (cid:28) R, (cid:12)(cid:12)(cid:12)(cid:12) J (cid:12)(cid:12)(cid:12)(cid:12) N, (cid:29) ξ, (39) J + = − √ J x + i J y ) , where J µ is the electromagnetic transition current de-fined by the diagrams of Fig. 3. The helicity amplitudes(39) are defined up to a phase ξ . The amplitudes are writ-ten in the c.m. frame of the nucleon and the photon, i.e.in the Roper-resonance rest frame. The 4-spinors presentin | R i , | N i are normalized as ¯ RR = m R ε R , ¯ N N = m N ε N . IV. RESULTS AND COMPARISON WITH DATAA. Parameter fitting
In the calculation the helicity amplitudes A / and S / we use two variants for the free parameters, de-noted as (a) and (b), both typical for the CQM. Theywere only fitted to the A / JLab data [5, 6] without anyadditional adjustment to the S / data [we only take intoaccount the condition (19)]. One of the parameter setsgives the best description of the data in the ‘soft‘ regionwith 0 . Q . and the other one the optimaldescription in the whole measured interval including the‘hard‘ region of Q & /c .Note, we do not pretend that the non-relativistic modelfor the quark configurations is able to describe data forthe whole ‘hard‘ region. We only study the compatibil-ity of our predictions with the behavior of the data inthe transition region between the ‘soft‘ and the ‘hard‘regimes (a relativistic generalization of the model couldbe the next step to start from a ‘hard‘ variant which givesa realistic description of the data at low and moderatevalues of Q ).Our parameters are grouped into two sets: one set ofparameters is related to the 3 q components of the baryonsand the other set is connected with the N σ molecularcomponent. One of the quark model parameters is fixedby the strong constraints following from the Ward iden-tity [parameter b R , see Eq. (19)]. The additional freequark parameters b, b V and χ are adjusted to optimizethe description of the proton magnetic form factor in theconsidered region of Q , including the intermediate val-ues 0.5 . Q . /c , and of the above mentionedsubset of data on the helicity amplitude A / . Two ofthese fitted parameters, b and b V , should in addition havevalues which are typical for the quark core radii of the nu-cleon and a vector meson with b ≈ b V ≈ χ should not be smaller than the charac-teristic scale ∼ (1 – 1.5) m N associated with short-rangeeffects in eN scattering. These additional constraints onthe parameters b , b V and χ sufficiently limit the rangeof allowed values. Finally we arrive at the following twooptimal sets of quark component parameters:(a) a ‘hard‘ variant b = 0 .
48 fm , y = b V b = 0 . , χ = 1 . m N b R = 0 . b (40)adjusted to the data of A / with taking into account the‘hard‘ region of Q & /c and (b) a ‘soft‘ variant b = 0 .
54 fm , y = b V b = 0 . , χ = 4 m N b R = 0 . b , (41)fitted to the A / data with 0 . Q . /c .The set of parameters related to the molecular compo-nent includes the mixing parameter θ , the scale parame-ters Λ M , Λ N and the parameter λ entering in the vertexfunction of the Roper. Further parameters linked to the σ are the mass M σ , the width Γ σ and the strong couplingconstant g σNN . The parameters Λ M ≈ Λ N ≈ λ is fixed through the orthogonality con-dition h R | N i = 0 (finally fitted at λ = 2 . σ resonance we take values which are reasonable [2] (awide range of values is given by M σ = (0 . − .
2) GeV,Γ σ = (0 . −
1) GeV and g σ NN ≈ A / results in the following set of molecular parameters:Λ M = 1 GeV , Λ N = 0 . ,M σ = 0 . ± .
