Q^2-evolution of nucleon-to-resonance transition form factors in a QCD-inspired vector-meson-dominance model
aa r X i v : . [ h e p - ph ] O c t Phys. Rev. D , 073007 (2007)c (cid:13) http://link.aps.org/abstract/PRD/v76/e073007 Q -evolution of nucleon-to-resonance transition form factorsin a QCD-inspired vector-meson-dominance model G. Vereshkov ∗ and N. Volchanskiy † Research Institute of PhysicsSouthern Federal University344090, Rostov-na-Donu, Russia
We adopt the vector-meson-dominance approach to investigate Q -evolution of NR -transitionform factors ( N denotes nucleon and R an excited resonance) in the first and second resonance re-gions. The developed model is based upon conventional γNR -interaction Lagrangians, introducingthree form factors for spin-3 / / ρ (770) and ω (782). Correct high- Q form factorbehavior predicted by perturbative QCD is due to phenomenological logarithmic renormalization ofelectromagnetic coupling constants and linear superconvergence relations between the parametersof the meson spectrum. The model is found to be in good agreement with all the experimental dataon Q -dependence of the transitions N ∆(1232), NN (1440), NN (1520), NN (1535). We present fitresults and model predictions for high-energy experiments proposed by JLab. Besides, we makespecial emphasis on the transition to perturbative domain of N ∆(1232) form factors. PACS numbers: 13.40.Gp, 12.40.Vv, 14.20.Dh, 14.20.Gk
I. INTRODUCTION
Substantial experimental efforts have been made in re-cent years to measure Q -dependence of baryon transi-tion form factors via resonant inelastic eN -scattering.In particular, the data on the transitions N ∆(1232), N N (1440),
N N (1520),
N N (1535) were obtained up tosquared momentum transfer Q = 6 GeV in exclusiveexperiments carried out in such facilities as JLab, MIT-Bates, MAMI and others [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22] (see also review[23]). Although there is a total absence of exclusive dataabove Q = 6 GeV , a large body of inclusive measure-ments by SLAC is available for the resonance ∆(1232)(up to almost 10 GeV ) and N (1535) (up to 21 GeV )(see [24], review [25] and references therein). Besides,new high- Q exclusive measurements are proposed byJLab [26]. This study of pion and η -meson electropro-duction is supposed to provide a source of informationabout Q -behavior of the ∆(1232) and N (1535) multi-pole moments up to Q = 14 GeV . Also, there is adiscussion motivating possible experiments dealing withresonance excitation in the reaction πN → e + e − N [27]that is a crossed channel of pion electroproduction. Allthese experimental prospects as well as recent numer-ous high-precision measurements call for careful theoret-ical examination and interpretation of the data available.Our work is an example of this effort. ∗ Electronic address: [email protected] † Electronic address: [email protected]
A starting point of such an interpretation is the par-tial wave analysis (PWA) of both non-polarized and po-larized exclusive data, which provide Q -dependence ofthree helicity amplitudes A / ( Q ), A / ( Q ), S / ( Q )for spin-3 / A / ( Q ), S / ( Q ) for spin-1 / A T = ( A / + A / ) / can be extracted from inclusivedata.) Detailed discussions of amplitude-extraction tech-niques and their model dependence is given in the reviews[23, 28].Helicity amplitudes encode information about thespace structure of nucleon excitations and physical na-ture of baryon transition form factors. The underlyingfundamental theory of N R -transitions is quantum chro-modynamics (QCD). In the non-perturbative domain,however, ab initio
QCD calculations are not currentlyfeasible, because of their extreme complexity. Even nu-meric calculations [29, 30, 31] utilizing lattice-QCD tech-niques are hindered by computer power available now.The best presentday lattice calculations are obtained forpion mass above 0 . Q ) behavior of resonance formfactors [25], still there is not clear indication of the on-set of hard scattering processes in the existing experi-mental data base (except for the inclusive data on the N N (1535)-transition [25]).Since QCD itself is currently not able to provide a com-prehensive treatment of resonant helicity amplitudes, alot of phenomenological models have emerged. Some ofthe most prominent approaches are: quark shell modelssuch as chiral bag model [33] and numerous constituentquark models (single quark transition model [34], hyper-central model [35], model with two-body exchange cur-rents [36] and so forth); soliton models (Skyrmion models[37, 38], linear σ -model [39, 40, 41], chiral chromodielec-tric model [40, 41], chiral quark-soliton model [42]); alge-braic approach [43]; generalized parton distributions (seereviews [28, 44, 45, 46, 47]); chiral effective field theories[48, 49]. The majority of these models are in quantita-tive agreement with the experimental data points just atsmall Q < . .In this paper our main objective is to demonstratethat effective-field theory incorporating vector-meson-dominance (VMD) effects can reproduce Q -evolution ofresonant helicity amplitudes in both perturbative andnon-perturbative domains. So far VMD models havebeen successfully applied to address the same problemof Q -evolution mostly in elastic eN -scattering [50, 51,52, 53, 54, 55, 56], the lowest-resonance electroproduc-tion [57, 58], and deeply virtual Compton scattering [59].All the versions of the VMD model contain several am-biguities:1. The choice of the vector-meson spectrum.
Themajority of the models take into account onlythe lightest vector mesons ρ (770), ω (782), φ (1020)[50, 51, 52, 53, 54, 60] and seldom ρ (1450) [55, 56], ω (1420) [56]. Only the model [60] includes all thevector mesons reported by experimentalists beforeits publication.2. The choice of the interaction Lagrangian.
Theproblem of the interaction-Lagrangian symmetriesbecomes important in the description of high-spinresonances. First of all, most Lagrangians cur-rently in use could break free-field subsidiary con-straints reducing nonphysical degrees of freedom ofthe Rarita-Schwinger field [61], which result in dif-ferent pathologies, e.g., excitation of superluminalmodes [62]. This could be avoided in the theorywith additional symmetries such as gauge invari-ance of the resonance field [63].3.
The way to impose high- Q behavior predicted bypQCD on Lagrangian form factors. In the mod-els involving only ground-state vector mesons andtheir first excitations, agreement with pQCD isdue to artificial suppression of meson-spectrum pa-rameters by power corrections [50, 51, 52, 53, 54,55, 56]. Obviously, such a suppression of photon-meson couplings disagrees with quantum field the-ory, since parameters of vector-meson spectrum canbe renormalized by only slight logarithmic func-tions. Another way to fulfill asymptotic pQCD-constraints is by linear superconvergence relationsbetween meson parameters [64]. 4.
The way to treat logarithmic corrections.
Loga-rithmic dependencies in form factors are an es-sential feature of the model. They are necessaryto incorporate logarithmic corrections to pQCD-asymptotes, though not calculated directly yet.5.
Inclusion of continuum contributions.
An infinitenumber of virtual intermediate multihadron statesgive rise to continuum contributions to form fac-tors. Most VMD models include only the 2 π -cutassociated with the lightest isovector intermediarystate [52, 53, 54]. Also some models take into ac-count K ¯ K and ρπ continua [65] or 3 π -continuumand effective inelastic cuts [60].In this paper we build up a VMD model satisfyingasymptotic constraints predicted by pQCD (that’s why,we refer to it as “QCD-inspired”). In the perturbativedomain, QCD expects resonant helicity amplitudes tohave power-logarithmic asymptotes and fall faster thanthe dipole (ground-state-meson) model predicts. We pre-fer to impose correct asymptotic behavior on form factorsby superconvergence relations in the manner of the paper[60], rather than by invoking unphysical power suppres-sion. As we show in Sec. III, this requires the model toinclude at least four vector mesons. Nevertheless, thisdoes not lead to a dramatic increase in the number offree parameters. For example, in the simplest four-mesonmodel, vector-meson spectrum comprises only one inde-pendent parameter, and even this model is in accord withthe data at all Q with high accuracy (except for the res-onance N (1440), whose peculiar structure can be repro-duced in the model with at least five mesons). Our VMDmodel differs, however, from that of Ref. [60], becausethe significant feature of our calculation is phenomeno-logical logarithmic renormalization of the parameters ofthe vector-meson spectrum. Logarithmic renormaliza-tion is essential to comply with both power and logarith-mic pQCD-behavior, which we discuss in Sec. II C 2. An-other difference of this work from Ref. [60] is that we ne-glect continuum contributions to transition form factors,since this simplest (tree-level) parametrization is foundto describe satisfactorily all the experimental data. Thefollowing discussion is constrained to the calculation ofthe vector transition form factors for the first four low-lying baryon resonances. Application of the model todescription of the nucleon axial and elastic form factorsis the topic of our further publications [66].In this paper we make use of the traditional Lagrangian(Eqs. (1), (3)) of the γ ∗ N R -interaction for spin-3 / Q -evolution of form fac-tors [49, 69]. In the following, we confine ourselves toworking with only this inconsistent but popular interac-tion, since it enables us to demonstrate the validity ofVMD approach in physics of transition form factors atthe entire range of Q . We are going, however, to discussalternative couplings in our further publication [70].The remainder of this paper proceeds as follows. Sec-tion II comprises Lagrangians and corresponding to themcross-section formulas. Also we present a detailed discus-sion of how to bring an effective-field-theory model intoaccordance with pQCD-predictions. The next Sec. IIIlays out our VMD model, including superconvergence re-lations and logarithmic renormalization. Section IV con-tains fits, model predictions, and discussion of these re-sults with the emphasis on the transition to pQCD regimeof the N ∆(1232) form factors. Finally, Sec. V is a sum-mary of our main conclusions as well as possible areas ofextension of the model and improvement to it. The tech-nical details of our calculations, concerning the choice ofhelicity-amplitude signs, can be found in the appendix. II. PHENOMENOLOGICAL MODEL OFBARYON ELECTROPRODUCTION
In this rather long section we write down conventional γN R -vertexes and define resonant helicity amplitudes bytheir relations to observables, i.e., differential cross sec-tions. Starting from these formulas, it is quite easy tocompute relations between phenomenological-model formfactors, comprised in the γN R -vertexes (3) and (4), andhelicity amplitudes (or any other quantities traditionallyused to describe resonant eN -scattering). However, animportant step in the calculations is the choice of am-plitude phases, which is discussed in the appendix. Itshould be stressed that the choice of amplitude phasescould strongly influence the quality of fits to experimen-tal data. Besides, to determine amplitude phases is nota straightforward task if one uses simple factorized cross-section formulas, not involving amplitude interference, asit is, e.g., in Ref. [69] and this paper (see the appendix).Also in this section we extract Lagrangian form factorsfrom the experimental data on helicity amplitudes anddiscuss their asymptotic behavior. A. Matrix elements and vertex operators
To discuss the underlying physics of the nucleon-to-resonance transition form factors, it seems reasonable tocombine the results and approaches of quantum chromo-dynamics and effective-field-theory (EFT) models. More-over, this synthesis is mathematically inescapable. Asquark confinement is an essential feature of QCD, anyamplitude of physical process presented by a functionalintegral over the space of the quark and the gluon fieldscan be equally expressed as the integral over hadron de-grees of freedom [71]. This problem, however, is too com-plicated to be applied directly in the nonperturbative do-main of QCD. In such a situation it is EFT that exposeslimitations of the vertexes of effective hadron interac-tions — the objects of QCD calculations. For example, EFT provides the matrix elements for the electroproduc-tion of spin-3 / M ( γ ∗ N → R ) = h R | ¯ u µR ( p ′ )Γ µνλ ( q, p, p ′ ) ×× (cid:0) q ν e λ ( q ) − q λ e ν ( q ) (cid:1) u N ( p ) | N i , (1)where q = p ′ − p is the 4-momentum transfer; u µR ( p ′ ) , u N ( p ) are the resonance vector-spinor andthe nucleon spinor; e ν ( q ) is the photon polarization;Γ µνλ ( q, p, p ′ ) is the vertex operator, which is antisymmet-ric on the last two indices. To write the matrix elementfor the transition N → spin-1 / µ in Eq. (1): M ( γ ∗ N → R ) = h R | ¯ u R ( p ′ )Γ νλ ( q, p, p ′ ) ×× (cid:0) q ν e λ ( q ) − q λ e ν ( q ) (cid:1) u N ( p ) | N i . (2)The next important step to build up an EFT modelof baryon electroproduction is to decompose vertexesΓ µνλ ( q, p, p ′ ), Γ νλ ( q, p, p ′ ) in terms of the particular spin-tensor basis. The scalar coefficients of this expansion areform factors, their number being equal to the number ofbasic elements. We note that the basis should be pos-tulated in both QCD and EFT. In QCD the decompo-sition made in terms of quark correlators yields directlyform factors in the domain of perturbative QCD [72]. InEFT form factors can be evaluated by means of eitherthe dispersion relation approach or VMD model. Thelinking idea of these two methods — QCD and EFT —is quark-hadron duality, i.e., the form factor asymptotescalculated in both pQCD and EFT must be the same.To understand the physical origin of electromagneticform factors, one should address two aspects of the prob-lem. First of all, the reliable physical arguments fix-ing mathematical structure of the vertexes Γ µνλ ( q, p, p ′ ),Γ νλ ( q, p, p ′ ) should be discussed. In the phenomenologyof spin-vector resonance electroproduction the followingvertex is often in use [49, 69]:Γ µνλ ( q, p, p ′ ) = 12 ( g µν g λσ − g µλ g νσ ) ×× (cid:18) C ( q ) M N γ σ + C ( q ) M N p σ ′ + C ( q ) M N p σ (cid:19) γ R , (3)where C ( q ) , C ( q ) , C ( q ) are phenomenological formfactors in the notation of Ref. [69]; γ R = γ for R =∆(1232) and γ R = 1 for R = N ∗ (1520). From this pointon, we label form factors in such a manner to unify thenotation of the non-spin-flip and spin-flip amplitudes: C ( Q ) ≡ F ( Q ) , C ( Q ) ≡ F ( Q ) ,C ( Q ) ≡ F ( Q )(this notation is similar to that of the elastic Dirac F ( Q ) and Pauli F ( Q ) form factors).The theory of the J = 1 / νλ ( q ) =12 (cid:18) G ( q ) M ( γ ν q λ − γ λ q ν ) − G ( q ) M σ νλ (cid:19) γ R , (4)where σ νλ = ( γ ν γ λ − γ λ γ ν ); γ R = γ for R = N (1535)and γ R = 1 for R = N (1440); G ( Q ), G ( Q ) are, re-spectively, the non-spin-flip and spin-flip form factors; M = M N is the normalization factor (there is anotherconvention M = M R + M N [69] but, in our opinion, itis inconvenient especially when the baryons are off themass shell).It should be noted that there are two kinds of Q -dependent functions in any definition of the vertex op-erator. The functions of the first kind are multiplicativefactors fixed by the structure of the vertex itself. The sec-ond kind is form factors F α ( Q ) , G α ( Q ). If one treatsform factors as just phenomenological objects, Eqs. (3)and (4) define the general model since three arbitraryfunctions F ( Q ), F ( Q ), F ( Q ) are used to describethree observables A / ( Q ), A / ( Q ), S / ( Q ) and twofunctions G ( Q ), G ( Q ) are used to describe two am-plitudes A / ( Q ), S / ( Q ). But to evaluate form fac-tors in the framework of any particular dynamics model,care must be taken in choosing the first kind functions de-pendent on kinematic variables. However, in this paperwe deal with only conventional models defined by Eqs.(3) and (4) and skip the discussion of their mathematicalstructure.The second aspect of the physical origin of form factorsis the modeling the functions F α ( Q ) , G α ( Q ). In thispaper, to evaluate form factors, we adopt the VMD ap-proach. The agreement of this model with quark-hadronduality (i.e., pQCD asymptotic behavior) is due to the su-perconvergent relations between meson-spectrum param-eters and logarithmic renormalization of effective cou-pling parameters. B. Helicity amplitudes and cross sections —notation
The pairs of Eqs. (1), (3) and (2), (4) allow to com-pute photoabsorption amplitudes. The differential cross section of the on-shell resonance electroproduction is ex-pressed in terms of these amplitudes as follows: dσdQ ( eN → eR ) = αM N ( M R − M N )2 Q ( s − M N ) (1 − ε ) ×× h | ˜ S / ( Q ) | ε ( s, Q ) + | A T ( Q ) | i , (5)where ε ( s,Q ) = (cid:26) Q + 2 Q ( M R + M N ) + ( M R − M N ) s − M N )( s − M R ) − sQ ] (cid:27) − is the virtual photon polarization parameter; | ˜ S / ( Q ) | = | S / ( Q ) | (cid:20) Q + M R − M N ) M N Q (cid:21) − / is the amplitude for the absorption of a longitudinallypolarized photon in the normalization that we use fromnow on. The transverse helicity amplitude is A T ( Q ) ≡ A / ( Q ) for spin-1 / | A T ( Q ) | = | A / ( Q ) | + | A / ( Q ) | (6)for spin-3 / d σdQ dW ( eN → eR → eN + mesons) = αM N ( W − M N )2 Q ( s − M N ) [1 − ε ( s, Q , W )] ×× n | ˜ S / ( Q , W ) | ε ( s, Q , W ) + | A T ( Q , W ) | o π − M R Γ R ( W − M R ) + M R Γ R , (7)where W = X a p ′ ( a ) ! is the squared invariant mass of the final hadron state; M R , Γ R are the Breit-Wigner mass and the total width of resonance; ε ( s, Q , W ) = (cid:20) Q + 2 Q ( W + M N ) + ( W − M N ) s − M N )( s − W ) − sQ ] (cid:21) − . (8)In addition to helicity amplitudes for ∆(1232), we willalso use magnetic dipole form factor in the Jones-Scadronconvention [74] G ∗ M ( Q ) = − (cid:20) M N ( M − M N )2 πα ( M ∆ + M N ) (cid:21) / A / + √ A / [ Q + ( M ∆ − M N ) ] / , the ratio R EM between electric quadrupole and magneticdipole multipoles R EM ( Q ) = − G ∗ E G ∗ M = A / − √ A / A / + √ A / , and the ratio R SM of Coulomb quadrupole multipole tomagnetic dipole one R SM ( Q ) = − | q | M ∆ G ∗ C G ∗ M = √ S / A / + √ A / , where q is the photon 3-momentum with the modulus | q | = Q + Q − / M , Q ± = h Q + ( M ∆ ± M N ) i / , M = M ∆ in the rest frame of the ∆ and M = M N in thelaboratory frame, in which the initial nucleon is at rest.Note that the amplitude S / is not a Lorentz scalar inthe convention utilized by experimentalists [75]. In thefollowing we use the lab frame to calculate this quantity. C. Helicity amplitudes and extracted form factors