05 GeV , Γ σ = 0 . ± .
25 GeV ,g σ NN = 5 . (42)The mixing parameter θ is fixed in the low energy re-gion (0 . Q . /c ) of A / , where the molecularcomponent is optimized to reproduce the differnece be-tween the 3 q contribution and the JLab data. We obtainsin θ = 0.6 and 0.7 for sets (a) and (b) respectively. Thecomplete results for the parameters should be consideredpreliminary and be tested seriously in further applica-tions.It is important to remark that in the evaluation of thehelicity amplitudes we use the free σ meson propagator(as some kind of approximation), while in case of thestrong Roper decay R → N + 2 π we had to use theBreit-Wigner σ -meson propagator. The sensitivity of theresults to a variation of the σ meson mass from 0.45 to0.55 GeV gives a variation of the helicity amplitudes upto 10%. The sensitivity of the strong decay Γ( R → N +2 π ) to a variation of Γ σ is discussed in Sec.IVC. In fact,more precise data on Γ( R → N + 2 π ) can give a new,additional constraint on Γ σ . B. Helicity amplitudes
The calculated helicity amplitudes A / and S / areshown in Figs. 6a,b [using the parameter sets (a) and (b)respectively]. We also show separately the contributionsto the amplitude from the quark and the hadron moleculecomponents (dashed and dashed-dotted curves, respec-tively). The comparison with the standard P modelcalculation with a fixed value for the vector-meson radius b V = 0 . b (dotted curves) demonstrates the following: asmooth transition from the P γRN vertex (Fig. 1a) tothe parton-like one (Fig. 1b) using a Q -dependent vec-tor meson radius b V ( Q ) → P model results at moderatevalues of Q . Q (GeV /c ) -50050100 A / ( - G e V - / ) a Q (GeV /c ) -50050100 A / ( - G e V - / ) b Q (GeV /c ) S / ( - G e V - / ) a Q (GeV /c ) S / ( - G e V - / ) b FIG. 6: Helicity amplitudes A / (top panels) and S / (bottom) for two variants of the model parameters, ‘hard‘ (a, leftpanels) and ‘soft‘ (b, right panels), in comparison to JLab data [5, 6]. Dotted curves — the quark core excitation amplitudes | q i + γ ∗ → | q ∗ i calculated in the framework of the standard ‘ P + VMD’ model with a fixed vector meson radius b V = y b .Dashed curves — the same amplitudes calculated in a modified ‘ P + VMD’ model with a Q -dependent scale parameter y = y e − Q /χ for the vector meson radius b V = yb . Dashed-dotted curves — helicity amplitudes for the electroexcitationof the hadron molecule N + σ . Solid curves — the full calculation of A / and S / in terms of a combined structure R = cos θ | q ∗ i + sin θ | N + σ i . For comparison, the valence quark contribution to A / calculated in Ref. [20] on the basis of acovariant spectator model is also shown (the dashed-double-dotted curve in the left top panel). The quark core component of R plays the main rolein the electroproduction of the Roper resonance for this Q region ( Q & /c ). For small valuesof Q . , where the contribution of the mesoncloud should also be important, it can be effectivelytaken into account in the framework of P - and VMDmodels. However, such a model overestimates the trans-verse amplitude A / in the region 0.5 . Q . (the dashed line in Fig. 6). The description of the JLabdata [5, 6] on A / can be considerably improved if one takes a combined structure for the Roper in the form of | R i = cos θ | q ∗ i +sin θ | N + σ i . The adjustable parameter θ fitted to the JLab data in this region is about cos θ = 0.8[for the ‘hard‘ variant (a)] or cos θ = 0.7 [for set (b)], inboth cases indicating an admixture of N σ component ofabout 50%.The ‘hard‘ version (a) looks more plausible in the de-scription of both amplitudes A / and S / , while set(b) only represents a fit to the soft- Q (up to Q ≈ /c ) behavior of the transverse amplitude A / . In0the soft- Q region the contribution of the pion cloud andthe influence of the coupled channel ∆ + π are impor-tantd [6, 21–24]. Both effects should be taken into ac-count in further detailed calculations. C. Decay widths