1. The P (1232) , D (1520) The amplitudes for the electroproduction of spin-3 / A / ( Q ) = ∓ (cid:20) πα ( Q + ( M R ∓ M N ) ) M N ( M R − M N ) (cid:21) / (cid:20) M N ( M R ± M N ) F ( Q ) ++ 12 (cid:0) M R − M N − Q (cid:1) F ( Q ) + 12 (cid:0) M R − M N + Q (cid:1) F ( Q ) (cid:21) , (9) A / ( Q ) = − √ (cid:20) πα ( Q + ( M R ∓ M N ) ) M N ( M R − M N ) (cid:21) / (cid:20) M N M R ( Q ± M N ( M R ± M N )) F ( Q ) −− (cid:0) M R − M N − Q (cid:1) F ( Q ) − (cid:0) M R − M N + Q (cid:1) F ( Q ) (cid:21) , (10)˜ S / ( Q ) ≡ S / ( Q ) (cid:20) Q + M R − M N ) M N Q (cid:21) − / = ± r (cid:20) πα ( Q + ( M R ∓ M N ) ) M N ( M R − M N ) (cid:21) / ×× Q (cid:20) M N F ( Q ) + M R F ( Q ) + Q + M R + M N M R F ( Q ) (cid:21) . (11)In Eqs. (9)–(11) the top signs in ± and ∓ refer to the case of the ∆(1232), while the bottom ones are for the N (1520). The phases of the amplitudes are chosen under some extra assumptions (see the appendix). The amplitudes A / ( Q , W ), A / ( Q , W ), ˜ S / ( Q , W ) for the off-shell electroproduction can be obtained from Eqs. (9)–(11) bythe substitution M R → W . To extract the form factors from the experimental data on photoabsorption amplitudes,one should simply resolve Eqs. (9)–(11) with respect to F ( Q ), F ( Q ), F ( Q ). The result is F ( Q ) = − (cid:20) M N ( M R − M N ) πα [ Q + ( M R ∓ M N ) ] (cid:21) / M R [ ± A / ( Q ) + √ A / ( Q )] M N [ Q + ( M R ± M N ) ] , (12) F ( Q ) = ± (cid:20) M N ( M R − M N ) πα [ Q + ( M R ∓ M N ) ] (cid:21) / Q + ( M R + M N ) ][ Q + ( M R − M N ) ] ×× "(cid:2) Q + ( M R ∓ M N ) (cid:3) A / ( Q ) + M R M N (cid:2) ± A / ( Q ) − √ A / ( Q ) (cid:3) ++ r M R ˜ S / ( Q ) Q (cid:0) Q + M R − M N (cid:1) , (13) F ( Q ) = (cid:20) M N ( M R − M N ) πα [ Q + ( M R ∓ M N ) ] (cid:21) / M R [ Q + ( M R + M N ) ][ Q + ( M R − M N ) ] ×× " √ A / ( Q ) ∓ A / ( Q ) ± r
32 ˜ S / ( Q ) M R Q ( Q − M R + M N ) . (14)Available experimental data [1, 2, 3, 4, 5, 6, 24, 25, 76,77, 78, 79] on the γ ∗ N → ∆(1232) transition are depictedin Figs. 1, 2, 3. The extracted Lagrangian form factors F α ( Q ) are pictured in Fig. 4. The Fig. 5 presents the setof experimental data on the γ ∗ N → N (1520) amplitudes[69, 77, 78, 79, 80]. The form factors extracted from thisdata are in Fig. 6. The fit of the amplitudes (9)–(11)to the experimental data (the solid and dashed lines inall figures) is carried out in the framework of the QCD-inspired VMD model (see Secs. III, IV).
2. Quark-hadron duality and high- Q form-factor behavior The phenomenological model to interpret experimen-tal data should obey the general implications of thequark-hadron duality. It makes the amplitudes A / ( Q ), A / ( Q ), ˜ S / ( Q ) and the form factors F α ( Q ) to takeon some specific properties in accordance with the originof the form factors on both quark and hadron levels.At very high momentum transfer, pQCD predicts thescaling behavior of the photoabsorption amplitudes to be[81] A / ( Q ) ≃ C A (1 / Q , A / ( Q ) ≃ C A (3 / Q , ˜ S / ( Q ) ≃ C S Q , (15)where C A (1 / , C S , C A (3 / are constants or slight loga-rithmic functions of Q . This brings up the question ofwhether it is possible to obtain the power asymptotes ofthe form factors from those of the amplitudes (15). Theanalysis of Eqs. (9)–(11) and (12)–(14) shows that onlytwo form factors of the vertex (3) have uniquely deter-mined asymptotes but the exponent of the third formfactor asymptote is bounded below: F ( Q ) ≃ C Q , F ( Q ) ≃ C Q ,F ( Q ) ≃ C Q p , p > . (16)To fit the experimental data, we suppose that F ( Q ) ∼ Q − for Q → ∞ .In what follows we will exploit the three aspects ofquark-hadron duality:1. In the asymptotic region of Q ≫ M R resonanceelectroproduction is described in the framework of QCD by only two independent form factors — thenon-spin-flip F ( Q ) and the spin-flip F ( Q ). For Q → ∞ the transverse helicity amplitudes are pro-portional to different form factors while the ratio oftheir asymptotes is A / ( Q ) A / ( Q ) ∼ F ( Q ) F ( Q ) ∼ Q . (17)2. The asymptotic constrains must be imposed on theform factors so that the asymptotic scaling relation R SM → const were valid. The longitudinal ˜ S / ( Q ) and transverse A / ( Q ) amplitudes are proportional to the same form factor F ( Q ) at high Q . The first two statements are due to the baryon helic-ity conservation at high Q . The third one arises fromthe fact that the absorption of a longitudinally polarizedphoton is asymptoticly a non-spin-flip interaction.Taking all above considerations into account, one caneasily obtain the asymptotes of the photoabsorption am-plitudes (9)–(11): A / ( Q ) = − √ N M N M R Q F ( Q ) , (18) A / ( Q ) = ± N Q F ( Q ) , (19)˜ S / ( Q ) = ± N M N Q F ( Q ) , (20)where N = (cid:20) παM N ( M R − M N ) (cid:21) / .In Eqs. (18)–(20) it is assumed that the following in-equalities hold for Q → ∞ : F ( Q ) ≫ M R M N (cid:12)(cid:12) F ( Q ) − F ( Q ) (cid:12)(cid:12) , (cid:12)(cid:12) F ( Q ) − F ( Q ) (cid:12)(cid:12) ≫ M N ( M R ± M N ) Q (cid:12)(cid:12) F ( Q ) (cid:12)(cid:12) ,F ( Q ) ≫ (cid:12)(cid:12)(cid:12)(cid:12) M R M N F ( Q ) + Q M R M N F ( Q ) (cid:12)(cid:12)(cid:12)(cid:12) . (21)The first of the inequalities (21) is true by virtue of theform-factors asymptotes (16), derived from the asymp-totic pQCD-predictions (15). But the last two of the in-equalities (21) are valid only with regard to logarithmicrenormalization. Note that logarithmic renormalizationis inescapable due to the following reasons. There are at A / , A / , S / , G e V - / Q , GeV A A S A / , A / , S / , G e V - / Q , GeV A A S A T , G e V - / Q , GeV A T , G e V - / Q , GeV FIG. 1: Helicity amplitudes of the transition γ ∗ N → ∆(1232). The dashed curves correspond to fit F1 with one-parameterlogarithmic renormalization (39), the solid curves to fit F2 with two-parameter renormalization (40). The data points aredenoted as follows: (cid:7) [77], • [78], ◦ [24], (cid:4) [1], (cid:3) [25], ◭ [3], ◮ [4], N [2], H [76], ⋆ [6], filled pentagon [5], filled hexagon [79]. R E M Q , GeV R S M Q , GeV FIG. 2: Ratios R EM and R SM for the ∆(1232). Fit curves for R EM are prolonged in the domain of proposed upgraded-JLabexperiments. Curves and data are the same as in Fig. 1. G * M / G D Q , GeV FIG. 3: ∆(1232) magnetic form factor G ∗ M normalized by three times the dipole form factor. We use the standard definition G D = (1 + Q / . − . Fit curves are prolonged in the domain of proposed upgraded-JLab experiments. Curves and data arethe same as in Fig. 1. F , F , F Q , GeV F F F F , F , F Q , GeV F F F FIG. 4: Form factors of the model (3) extracted from the data on the transition γ ∗ N → ∆(1232). Curves and data are thesame as in Fig. 1. least two chromodynamic quark subprocesses contribut-ing to the resonance electroproduction in the asymptoticregion: (1) a single-quark transition to an excited state;(2) 4-momentum exchange between valence quarks. Ifthese processes are short-distance and non-spin-flip, theamplitude A / ( Q ) is proportional to the third power ofthe strong coupling constant. Granting this considera-tion and the asymptotic relation A / ( Q ) ∼ Q F ( Q ),it is easily seen that the non-spin-flip transition form fac-tors obey quark counting rule F ( Q ) ∼ (cid:20) α s ( Q ) Q (cid:21) n ∼ Q n ln n Q / Λ ,n = n val − n ex = 3 , (22)where n val = 3 is the number of the valence quarks; n ex = 1 is the number of the excited quarks; Λ ≈ Λ QCD =0 . ± .
025 GeV is the QCD scale parameter [83].Modifying the asymptotes (16) with the small parame-ter ln − Q / Λ ≪
1, one readily imposes them to satisfyall the inequalities (21): F α ( Q ) ≃ (cid:18) M N Q (cid:19) p α · f α ln n α Q / Λ , (23)where p = 3, p = p = 4, n > n > n , n ≃ f α aredimensionless parameters.The asymptotic relation among the spin-flip and non-spin-flip form factors F ( Q ) F ( Q ) ∼ ln n Q / Λ Q , n = n − n , (24) A p / , A p / , S p / , G e V - / Q , GeV S p A p A p FIG. 5: Helicity amplitudes of the transition γ ∗ p → N (1520). The dashed curves correspond to fit F1 with one-parameterlogarithmic renormalization (39), the solid curves to fit F2 with two-parameter renormalization (40). The data points aredenoted as follows: (cid:7) [80], • [78], (cid:4) [77], N [69], H [79]. F p , F p , F p Q , GeV F p F p F p F p , F p , F p Q , GeV F p F p F p FIG. 6: Form factors of the model (3) extracted from the data on the transition γ ∗ p → N (1520). Curves and data are thesame as in Fig. 5. resembles that among the elastic Pauli and Dirac form factor [84] [82].
3. The P (1440) , S (1535) The amplitudes for the electroproduction of spin-1 / A / ( Q ) = √ (cid:20) πα ( Q + ( M R ∓ M N ) ) M N ( M R − M N ) (cid:21) / (cid:2) Q G ( Q ) + M N ( M R ± M N ) G ( Q ) (cid:3) , (25)˜ S / ( Q ) ≡ S / ( Q ) (cid:20) Q + M R − M N ) M N Q (cid:21) − / == ∓ (cid:20) πα ( Q + ( M R ∓ M N ) ) M N ( M R − M N ) (cid:21) / (cid:2) ( M R ± M N ) G ( Q ) − M N G ( Q ) (cid:3) Q. (26)0In Eqs. (25) and (26) the top signs correspond to the N (1440), while the bottom ones are for the N (1535); G ( Q )is the non-spin-flip form factor; G ( Q ) is the spin-flip form factor.The form factors extracted from the experimental data on the helicity amplitudes are G ( Q ) = (cid:20) M N ( M R − M N ) πα [ Q + ( M R ∓ M N ) ] (cid:21) / ×× Q + ( M R ± M N ) ] (cid:20) A / ( Q ) √ ∓ M R ± M N Q ˜ S / ( Q ) (cid:21) , (27) G ( Q ) = (cid:20) M N ( M R − M N ) πα [ Q + ( M R ∓ M N ) ] (cid:21) / ×× Q + ( M R ± M N ) ] (cid:20) M R ± M N M N A / ( Q ) √ ± QM N ˜ S / ( Q ) (cid:21) , (28)The experimental data on the photoabsorption ampli-tudes [7, 8, 25, 77, 78, 79, 80] and the extracted La-grangian form factors are pictured in Figs. 9 and 10 forthe transfer γ ∗ p → N (1440) and in Figs. 7 and 8 for the γ ∗ p → N (1535).The asymptotic behavior of the helicity amplitudespredicted by pQCD are A / ( Q ) ≃ C A Q , ˜ S / ( Q ) ≃ C S Q . (29)At high Q the dominant contribution to both the elec-tromagnetic amplitudes is to be come from only the non-spin-flip interactions. Granting this and substituting (29)into (25), one can easily obtain the asymptotic behaviorof the first form factor and the limitations on the asymp-tote of the second form factor: G ( Q ) ≃ C Q , G ( Q ) ≃ C Q p , p > . (30)Logarithmic renormalization of the form factor G ( Q )can be carried out with regard to the same quark count- ing rule as (II C 2). Besides, it is reasonable to suggestthe ratio G ( Q ) /G ( Q ) at high Q to be the same as inthe case of spin-vector resonance electroproduction (24).In the framework of the two latter considerations, onecan easily obtain the following form-factor asymptotesfor spin-1 / G α ( Q ) ≃ (cid:18) M N Q (cid:19) p α · g α ln n α Q / Λ ,p = 3 , p = 4; n > n , n ≃ . (31) D. Electromagnetic coupling constants
By definition, the electromagnetic constants of the res-onances are the amplitudes for the absorption of a realtransverse photon A / (0), A / (0). It is also possibleto define model electromagnetic constants, i.e., the La-grangian form factors at Q = 0. Some of such parame-ters within the models (3) and (4) are F (0) = (cid:20) M N πα ( M R − M N ) (cid:21) / M R M N ( M R ± M N ) ( ∓ A / − √ A / ) ,F (0) + F (0) = (cid:20) M N πα ( M R − M N ) (cid:21) / M R − M N ( − M N A / + M R √ A / ) ,G (0) = (cid:20) M N πα ( M R − M N ) (cid:21) / √ M N A / . (32)The form factors and observed values of the amplitudes at Q = 0 [77] are set out in Table I.1 A p / , S p / , G e V - / Q , GeV A p S p A p / , S p / , G e V - / Q , GeV A p S p FIG. 7: Helicity amplitudes of the transition γ ∗ N → N (1535). The solid curves correspond to fit S1-F1, dashed curves toS1-F2, dot-dashed curves to S2-F1, dotted curves to S2-F2 (see Sec. IV B 2). The data points are denoted as follows: (cid:7) [8], • [69], (cid:4) [77], (cid:3) [25], ◭ [7], N [79], H [78]. G p Q , GeV G p Q , GeV FIG. 8: Form factors of the model (4) extracted from the data on the transition γ ∗ N → N (1535). Curves and data are thesame as in Fig. 7. III. FORM FACTORS WITHIN VMD MODELA. Origin of the model
Vector-meson-dominance models being consistent withpQCD-predictions are well known to give a satisfactorydescription of existing experimental data on elastic eN -scattering [50, 51, 52, 53, 54, 55, 56, 60]. The universalphysical ground of VMD allows to apply its principlesto physics of the transition form factors. But the VMDmodels currently in use are suffering the drawback of tak-ing into account solely the ground-state vector mesons ρ (770), ω (782), φ (1020) [50, 51, 52, 53, 54] and seldom ρ (1450) [55, 56] and ω (1420) [56]. This cut-off of the me-son spectrum is usually motivated by the data on decaywidths Γ( V → e − e + ) testifying a photon to hadronizedominantly into the above mesons [50]. To join predic-tions of such VMD models with pQCD expectations, thehadronization amplitudes should be suppressed by powerand logarithmic functions.However, the truncation of the intermediary vectormesons spectrum and suppression of the amplitudes byartificial means are in conflict with physics of the processand beyond the framework of quantum field theory. Ac-tually, in the nonperturbative hadronic vacuum a photonexcites all modes of hadronic string, carrying the quan-tum numbers J P C = 1 −− . Thus, all the vector mesons2 A p / , G e V - / Q , GeV S p / , G e V - / Q , GeV FIG. 9: Helicity amplitudes of the transition γ ∗ N → N (1440). The solid curves correspond to the fit to the data sample S1,dashed curves to S2 (see Sec. IV B 3). The data points are denoted as follows: (cid:7) [78], • [79], (cid:4) [77], N [80], H [69]. G p Q , GeV G p Q , GeV FIG. 10: Form factors of the model (4) extracted from the data on the transition γ ∗ N → N (1440). Curves and data are thesame as in Fig. 9. TABLE I: Electromagnetic coupling constants [77]. A h (0) , F α (0) , G α (0) P ∆(1232) D N (1520) P N (1440) S N (1535) A ( p )1 / (0) , GeV − / − . ± . − . ± . − . ± .
004 0 . ± . A ( p )3 / (0) , GeV − / − . ± .
008 0 . ± .
005 — — F ( p )1 (0) 2 . ± .
079 2 . ± .
271 — — F ( p )2 (0) + F ( p )3 (0) − . ± . − . ± .
188 — — G ( p )2 (0) — — − . ± .
016 0 . ± . A ( n )1 / (0) , GeV − / . ± . − . ± .
009 0 . ± . − . ± . A ( n )3 / (0) , GeV − / . ± . − . ± .
011 — — F ( n )1 (0) − . ∓ . − . ± .
350 — — F ( n )2 (0) + F ( n )3 (0) 1 . ± . − . ± .