When the weight of N + σ component in the Roperresonance in terms of ∼ sin θ is fixed, the Roper decaywidth for the transition N +( ππ ) I =0 Swave can be calculated.The assumption that the quark part of the Roper justgives a very small contribution through a virtual tran-sition R → N + σ is justified in our quark model [see,e.g., our evaluation of the quark amplitude M qR → N + σ inEq. (B20) which goes to zero at y σ = 1 as it follows fromEq. (B22)]. Then the transition is described as the vir-tual decay of the molecular part to N + σ followed bythe σ → ππ decay. The diagram for such a mechanism isshown in Fig. 7. ππσ NR FIG. 7: R → N + ( π + π ) I =0 decay process via the σ -meson resonance. The probability | M fi | for the transition process ofFig. 7 contains the Breit-Wigner representation for theintermediate σ -meson state with | M fi | = g R σ N g σππ ˜Φ R ( k ) ( m N + m R ) − s ππ ( m σ − s ππ ) + m σ Γ σ ( s ππ ) , Γ σ ( x ) = Γ σ m σ √ x p x − m π p m σ − m π , x = s ππ ≡ k σ , (43)where k = p R − ω Nσ p N and the coupling constant g σππ is deduced from the two-pion decay width of the σ with g σππ = π Γ σ m σ (cid:16) − m π m σ (cid:17) − / . The coupling constant g R σ N of the hadron-molecular vertex is defined by thecompositeness condition (7).The result for the R → N + ( ππ ) I =0 Swave decay width ispresented by an integral over the variables of the phasespace volumeΓ R → Nσ ( ππ ) = 3 sin θ π m R ( m R − m N ) Z m π ds ππ s ππ | M fi | (44) × λ / ( m R , m N , s ππ ) λ / ( s ππ , m π , m π )with λ ( a, b, c ) = a + b + c − ab − bc − ac . The numerical value for Γ R → Nσ with g RσN = 6 . θ ≃ . R → Nσ ( ππ ) = (19 . − .
7) MeV . (45)where the lower and upper limits correspond to a vari-ation of the σ decay width Γ σ from 0.5 to 1 GeV, re-spectively (the variation of the σ -meson mass M σ =500 ±
50 MeV can only change the result within 10%).This should be compared to the PDG [2] valueΓ R → Nσ ( ππ ) ≈ (0.05 – 0.1) Γ tot R ( ≈ −
30 MeV) or the re-cent data [3] Γ R → Nσ = 71 ±
17 MeV. It is clear that thestrong Roper decay can serve as a constraint on Γ σ , how-ever present results for Γ R → Nσ ( ππ ) are compatible withall values of Γ σ .The pion decay width calculated for the quark part ofthe Roper resonance in the framework of our approach(Γ qR → πN ≃
36 MeV) is not as small as in the case ofEEM evaluations (Γ
EEM R → πN ≃ R → N + π ≈ (0.55– 0.75) Γ tot R . It is clear that considerable corrections toΓ qR → πN can come from the pion cloud contribution whichis neglected here. V. CONCLUSIONS
We suggested a two-component model of the light-est nucleon resonance R = N / + (1440) as a combinedstate of the quark configuration sp [3] X and the hadronmolecule component N + σ . This approach allows todescribe with reasonable accuracy the recent CLAS elec-troproduction data [5, 6] at low- and moderate valuesof Q with 0 ≤ Q . . In the model the R → N + ( ππ ) I =0 Swave transition process is interpreted asthe decay of a virtual σ meson in the N + σ component.The calculated decay width Γ R → Nσ ( ππ ) correlates wellwith the PDG value [2] and the recent CB-ELSA andA2-TAPS data [3].The weight of the N + σ component in the Roper withsin θ ≈ Q . This weight is also compatible withthe value of the helicity amplitude A / at the photonpoint and with the data on the R → N + ( ππ ) I =0 Swave decay width.However, our evaluations have shown that at low Q the contribution of the pion cloud to the amplitude A / can be considerable. For example, this is evident fromFig. 6a, where the discrepancy of our results and theCLAS data is about 1 – 1.5 experimental error bars.Still, this discrepancy is considerably smaller than in thecase of previous quark models: note the predictions ofthe valence quark covariant spectator model (the dashed-double-dotted curve in Fig. 6a adapted from Ref. [20]) orpredictions of the LF models in the same region of Q . /c .In this paper we tried to show that the descriptionof transition amplitudes in terms of parton-like mod-1els, which are very good at high Q , can be naturallytransformed into a description in terms of the ‘soft‘ vec-tor meson cloud. This smooth transition is achieved by’switching on’ a non-zero radius of the intermediate vec-tor meson. The vector meson V of finite size generates anon-local V qq interaction. This weakens the effect of theorthogonality of the spatial R and N wave functions inthe transition matrix element N + γ ∗ T → R , and the am-plitude A / . Resulting theoretical values, which matchthe data, are contrary to the standard predictions of LF-models, which lead to non-zero and (negative) large val-ues at the photon point.Further we plan to develop a relativistic version of thesuggested electroexcitation mechanism. Acknowledgments
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We use the standard definitions for the harmonic os-cillator wave functions ϕ S ( p , r ) = (4 πr ) / e − p r / , (A1) ϕ S ( p , r ) = r
32 (1 − p r ) ϕ S ( p , r ) ,ϕ P,m ( p , r ) = r p r ϕ S ( p , r ) √ πY m (ˆ p ) . The relative momenta in the quark (antiquark) systemswith numbering i=1,2,3,4,5 (see Figs. 1 and 8) are set upas κκκ = 12 ( p − p ) , κκκ = 13 ( p + p ) − p , κκκ ′ = 13 ( p + p ) − p , (A2) κκκκ M = 12 ( p − p ) , p M = p + p In the rest frame with P ′ R ( N ) = p + p + p = 0 we havethe relations p M = k , P N = p + p + p = − k , p − p = − k , κκκκ M = − κκκ ′ − k , p + p = − κκκ ′ − k (A3)which are used with m = m = m = m q = m N / ν = 2 , ν = 32 , m = m m m + m = m q ν ,m (12)3 = ( m + m ) m m + m + m = m q ν , (A4)in the calculation of matrix elements.
2. Quark configurations a. Baryons
The translationally invariant quark configurations s [3] X and sp [3] x , L = 0 are represented in terms ofharmonic oscillator wave functions (A1) depending onthe relative momenta (A2) asΨ N ( κκκκ , κκκκ ; Sz , T z )= ϕ S ( κκκκ , √ ν b ) ϕ S ( κκκκ , √ ν b ) ψ STN (124) , (A5)Ψ R ( κκκκ , κκκκ ′ ; Sz , T z )= (cid:20)r ϕ S ( κκκκ , √ ν b ) ϕ S ( κκκκ ′ , √ ν b ) (A6)+ r ϕ S ( κκκκ , √ ν b ) ϕ S ( κκκκ ′ , √ ν b ) (cid:21) ψ STN (123) . ψ STN for both configurations is de-scribed by the state vector ψ STN (124) = X µ t (cid:20)r
12 (1( Sz − µ ) µ | Sz ) × (1( T z − t ) t | T z ) | S = 1 , Sz − µ i | T = 1 , T z − t i + r δ µ , S z δ t , T z | S = 0 , i | T = 0 , i (cid:21) χ µ ξ t , (A7)where χ µ i ( ξ t i ) is the spinor (isospinor) of i-th quark, µ i ( t i ) is the spin (isospin) projection; i=1,2,4 for thenucleon and i=1,2,3 for the Roper. b. Mesons
1) Pseudoscalar ( π ) and scalar ( σ )Ψ π ( κκκκ π , t π ) = ϕ S ( κκκκ π , √ ν b π ) X µ µ ( µ µ | × X t t ( t t | t π ) χ µ χ µ ξ t ξ t (A8)Ψ σ ( κκκκ σ ) = ϕ S ( κκκκ σ , √ ν b σ ) √ ν b σ κ σ × r π X m (1 m − m | X µ µ ( µ µ | m ) Y − m ( ˆ κκκκ σ ) × X t t ( t t | χ µ χ µ ξ t ξ t (A9)2) Vectors ( ρ, ω )Ψ ρ ( κκκκ ρ , λ ρ , t ρ ) = ϕ S ( κκκκ ρ , √ ν b ρ ) X µ µ ( µ µ | λ ρ ) × X t t ( t t | t ρ ) χ µ χ µ ξ t ξ t (A10)(for the ω use the substitution( t t | t ρ ) → ( t t |
00) (A11)in Eq. (A10)).