225 — — G ( n )2 (0) — — 0 . ± . − . ± . TABLE II: PDG vector-meson masses. k m ( ρ ) k , GeV m ( ω ) k , GeV m k , GeV a ρ (770) 0.7755 ω (782) 0.78265 0 . ρ (1450) 1.459 ω (1420) 1.425 1 . ρ (1700) 1.720 ω (1650) 1.670 1 . ρ (1900) 1.885 ω (1960) b . ρ (2150) 2.149 ω (2145) b . a m k = h ( m ρ ) k + m ω ) k ) / i / is an averaged mass used in thefits for the second-region resonances (see Sec. IV B). b These isosinglet mesons are from “Further states” section. (at least the observed ones) should be incorporated inthe VMD model. Furthermore, the low values of thehadronization amplitudes is not an adequate cause todisregard of heavy mesons. This is due to the structureof the amplitudes for the transition eN → eR : A V ( k ) ( γ ∗ N → R ) = X α A ( V ) αk ( γ ∗ N → R ) = A ( γ ∗ → V ∗ k ) × X α A α ( V ∗ k N → R ) , (33)where A ( γ ∗ → V ∗ k ) is the amplitude for the transition ofa virtual photon to virtual vector meson; A α ( V ∗ k N → R )is the amplitude for the absorption of a virtual meson bynucleon; the “ α ” indexes the vertexes of meson-nucleoncoupling corresponding to independent form factors. Inthe case of high excited resonances, the photoabsorptionamplitudes A ( V ) αk ( γ ∗ N → R ) are not necessarily negligi- ble since small hadronization amplitudes are multipliedby arbitrary large meson-absorption amplitudes. Notethat it is the set of the amplitudes A ( V ) αk ( γ ∗ N → R ) [not A α ( V ∗ k N → R )] that is obtained by the fit to experimen-tal data [the meson-absorption amplitudes can be theneasily calculated if the A ( V ) αk ( γ ∗ N → R ) are known].Light unflavored mesons listed in Particle Data Grouptables [77] are grouped by near mass degeneracy into fivesinglet-triplet families (see Table II).In the general case, φ -mesons are other intermedi-aries in eN -interactions. However, to simplify the VMDmodel, we neglect their contribution to transition formfactors due to the following reasons. In the case of idealsinglet-octet mixing corresponding to the quark content φ = ¯ ss , these mesons interact only with the strange com-ponent of the nucleon which is suppressed with respectto nonstrange quark content. The difference between ac-tual and ideal mixing is also suppressed by small param-eters, and to the first approximation in these parametersit is possible not to take into account coupling between φ -mesons and ud -component of the nucleon.So, to the extent that φ -mesons contributions can beneglected, the transition form factors are specified by dis-persionlike expansions with poles at meson masses. Theexpansion coefficients are the amplitudes A ( V ) αk ( γ ∗ N → R ) [in the following we use this designation only for R = ∆(1232) , N (1520) and B ( V ) αk ( γ ∗ N → R ) for R = N (1440) , N (1535)]. Having regard for the isotopic sym-metry of strong interactions, all the transition form fac-tors are given by the sum over isosinglet and isovectorcontributions:∆(1232) , N (1520) : F ( p,n ) α ( Q ) = 12 K X k =1 " A ( ω ) αk ( Q ) m ω ) k Q + m ω ) k ± A ( ρ ) αk ( Q ) m ρ ) k Q + m ρ ) k , (34) N (1440) , N (1535) : G ( p,n ) α ( Q ) = 12 K X k =1 " B ( ω ) αk ( Q ) m ω ) k Q + m ω ) k ± B ( ρ ) αk ( Q ) m ρ ) k Q + m ρ ) k . (35)Because of the value of the ∆(1232) isospin, ρ -mesons areonly intermediaries in the N ∆-coupling, i.e., A ( ω ) αk ( Q ) =0. Dispersionlike expansions of the form factors are pre-dicted by the foundations of quantum field theory, thatare taken into account by the dispersion relation ap-proach. In the one-meson exchange approximation andin the limit of narrow-width mesons, the expansion coef-ficients A ( V ) αk ( Q ), B ( V ) αk ( Q ) are constants. But in Eqs.(34) and (35) they are supposed to be logarithmic func-tions of Q . It has been pointed out above in the Sec.II C 2 that logarithmic renormalization of the form fac-tors is demanded by quark-hadron duality. To this must be added that the logarithmic renormalization is also im-posed by short-distance quark-gluon processes influenc-ing the photon transition to mesons inside nucleon, i.e., at Q > R − N = (0 . . Logarithmic factors at expan-sion coefficients absorb in phenomenological fashion theeffects of the renormalization of the strong coupling con-stant and Q -evolution of the parton distribution func-tions.4 B. Asymptotic behavior of the dispersionlikeexpansions
At high Q the form factors (34) and (35) should joinpQCD-predictions (23) and (31). This property requiresthe expansion coefficients A ( ω, ρ ) αk ( Q ), B ( ω, ρ ) αk ( Q ) to obeya number of relations that we refer to from now on as thesuperconvergence relations (SRs).Since logarithmic renormalization has a bearing to onlythe QCD effects taking place inside nucleon, it seemsjustified to suppose that logarithmic Q -dependence ofthe form-factor expansion coefficients A ( V ) αk is univer-sal function L ( V ) A α ( Q ) independent of meson family in-dex k = 1 , , ..., K (similarly L ( V ) B α is Q -dependence ofthe amplitudes B ( V ) αk up to numeric factors). Then, theisosiglet and isotriplet running electromagnetic couplingparameters can be represented in the following form: K X k =1 A ( ω, ρ ) αk ( Q ) ≡ κ ( ω, ρ ) α ( Q ) = κ ( ω, ρ ) α (0) L ( ω, ρ ) A α ( Q ) , K X k =1 B ( ω, ρ ) αk ( Q ) ≡ κ ( ω, ρ ) α ( Q ) = κ ( ω, ρ ) α (0) L ( ω, ρ ) B α ( Q ) , (36)where κ ( ω, ρ ) α (0) = F ( p ) α (0) ± F ( n ) α (0) ,κ ( ω, ρ ) α (0) = G ( p ) α (0) ± G ( n ) α (0) (37)are the values of the parameters at Q = 0. The loga-rithmic functions L ( V ) A α ( Q ) and L ( V ) B α ( Q ) are known inthe static and asymptotic limit: L ( V ) α ( Q ) → ( , Q → ,C ( V ) α ln n α Q / Λ , Q → ∞ . (38) The most simple interpolation function retaining theasymptotic behavior (38) is L ( V ) α ( Q ) = 1 + C ( V ) α ln n α (cid:18) Q Λ (cid:19) . (39)Another possibility is L ( V ) α ( Q ) = (cid:20) h ( V ) α ln (cid:18) Q Λ (cid:19) + k ( V ) α ln (cid:18) Q Λ (cid:19)(cid:21) n α / . (40)The interpolation function (40) effectively takes into ac-count effects of nonleading pQCD-logarithms.The expansion coefficients are proportional to the run-ning coupling parameters. The numeric dimensionlessfactors of proportionality are denoted as follows: A ( V ) αk ( Q ) κ ( V ) α ( Q ) = a ( V ) αk = const, K X k =1 a ( V ) αk = 1 , B ( V ) αk ( Q ) κ ( V ) α ( Q ) = b ( V ) αk = const, K X k =1 b ( V ) αk = 1 . (41)Now the form factors (34) and (35) can be expressed interms of the parameters introduced above:∆(1232) , N (1520) : F ( p,n ) α ( Q ) = 12 " κ ( ω ) α ( Q ) K X k =1 a ( ω ) αk m ω ) k Q + m ω ) k ± κ ( ρ ) α ( Q ) K X k =1 a ( ρ ) αk m ρ ) k Q + m ρ ) k , (42) N (1440) , N (1535) : G ( p,n ) α ( Q ) = 12 " κ ( ω ) α ( Q ) K X k =1 b ( ω ) αk m ω ) k Q + m ω ) k ± κ ( ρ ) α ( Q ) K X k =1 b ( ρ ) αk m ρ ) k Q + m ρ ) k , (43)where κ ( ω ) α ( Q ) ≡ /Q and set the coefficients preceding Q − , Q − (and Q − inthe case of F , , G ) equal to zero. The constraints ob-tained in the fashion described are the SRs between themeson parameters a ( V ) αk , b ( V ) αk .The set of the SRs between the parameters of the tran- sition N → ∆(1232) are α = 1 : K X k =1 a ( ρ )1 k = 1 , K X k =1 a ( ρ )1 k m ρ ) k = 0 , K X k =1 a ( ρ )1 k m ρ ) k = 0; (44)5 α = 2 , K X k =1 a ( ρ ) αk = 1 , K X k =1 a ( ρ ) αk m ρ ) k = 0 , K X k =1 a ( ρ ) αk m ρ ) k = 0 , K X k =1 a ( ρ ) αk m ρ ) k = 0 . (45)In the case of the transition N → N (1520) the nonvan-ishing isosinglet contributions to the form factors lead tosome more SRs between the parameters of ω -mesons: α = 1 : K X k =1 a ( ω )1 k = 1 , K X k =1 a ( ω )1 k m ω ) k = 0 , K X k =1 a ( ω )1 k m ω ) k = 0; (46) α = 2 , K X k =1 a ( ω ) αk = 1 , K X k =1 a ( ω ) αk m ω ) k = 0 , K X k =1 a ( ω ) αk m ω ) k = 0 , K X k =1 a ( ω ) αk m ω ) k = 0 . (47)The SRs similar to (44)–(47) are valid for the parameters b ( ω, ρ ) αk of the form factors for the N → N (1440) and N → N (1535) transitions. IV. DATA ANALISIS. DISCUSSION ANDPREDICTIONSA. The ∆(1232)
From the point of view of helicity-amplitude fitting,the ∆(1232) resonance offers an important simplification:its excitation via electroproduction off nucleon is only byphoton and ρ -mesons, which halves the number of disper-sionlike expansion coefficients. Besides, the data on the∆(1232) helicity amplitudes is much more vast and pre-cise compared to the data sets on other resonant ampli-tudes. All that allows the form factors F ( Q ), F ( Q ), F ( Q ) to be extracted to a high accuracy.In the case of the ∆(1232), the described VMD modelgives the following expressions for the form factors F ( Q ) = F ( exp )1 L ( ρ )1 ( Q ) K X k =1 a ( ρ )1 k m ρ ) k Q + m ρ ) k ,F ( Q ) = F (0) L ( ρ )2 ( Q ) K X k =1 a ( ρ )2 k m ρ ) k Q + m ρ ) k ,F ( Q ) = F ( exp )23 − F (0) L ( ρ )3 ( Q ) K X k =1 a ( ρ )3 k m ρ ) k Q + m ρ ) k , (48) where F ( exp )1 = 2 . ± . F ( exp )23 ≡ F (0) + F (0) = − . ± .
452 — measured electrodynamic parameters; L ( ρ )1 , , ( Q ) — logarithmic functions satisfying the asymp-totes (38).The number of the form-factor poles K is boundedabove by the cut-off of the ρ -meson spectrum and belowby the number of the SRs (44) and (45). In this simplestmodel K = 4 dealing with only the first four ρ -mesonsfrom Table II, all the parameters a ( ρ ) αk , α = 2 , a ( ρ )2 k = a ( ρ )3 k . From this point on,we restrict the discussion to the specific case of the modelwith K = 4, in order to reduce the number of free param-eters. However, as it was pointed out in Sec. III, thereare no physical reasons to cut off the meson spectrumartificially. In fact, the spectrum should be truncated athighly excited vector states with widths exceeding inversehadronization time Γ V > T − g ≃ .