Appendix B: Meson-baryon coupling
The meson-baryon vertex generated by the effectivepair-creation operator V eff ¯ qq is schematically sketched inFig. 8. Performing a recoupling of quark (antiquark) vari-ables in the matrix element h M |h N | V eff ¯ qq | N ( R ) i ( M = π, σ, ρ, ω ), omitting the isospin part and other trivialfactors ( g q m q , g q m q , δ ( p + p ), etc.) we obtain for thenon-trivial spin part of the effective πqq and ρqq verticesthe following expressions: πqq : σ · p , ρqq : ( − p + i [ σ × p ]) · ǫǫǫǫ ( λ ρ ) , (B1) where ǫ ( λ ρ ) µ = (cid:8) ǫ ( λ ρ )0 , ǫǫǫǫ ( λ ρ ) (cid:9) (B2)is the ρ meson polarization vector and σ is the vector ofquark spin Pauli matrices. Expressions in Eq. (B1) areonly acceptable in the rest frame of the initial baryon N ( R ), in which case p = − k and p = − p (see Fig 8).For the 3rd quark with a non-zero momentum p = 0both expressions do not satisfy the Galilean invarianceand the second expression in Eq. (B1) does not corre-spond to the elementary ρqq vertex ¯ u ( p ) γ µ u ( p ) ρ µ ( p − p ). Thus it does not lead to a conserved current in theVMD model. p p p p p p p p p + p = kP = kP = 0N,R N FIG. 8: Quark diagram of the P model for the meson-baryoncoupling. It is possible to improve the expressions (B1) in an ac-ceptable form without changing them in the rest framewhere they were deduced in the P ansatz. Such cor-rections are only possible with the substitutions p → p ± p , which become identities for p = 0, i.e. in therest frame of the 3-rd spectator quark.In our calculations we use the following corrected formof Eq. (B1): πqq : σ · ( p − p ) , (B3) ρqq : ( E + E ) ǫ ( λ ρ )0 − (cid:8) ( p + p ) − i [ σ × ( p − p )] (cid:9) · ǫǫǫǫ ( λ ρ ) (cid:1) . These expressions satisfy Galilean invarianceand are well correlated with the Feynman amplitudes¯ u ( p ) γ u ( p ) π ( p − p ) and ¯ u ( p ) γ µ u ( p ) ρ µ ( p − p ), re-spectively (in the non-relativistic approximation). Herewe show that using such corrected form of Eq. (B1)one can obtain realistic values for the coupling constants ρN N , ωN N , σN N and the nucleon magnetic moments µ p , µ n . We further predict the non-diagonal couplings πN R , σN R , ρN R , ωN R starting from a single con-stant γ = g q m q normalized to the well established value g π NN = 13.5.
1. Diagonal N → N transitions Substituting wave functions (A8) – (A10) into Eq. (11)and taking into account the modification (B3) of Eq. (B1)4gives after some algebra the following expressions for the N → N + M ( N + M → N ) amplitudes: T qN → N + π = 3 nr h Ψ π , k |h Ψ N , − k | V effq | Ψ N , i nr = 53 (cid:18) g q m q (cid:19) n / ( y π ) (8 πb ) / y / π e − ζ ( y π ) k b / × h T ′ z | τ t π | T z ih S ′ z | σ · k m q | Sz i , (B4) T qN → N + σ = 3 nr h Ψ σ , k |h Ψ N , − k | V effq | Ψ N , i nr = − (cid:18) g q m q (cid:19) y σ √ m q b ) 1 n / ( y σ ) (8 πb ) / y / σ × (cid:18) f ( y σ ) k b (cid:19) e − ζ ( y σ ) k b / δ S ′ z ,S z δ T ′ z ,T z , (B5) T qρ + N → N = 3 nr h Ψ N , | V effq | Ψ N , − k i| Ψ ρ , k , λ ρ , t ρ i nr = − (cid:18) g q m q (cid:19) n / ( y ρ ) (8 πb ) / y / ρ e − ζ ( y ρ ) k b / ×h T ′ z | τ t ρ | T z i (cid:20)(cid:0) k · ǫ ( λ ρ ) m q n ( y ρ ) (cid:1) δ S ′ z ,S z − h S ′ z | i [ σ × k ] · ǫ ( λ ρ ) m q | Sz i (cid:21) , (B6) T qω + N → N = 3 nr h Ψ N , | V effq | Ψ N , − k i| Ψ ω , k , λ ω i nr = − (cid:18) g q m q (cid:19) n / ( y ω ) (8 πb ) / y / ω e − ζ ( y ω ) k b / (B7) × δ T ′ z ,T z (cid:20)(cid:0) k · ǫ ( λ ω ) m q n ( y ω ) (cid:1) δ S ′ z ,S z − h i [ σ × k ] · ǫ ( λ ω ) m q i S (cid:21) . The parameter b is the r.m.s. radius of the quark con-figuration 0 s which is used for the nucleon. The mesonradius b M is related through the relative value y M = b M b , M = π, σ, ρ, ..., (B8)and we use the notations n ( y ) = 1 + 23 y , ζ ( y ) = 1 + 5 y / n ( y ) ,f ( y ) = 1+ y / n ( y ) . (B9)If b R = b we also use another relative variable y R = b R b and then Eq. (B8) should be generalized as n ( y ) → N ( y, y R ) = 12 (1 + y R ) + 23 y. (B10)The strength parameter of the P model, γ = g q m q , isfixed as usual by normalizing the value of the πN N cou-pling constant to g πNN = 13 .
5. From (B4) it follows that g qπ NN = 53 m N m q (cid:18) g q m q (cid:19) n / ( y π ) × h (8 πb ) / y / π m N p M π i (B11)and for a typical value of b = 0.5 fm one obtains γ ≃ σN N coupling constant g qσ NN = 3 (cid:18) g q m q (cid:19) y σ √ m q b ) 1 n / ( y σ ) × h (8 πb ) / y / σ m N p M σ i (B12)we get g qσ NN = g πNN m N b √ √ r M σ M π y / σ y / π n / ( y π ) n / ( y σ ) . (B13)taking g qπ NN = g π NN . For typical CQM values of b =0.5 fm and y σ = y π = 1 this expression gives a realisticvalue for the coupling constant with g qσ NN = 0 . g π NN =3.54.For the ρN N coupling constant defined in Eq. (B6) as g qρ NN = 13 (cid:18) g q m q (cid:19) n / ( y ) h (8 πb ) / y / m N p M V i (B14)substitution of the value g q m q deduced from Eq. (B11)gives g qρ NN = g π NN r M ρ M π y / ρ y / π n / ( y π ) n / ( y ρ ) (B15)and for y ρ = y π = 1 one obtains the realistic value g ρNN = g qρNN = 0 . g πNN = 6.33.Comparing Eqs. (B6) and (B8) one can see that inthis approach the ωN N - and ρN N couplings are linkedby the standard relation g qω NN = 3 g qρ NN (B16)which corresponds to “ideal mixing” usually used in theVMD model.