1. Fit results
In the simplest model with K = 4 incorporating onlythe first four ρ -mesons, all the parameters a ( ρ )2 k are de-termined by the four SRs (45). Also the parameters a ( ρ )1 k satisfy three SRs (44) that allow one parameter to beadjusted freely. Another one independent parameter ofthe model is either F (0) or F (0). Besides, electrody-namic parameters F (0), F (0) + F (0) and the scale Λcan be varied, so that not to go beyond experimentalerrors. Logarithmic renormalization is taken into ac-count within the models with the simplest one-parameter(39) and two-parameter (40) interpolation functions for n = 3, n = 1, n = 4. Thereby, the four-pole mod-els used to fit experimental data comprise 8 and 11 freeparameters, respectively, three of which are constrainedwithin experimental uncertainties. In the following werefer to corresponding fits as F1 (one-parameter inter-polation functions) and F2 (two-parameter interpolationfunctions).The adjusted parameters are set out in Table III. Thecorresponding curves are depicted in Figs. 1, 2, 3, 4 incomparison with experimental data points collected fromthe papers [1, 2, 3, 4, 5, 6, 24, 25, 76, 77, 78, 79]; thecurves of the magnetic transition form factor G ∗ M ( Q )and the ratio R EM ( Q ) are displayed up to 15 GeV ,since the high-energy measurements in this region areproposed by JLab Hall C collaboration [26].Distinctions between the models with (39) and (40)are clearly seen in Figs. 2 and 4 displaying the ratios R EM , R SM and extracted form factors. The model F1with one-parameter logarithmic renormalization tends tounderestimate significantly the magnitude of the Mainzdata [3, 4] on R SM in the quasistatic domain. While thenine-parameter fit F2 does not suffer from this flaw, it,however, predicts electric quadrupole moment to change6sign at 5 . , which contradicts the highest to dateJLab data point indicating no sign change up to 6 GeV [1].It should be noted at this point, that any realizationof VMD model involving logarithmic renormalization re-quires putting forward reliable hypothesis about the wayto introduce logarithmic corrections. In the developedframework, such an arbitrary treatment of logarithmicinterpolation functions originates in part from the suppo-sition of the values of the exponents n , n , n . This sup-position is necessary only until the proper calculations ofthe helicity amplitude asymptotes including logarithmiccorrections are carried out. However, only an improvingexperimental data seems to be an ultimate solution to theproblem that could rule out some interpolation formulasand reduce discrepancy between fits making use of therest allowed ones. In this regard, the quasistatic domainis as important as proposed high-energy JLab measure-ments [26]. For example, the current errors of the he-licity amplitude extraction do not exclude two types of Q -evolution of the form factors F and F near the pho-ton point: monotonous falloff and the Q -behavior withthe derivative changing sign. It may be shown that themodel with K = 4 and logarithmic renormalization (39)is capable of reproducing the first regime only. There-fore, the future measurements could prove such a modelto be inadequate in the limit of large distances. Also theexperiments at small Q might reveal the sign change ofthe form factors F and F . The zero of the form factorscan be reproduced only in the model with the number of ρ -mesons K > S / (0) are 0 .
011 GeV − / for the fit F1 and0 .
017 GeV − / for the fit F2.A good agreement of the four-pole models with exper-imental data ( χ / DOF ≈ . − .
0) testifies that physicsof the transition form factors can be formulated in termsof the QCD-inspired VMD model which deals with all ex-cited states of the ρ (770) and involves logarithmic renor-malization and SRs between parameters of meson spec-trum. It is remarkable that this good fit is possible inthe model with the minimal number of free parameters.
2. Problem of the transition to pQCD
The challenge to observe the onset of asymptotic evo-lution of resonant helicity amplitudes is one of the goalsinspiring experimentalists to carry out high- Q measure-ments [26]. However, the transition to perturbative do-main is unlikely to manifest itself anyway in the cur-rent world data base on N ∆(1232)-transition form fac-tors. This fact is clearly exemplified by recent exclusiveJLab data on R EM [1] and inclusive SLAC data [24, 25]on transverse helicity amplitude A T . JLab results on R EM depicted in Fig. 2 evidence that this ratio remainssmall and negative up to 6 GeV , while the perturba-tive asymptote is R EM → +1. The inclusive data are TABLE III: Fit parameters (spin-3/2 resonances). Depen-dent parameters are tabulated in the bottom part of the ta-ble ( a ( ρ,p )3 k are not presented, since a ( ρ,p )2 k = a ( ρ,p )3 k in the 4-polemodel). F1 is a fit with logarithmic functions (39); F2 is a fitwith logarithmic functions (40).∆(1232) N (1520)F1 F2 F1 F2 χ / DOF 1 .
98 1 .
63 3 . . a . . a a ( ρ,p )14 − . − .
240 0 . − . F ( p )1 (0) 2 .
046 2 .
052 2 .
970 2 . F ( p )2 (0) − . − . − . − . F ( p )2 (0) + F ( p )3 (0) − . − . − . − . .
190 0 .
190 0 .
190 0 . C ( ρ,p )1 .
010 — 0 .
014 — C ( ρ,p )2 .
021 — 0 — C ( ρ,p )3 .
003 — 0 — h ( ρ,p )1 — − .
007 — 0 . k ( ρ,p )1 — 0 .
014 — 0 . h ( ρ,p )2 — − .
338 — 0 k ( ρ,p )2 — 0 .
053 — 0 h ( ρ,p )3 — − .
278 — 0 k ( ρ,p )3 — 0 .
052 — 0 a ( ρ,p )11 .
870 2 .
041 1 .
564 1 . a ( ρ,p )12 − . − . − . − . a ( ρ,p )13 .
756 3 . − . − . a ( ρ,p )21 .
101 2 .
101 2 .
142 2 . a ( ρ,p )22 − . − . − . − . a ( ρ,p )23 .
033 4 .
033 3 .
146 3 . a ( ρ,p )24 − . − . − . − . a This is the value of χ / DOF recalculated with data points at1 GeV and 1 .
45 GeV being excluded from the data set. Thesepoints disagree significantly with others, which is seen in Fig. 5. obtained up to almost 10 GeV and exhibit a trend todecrease more rapidly than 1 /Q predicted by pQCD,though experimental uncertainties are quite considerable.This is readily seen on the right panel of Fig. 11 thatshows transverse amplitude normalized by its asymptote.The studies of the transition form factors for Q <
14 GeV proposed by Jefferson Laboratory [26] seem tobe of great importance as the meeting ground betweenpredictions made by both baryon-meson and quark-parton physics, which is a new side of quark-hadronduality. It is a well-established fact that asymptoticquark-parton description of inclusive deep-inelastic eN -scattering is adequate for Q ≫ T − g , where T g =(1 . − . − is a space-time scale of nonpertur-bative quark-gluon fluctuations. However, the transi-tion to pQCD in exclusive resonant process eN → e ∆is shifted to higher momentum transfers, which is clearlyillustrated by the data depicted in Figs. 11. This couldbe explained by qualitative estimates as follows. Energy7 R E M Q , GeV Q A T , G e V / Q , GeV FIG. 11: Asymptotic behavior of the ratio R EM and total transverse amplitude A T for the ∆(1232). Curves and data are thesame as in Fig. 1. transfer from electron to quark (parton) ∆ E ∼ Q / M N should be shared equally between all valence quarks par-ticipating in exclusive process and all the quark energy-exchange subprocesses must be hard. Hence, pQCD isthe correct description for exclusive reactions in the re-gion of Q ≫ T − g ≃ . − . . This does notcontradict the current experimental data. Moreover, itis just the domain where VMD model discussed in thispaper predicts R EM to cross zero and rise gradually. Itcould be regarded as a signal of the transition to pQCD(see Fig. 11, left panel).In baryon-meson physics transition form factors canbe described in the framework of the dispersion relationapproach or its simplest realization — the QCD-ispiredVMD model. Form factors represented as dispersionlikeexpansions have correct pQCD asymptotic behavior bythe construction, but do not approach them in the region Q <
10 GeV . Indeed, expanding dispersionlike formfactors in inverse powers of Q provides a quantitativecriterion for the transition to the asymptotic domain: Q ≫ Λ → Q =sup n > p α +1 α =1 , , " K X k =1 a ( ρ ) αk m n ( ρ ) k (cid:30) K X k =1 a ( ρ ) αk m p α ( ρ ) k n − p α , (49)where p = 3, p = p = 4 [see Eq. (23)]. Thereby, thescale of the transition to pQCD Λ H → Q is determined inthe VMD model by properties of the interactions betweenbaryons and vector mesons and, certainly, by the struc-ture of the vector-meson spectrum. The quark-hadronduality as an agreement of the predictions by pQCD andbaryon-meson models implies the following relation3 T − g ≃ Λ → Q . (50)The discussed models involving four vector mesons and logarithmic renormalization of the type (39) and (40)provide Λ → Q = 9 .
24 GeV and Λ → Q = 9 .
67 GeV respectively. However, one is forced to accept the factthat these values are affected by the aforementioned in-trinsic drawbacks of the model and large experimental er-rors, especially, at high momentum transfers. But we be-lieve that theoretical refinement of the model and, whatis likely to be even more important, future high- Q ex-clusive experiments followed by extraction of all helicityamplitudes would improve the situation substantially. Inthis regard it is worth mentioning that the four-pole mod-els are able to reproduce quite different rates of the tran-sition to pQCD, which can be proved by adding somehypothetic experimental data to the current data baseand fitting to it. This fact makes us expect that futuremeasurements of transition form factors will neither con-strain the range of validity of the QCD-inspired VMDmodel nor reduce the overall quality of fit. B. The second resonance region
The second resonance region covers the W range be-tween approximately 1.4 GeV and 1.6 GeV. It includesthree isospin 1 / N (1440), N (1520), N (1535).Though both ρ - and ω -mesons contribute to the excita-tion of these baryons, currently there is no measurementsof neutron helicity amplitudes, except for the photopro-duction data [77]. Thus, in the framework developed,it is hardly possible to distinguish reliably isovector andisoscalar contributions to the form factors. Because ofthis reason, in the following we neglect singlet-tripletmass splitting and suppose that ρ - and ω -mesons propa-gate in the nucleon medium identically, i.e., L ( ρ ) α ≡ L ( ω ) α .In such a model, proton transition form factors dependon half as many independent parameters as the form fac-8tors (42) and (43): N (1520) : F ( p ) α ( Q ) = F ( p ) α (0) L ( p ) α ( Q ) K X k =1 a ( p ) αk m k Q + m k ; (51) N (1440) , N (1535) : G ( p ) α ( Q ) = G ( p ) α (0) L ( p ) α ( Q ) K X k =1 b ( p ) αk m k Q + m k , (52)where L ( p ) α ≡ L ( ρ ) α ≡ L ( ω ) α is a logarithmic interpolationfunction of type (39) or (40), m k = ( m ω ) k + m ρ ) k ) / a ( p ) k satisfy the same set of the SRs (44)–(47) as a ( ρ ) k and a ( ω ) k do.In spite of the simplifications described above, the formfactors (51) and (52) are found to provide a good fit ofthe existing data in the second resonance region.