2. Non-diagonal transitions
Here the main objective is the calculation of thenon-diagonal baryon matrix elements for the transitions N + ρ → R and R → N + M . The values of the cou-pling constants have been fixed by Eqs. (B13), (B15) and(B16) on the basis of g π NN . We further use them in theexpressions for the non-diagonal transitions N + γ ∗ → R , R → N + π , R → N + σ , etc. substituting symbols g qσ NN and g qρ NN (and g qω NN with g qω NN = 3 g qρ NN ) insteadof the explicit expressions of the r.h.s. of Eqs. (B12)and (B14). Then the vector meson absorption amplitude5 M q ( λ V ) V + N → R = 2 m N √ M V T q ( λ V ) V + N → R is represented by thefollowing two-component column vector: M q ( λ ρ ) ρ + N → R M q ( λ ω ) ω + N → R ! = √ g qρNN e − ζ ( y ) k b / (cid:18) h T ′ z | τ t ρ | T z i δ T ′ z , Tz (cid:19) × (cid:26)(cid:20)(cid:0) ǫ ( λ )0 + n ˜k · ǫ ( λ ) m N n ( y ) (cid:1) p ( y, k )+ p ( y ) n ˜k · ǫ ( λ ) m N n ( y ) (cid:21) δ S ′ z , Sz + (cid:18) (cid:19) h S ′ z | i [ σ × k ] · ǫ ( λ ) m N | Sz i p ( y, k ) (cid:27) . (B17)This is the main result of our considerations. Here we usemomenta k = P − P ′ , ˜k = P + P ′ , related to momenta P , P ′ of initial and final baryon. Only in the rest framethey have the same values, ˜k = k . In the case b R = b thepolynomials p , and ζ, n, n also depend on y R . Theyare defined by the equations p ( y ) = P ( y, y R = 1) , p ( y, q ) = P ( y, y R = 1 , q ) ,n = N ( y R = 1) , N ( y R ) = 32 y R − ,ζ ( y ) = ˜ ζ ( y, y R = 1) , ˜ ζ ( y, y R ) = y R + ( y R − ) y y R + y ,P ( y, y R ) = 43 1+ y N ( y, y R ) , P ( y, y R , k ) =(1 − y R ) / y / N ( y, y R ) − y R (cid:18) y N ( y, y R ) (cid:19) k b , (B18)with N ( y, y R ) defined in Eq. (B10). The R → N + π and R → N + σ decay widths are defined by the matrix elements M qR → N + π = 3 h Ψ π , k |h Ψ N , − k | V effq | Ψ R , i = √ g qπ NN p ( y π , k ) (B19) × e − ζ ( y π ) k b / h S ′ z | σ · k | Sz ih T ′ z | τ t π | T z i , M qR → N + σ = 3 h Ψ σ , k |h Ψ N , − k | V effq | Ψ R , i = √ g qσ NN p ( y σ , k ) e − ζ ( y σ ) k b / δ S ′ z , Sz δ T ′ z , Tz (B20)with p ( y σ , k ) = −
23 1 − y σ n ( y σ ) − k b
27 (B21) × (cid:20) y σ / y σ / n ( y σ ) + k b y σ )(1+ y σ / n ( y σ ) (cid:21) . As in the case of vector mesons the polynomials p ( y π , k ) and p ( y σ , k ) do not vanish in the limit | k | → Z π = 2 y π / n ( y π ) , Z σ = 23 1 − y σ n ( y σ ) (B22)determine the amplitudes (B19) – (B20) for small valuesof ||
27 (B21) × (cid:20) y σ / y σ / n ( y σ ) + k b y σ )(1+ y σ / n ( y σ ) (cid:21) . As in the case of vector mesons the polynomials p ( y π , k ) and p ( y σ , k ) do not vanish in the limit | k | → Z π = 2 y π / n ( y π ) , Z σ = 23 1 − y σ n ( y σ ) (B22)determine the amplitudes (B19) – (B20) for small valuesof || k ||