1. The N (1520) Since the resonance N (1520) possesses spin 3 /
2, we usethe four-pole model (48). Logarithmic renormalization istaken into account by means of one-parameter (39) andtwo-parameter (40) interpolation functions (correspond-ing fits are denoted F1 and F2, respectively) for n = 3, n = 1, n = 4. In both cases, logarithmic dependency ofthe spin-flip form-factors F p , F p could be neglected as itdoes not manifest itself in the fit to the data in the exper-imentally acceptable domain Q < . Therefore,the models used to fit the experimental data comprise 6and 7 adjustable parameters: two electrodynamic param-eters F p (0), F p (0) + F p (0) constrained by photoproduc-tion data and one parameter F p (0) varied freely; an ex-pansion coefficient a ( p )14 ; QCD-scale Λ = 0 . − .
24 GeV;and one or two parameters of the interpolation func-tion L ( p )1 .The fit results are presented in Table III and Figs. 5,6. The large values of χ /DOF is likely to be attributedto the discrepant data points [78] in the region between Q = 1 GeV and Q = 2 GeV .It should be noted that the good fits to the exper-imental data on helicity amplitudes of two spin-vectorstates ∆(1232) and N (1520) are obtained in the unifiedapproach, based on four-pole dispersionlike form-factorexpansions (48) satisfying SRs (44)–(47). This is an evi-dence for validity of the VMD model in physics of high-spin nucleon excitations.
2. The N (1535) The resonances N (1535) and N (1440) are spin-1 / Q -dependenceof the form factors. So, the fit models F1 and F2 intro-duce 8 and 10 free parameters: two electrodynamic pa-rameters G p (0) and G p (0) (the second one is constrainedby photoproduction data), five parameters of functions L ( p )1 and L ( p )2 (QCD scale Λ is varied between 0 .
19 GeVand 0 .
24 GeV), one (in the case of four-meson fit F1) orthree (in the case of five-meson fit F2) of the expansioncoefficients b ( p ) αk .The available data points [7, 8, 25, 69, 77, 78, 79] aredivided into two samples fitted separately. The first datasample S1 is all the data with the analyses [69, 79] byI. G. Aznauryan et al. being excluded. The second onedenoted S2 includes the PDG average at photon point[77], inclusive data from [25] in the region Q > ,where exclusive experiments have not yet been carriedout, and the most recent exclusive JLab data [7, 69, 79].The reason to fit these data samples apart is that analysisfullfilled by L. Tiator et al. [78] predicts slightly morerapid falloff of transverse amplitude and more substantialrise of longitudinal one than the data points from [69, 79]exhibit (Fig. 7). In the model based on vertex (4), thiscontradiction manifests itself in the most obvious way asthe discrepancy in the extracted data on the form factor G p , depicted in Fig. 8. Nevertheless, the four- and five-pole models provide good fits to both data samples.The fit parameters are set out in Table IV. The cor-responding curves are shown in Figs. 7 and 8. It is ofsome interest that the fits to both data samples give thevalue of b ( p )15 to be an order of magnitude less than b ( p )14 .It could be regarded as evidence that only the first fourmeson families contribute to the form factor G p of theresonance N (1535).
3. The N (1440) To fit the experimental data on the Roper resonance N (1440), we make use of the same model as that de-scribed in the previous subsection IV B 2. However, un-like the form factors of the transition N → N (1535),both nucleon-to-Roper form factors cross zero in the re-gion Q < . (Fig. 10). To incorporate this ef-fect is only possible in the models involving at least fivemesons, since the simplest four-pole form factors withcorrect power pQCD-asymptotes (30) are predicted bythe model with meson masses from [77] to be monotonousand to conserve the sign.As in the case of the N (1535), there is some discrep-ancy between results of the helicity-amplitude extractionmade in the framework of the MAID model [78] and JLabUIM [79, 80]. For instance, while the analysis [79, 80]indicates that the form factor G p crosses zero between0 . and 0 .
65 GeV , MAID calculations [78] shift9 TABLE IV: Fit parameters (spin-1/2 resonances). Dependentparameters are tabulated in the bottom part of the table. S1,S2 are data samples introduced in the text. F1 is a four-polefit; F2 is a five-pole fit. N (1535) N (1440)S1 S2 S1 S2F1 F2 F1 F2 F2 F2 χ / DOF 2 .
61 2 .
45 0 .
46 0 .
48 7 . . a . b ( p )14 .
860 1 .
082 1 .
964 2 . − . − . b ( p )15 — 0 .
048 — − . − . − . b ( p )25 — 1 .
064 — 0 .
872 5 .
814 9 . G ( p )1 (0) 0 .
417 0 .
407 0 .
179 0 . − . − . G ( p )2 (0) 0 .
210 0 .
210 0 .
210 0 . − . − . .
240 0 .
240 0 .
240 0 .
240 0 .
240 0 . h ( p )1 − .
001 0 . − . − . − . − . k ( p )1 .
009 0 .
012 0 .
057 0 .
054 0 .
095 0 . h ( p )2 − . − . − . − . − . − . k ( p )2 .
062 0 .
070 0 .
065 0 .
068 0 .
082 0 . b ( p )11 .
446 1 .
294 1 .
005 1 .
064 3 .
105 3 . b ( p )12 .
384 1 .
156 2 .
793 2 . − . − . b ( p )13 − . − . − . − .
564 8 .
687 11 . b ( p )21 .
142 2 .
563 2 .
142 2 .
487 4 .
441 5 . b ( p )22 − . − . − . − . − . − . b ( p )23 .
146 9 .
868 3 .
146 8 .
655 39 .
87 62 . b ( p )24 − . − . − . − . − . − . a This is the value of χ / DOF recalculated with data points at0 .
525 GeV and 1 .
45 GeV being excluded from the data sample.These points disagree significantly with others from the sample S1,which is seen in Fig. 9. the sign change to the domain between 0 .
75 GeV and0 . . That’s why we divide the data points into twosamples and fit to them separately. Both samples includePDG averages at photon point [77] and the data from [69]for Q > . , but in the region Q < . thefirst sample S1 takes into account just the analysis [78],and the second one S2 includes the data from [79, 80].The adjusted parameters are tabulated in Table IV.The corresponding helicity amplitudes and extractedform factors are depicted in Figs. 9 and 10. V. CONCLUSION
We have investigated
N R -form factors in the first andsecond resonance regions, utilizing effective-field theorywith 4–5 explicit vector-meson degrees of freedom. Tran-sition form factors in the model comprise 6–10 free pa-rameters for each resonance which have been fitted to ex-perimental data. All these parameters have clear physicalmeaning (low-energy electromagnetic constants, meson-baryon couplings, and phenomenological parameters oflogarithmic renormalization). This QCD-inspired VMD model is in good agreementwith the data available on resonant helicity amplitudein the first and second resonance regions. This suc-cess makes us believe that the model, though being phe-nomenological, provides an insight into the Q -evolutionof nucleon-to-resonance transitions. The basic physicalideas of the approach are as follows:1. The photon, propagating in the inside-nucleonmedium, excites all the modes of a hadronic string,carrying photon quantum numbers J P C = 1 −− .Thus, all the vector mesons should be, in principle,incorporated into the VMD model, which makesthe form-factors be dispersionlike expansions withpoles at meson masses.2. Short-distance quark-gluon processes contribute tothe hadronization of a photon into intermediarymesons inside nucleon, i.e., at Q > R − N =(0 . . The VMD model takes into accountthe small-scale dynamics by effective logarithmicrenormalization of electrodynamic coupling con-stants.3. The VMD model should be reconciled with pQCD,commonly believed to be the ab initio treatmentof resonant electroproduction at high momentumtransfers. To attain requisite asymptotic behav-ior of the form-factors is possible by imposing lin-ear superconvergence relations on the parametersof the vector-meson spectrum. Besides, logarith-mic renormalization of meson-baryon parametersis essential at this point, as it allows to includelogarithmic corrections to pQCD-asymptotes andto make the asymptotes of spin-flip and non-spin-flip phenomenological form factors match with theircounterparts at quark level.It should be noted that inclusion of all the vectormesons appears to be impossible in the framework ofthe VMD model tested in this paper, as it could over-parametrize fit to the data. However, even the simplestmodels with four and five lightest vector mesons, whichintroduce only one and three independent meson-baryoncoupling, respectively, is found to be in accord with allthe experimental data analyzed. In our opinion, it sup-ports the notion that the aforementioned effects makea major contribution to Q -evolution of the nonstrangeresonance excitation.Further work, however, needs to be done to improvevector-meson-dominance model of the nucleon transi-tion form factors. This improvement should includeboth theoretical refinement and new experiments. Fromthe theoretical point of view, it seems to be impor-tant to test alternative Lagrangians of nucleon interac-tions with high-spin resonances [70] and to carry outpQCD-calculations of logarithmic corrections to form-factor asymptotes. Also, it seems reasonable to extendthe model to be directly compared with experimental0data on eN -scattering observables, for the extractionof helicity amplitudes is known to be model dependent[67]. Future experiments in both quasistatic and high- Q regions will provide important information imposingconstraints on phenomenological logarithmic renormal-ization and, especially, on the contributions of nonleadingpQCD-logarithms. Acknowledgments
We are grateful to A. V. Beylin and V. I. Kuksa fortheir reading the draft of this paper, useful comments,and invaluable criticisms.
APPENDIX: HELICITY-AMPLITUDE PHASES
Helicity amplitudes are defined as matrix elements ofthe electromagnetic current operator calculated betweenthe initial nucleon and final resonance states. As thepolarized states of spinor fields always contain arbitraryphase, helicity amplitudes are also defined up to phases,that cannot be calculated in the extraction based on thesimple cross section formula (5). That’s why we put for-ward three additional empirical criteria that could fixamplitude phases:1. electrodynamic parameters should be within a fac-tor of ten;2. form-factor expansion coefficients should have anorder of magnitude of 1 or less;3. convergence and quality of the fit.These criteria are, obviously, “fit-dependent”, i.e.,phases might depend upon specific features of the model,such as the choice of logarithmic interpolation functions,the meson spectrum, etc. However, we found them to beable to fix the phases reliably, at least for the resonances∆(1232), N (1520), N (1535).In the scope of this paper, only the ratios of the phasesare important and one of the amplitude phases can befixed arbitrarily. We choose the phase of the amplitude A / .
1. The ∆(1232)
In the case of the ∆(1232), the ratio of the phases of thetransverse amplitudes is determined by the first criterion.Indeed, amplitudes (9) and (10) give F ( p )1 (0) ≈ . F ( p )2 (0) + F ( p )3 (0) ≈ . F ( p )1 (0) ≈ . F ( p )2 (0)+ F ( p )3 (0) ≈ .
62. This option does not fulfill the first criterion. Besides, itis found impossible to fit the experimental data in thefour-pole model with such phases, while this model canreproduce observed Q -evolution of helicity amplitudes(9)–(11).The sign of the longitudinal amplitude S / is fixed bythe third criterion, since the four-pole model with thesign of the S / opposite to that of (11) crucially under-estimates the ratio R SM predicting it to be of an order of0.001 or less for Q > .
2. The N (1520) For N (1520), the first criterion is not effective, sinceall the isovector and isoscalar electromagnetic parametersare of the same order of magnitude, regardless of the ratioof the transverse amplitudes phases. Therefore, phasesof the amplitudes (9)–(11) are chosen by the quality ofthe fits. All the alternatives to formulas (9)–(11) lead toa several-fold increase in χ /DOF compared to the bestvalues, obtained in the model (9)–(11).
3. The N (1535) The values of χ /DOF obtained in the fits with thesign of S / being opposite to that of (26) are 1.2 (S1-F1), 1.34 (S1-F2), 2.67 (S2-F1), 2.38 (S2-F2). The firsttwo values are more than twice as large as those set outin Table IV. However, the difference in the quality ofthe fits to the data sample S2 is subtle. Nevertheless,these fits converge for some of the form-factor expansioncoefficients being of an order of 10, which dissatisfies thesecond criterion.
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525 GeV and 1 .
45 GeV being excluded.So, the choice of amplitude signs for the N (1440) essen-tially depends on the particular sample of experimentaldata. We choose the signs which allow the best fit to thesample S2, since it includes the most recent results of he-licity amplitude extraction [69, 79, 80] from the data onone and two pion electroproduction.1 [1] M. Ungaro et al. (The CLAS Collaboration), Phys. Rev.Lett. , 112003 (2006), arXiv:hep-ex/0606042v1.[2] V. V. Frolov et al., Phys. Rev. Lett. , 45 (1999),arXiv:hep-ex/9808024v1.[3] S. Stave et al. (A1 Collaboration), Eur. Phys. J. A ,471 (2006), arXiv:nucl-ex/0604013v2.[4] N. F. Sparveris et al., Phys. Rev. Lett. , 022003 (2005),arXiv:nucl-ex/0408003v1.[5] D. Elsner et al., Eur. Phys. J. A , 91 (2006),arXiv:nucl-ex/0507014v2.[6] J. J. Kelly et al. (Jefferson Laboratory E91011 and HallA Collaborations), Phys. Rev. Lett. , 102001 (2005),arXiv:nucl-ex/0505024v1.[7] C. S. Armstrong et al., Phys. Rev. D , 052004 (1999),arXiv:nucl-ex/9811001v5.[8] R. Thompson et al. (The CLAS Collaboration), Phys.Rev. Lett. , 1702 (2001), arXiv:hep-ex/0011029v1.[9] C. Kunz et al. (MIT-Bates OOPS collaboration), Phys.Lett. B , 21 (2003), arXiv:nucl-ex/0302018v2.[10] H. Egiyan et al. (The CLAS Collaboration), Phys. Rev.C , 025204 (2006), arXiv:nucl-ex/0601007.[11] F. Kalleicher, U. Dittmayer, R. W. Gothe, H. Putsch,T. Reichelt, B. Schoch, and M. Wilhelm, Z. Phys. A359 ,201 (1997).[12] P. Bartsch et al., Phys. Rev. Lett. , 142001 (2002),arXiv:nucl-ex/0112009v1.[13] Th. Pospischil et al., Phys. Rev. Lett. , 2959 (2001).[14] K. Joo et al. (The CLAS Collaboration), Phys. Rev. Lett. , 122001 (2002).[15] K. Joo et al. (CLAS Collaboration), Phys. Rev. C ,032201(R) (2003), arXiv:nucl-ex/0301012v2.[16] K. Joo et al. (The CLAS Collaboration), Phys. Rev. C , 042201(R) (2004), arXiv:nucl-ex/0407013v2.[17] C. Mertz et al., Phys. Rev. Lett. , 2963 (2001),arXiv:nucl-ex/9902012v2.[18] M. Ripani et al. (CLAS Collaboration), Phys. Rev. Lett. , 022002 (2003), arXiv:hep-ex/0210054v1.[19] A. Biselli et al. (CLAS Collaboration), Phys. Rev. C ,035202 (2003), arXiv:nucl-ex/0307004v1.[20] G. A. Warren et al. (The M.I.T.-Bates, OOPS, andFPP Collaborations), Phys. Rev. C , 3722 (1998),arXiv:nucl-ex/9901004v1.[21] R. De Vita et al. (CLAS Collaboration), Phys. Rev. Lett. , 082001 (2002), arXiv:hep-ex/0111074v2.[22] G. Blanpied et al. (The LEGS Collaboration), Phys. Rev.C , 025203 (2001).[23] V. D. Burkert and T.-S. H. Lee, Int. J. Mod. Phys. E13 ,1035 (2004), arXiv:nucl-ex/0407020v1.[24] L. M. Stuart et al., Phys. Rev. D , 032003 (1998),arXiv:hep-ph/9612416v1.[25] P. Stoler, Phys. Rep. , 103 (1993).[26] P. Stoler, URL .[27] M. F. M. Lutz, B. Friman, and M. Soyeur, Nucl. Phys.A , 97 (2003).[28] V. Pascalutsa, M. Vanderhaeghen, and S. N. Yang, Phys.Rept. , 125 (2007), arXiv:hep-ph/0609004v5.[29] C. Alexandrou, Ph. de Forcrand, and A. Tsapalis, Phys.Rev. D , 094503 (2002), arXiv:hep-lat/0206026v3.[30] C. Alexandrou, Ph. de Forcrand, Th. Lippert, H. Neff,J. W. Negele, K. Schilling, W. Schroers, andA. Tsapalis, Phys. Rev. D , 114506 (2004), arXiv:hep- lat/0307018v2.[31] C. Alexandrou, Ph. de Forcrand, H. Neff, J. W. Negele,W. Schroers, and A. Tsapalis, Phys. Rev. Lett. ,021601 (2005), arXiv:hep-lat/0409122v1.[32] A. W. Thomas, Nucl. Phys. B (Proc. Suppl.) , 50(2003), arXiv:hep-lat/0208023v1.[33] D. H. Lu, A. W. Thomas, and A. G. Williams, Phys.Rev. C , 3108 (1997).[34] V. D. Burkert, R. De Vita, M. Battaglieri, M. Ripani, andV. Mokeev, Phys. Rev. C , 035204 (2003), arXiv:hep-ph/0212108v1.[35] M. De Sanctis, M. M. Giannini, E. Santopinto, andA. Vassallo, Eur. Phys. J. A , 81 (2004), arXiv:nucl-th/0401029v2.[36] A. J. Buchmann, Phys. Rev. Lett. , 212301 (2004),arXiv:hep-ph/0412421v1.[37] H. Walliser and G. Holzwarth, Z. Phys. A , 317(1997), arXiv:hep-ph/9607367v2.[38] E. Braaten, S.-M. Tse, and C. Willcox, Phys. Rev. Lett. , 2008 (1986).[39] P. Alberto, M. Fiolhais, B. Golli, and J. Marques, Phys.Lett. B , 273 (2001), arXiv:hep-ph/0103171v2.[40] M. Fiolhais, B. Golli, and S. ˇSirca, Phys. Lett. B373 ,229 (1996), arXiv:hep-ph/9601379v1.[41] L. Amoreira, P. Alberto, and M. Fiolhais, Phys. Rev. C , 045202 (2000), arXiv:hep-ph/0009151v1.[42] A. Silva, D. Urbano, T. Watabe, M. Fiolhais, andK. Goeke, Nucl. Phys. A675 , 637 (2000), arXiv:hep-ph/9905326v2.[43] M. D. Slaughter, Nucl. Phys.
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