We study the electromagnetic transitions of the Roper N(1440) resonance. Our
results, when combined with the previously obtained for the mass and the pionic
strong decay widths of the Roper, show that within a non-relativistic
constituent quark model scheme, a comprehensible understanding of the Roper
phenomenology can be achieved. They also seem to support the view of the Roper
as a radial excitation of the nucleon, though more experimental data are needed
to reach a definitive conclusion.Comment: 14 pages, 4 figures (7 postscript files). Some referencess adde

Barnes

Brown

Cano

Capstick

Cardarelli

Carlson

Dong

F. Cano

Gavela

Gerhardt

Glozman

Guichon

P. González

Silvestre-Brac

Physics Letters B

English

arXiv

Crossref

A consistent explanation of the Roper phenomenology

We study the electromagnetic transitions of the Roper N(1440) resonance. Our results, when combined with the previously obtained for the mass and the pionic strong decay widths of the Roper, show that within a non-relativistic constituent quark model scheme, a comprehensible understanding of the Roper phenomenology can be achieved. They also seem to support the view of the Roper as a radial excitation of the nucleon, though more experimental data are needed to reach a definitive conclusion

Cano Vargas, Florencio

González Marhuenda, Pedro

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

arXiv:nucl-th/9804071v2  15 May 1998A CONSISTENT EXPLANATION OF THEROPER PHENOMENOLOGYF. Cano and P. Gonza´lezDepartamento de F´ısica Teo´rica and IFIC,Centro Mixto Universidad de Valencia-CSIC46100 Burjassot (Valencia), SpainAbstractWe study the electromagnetic transitions of the Roper N(1440) res-onance. Our results, when combined with the previously obtained forthe mass and the pionic strong decay widths of the Roper, show thatwithin a non-relativistic constituent quark model scheme, a compre-hensible understanding of the Roper phenomenology can be achieved.They also seem to support the view of the Roper as a radial excitationof the nucleon, though more experimental data are needed to reach adefinitive conclusion.PACS: 12.39.Jh, 12.40.Vv, 13.40.Gp, 13.40.HqKeywords: Non-relativistic quark models, Roper, electromagnetic decays.cano@xaloc.ific.uv.esgonzalep@evalvx.ific.uv.es1Baryon resonances and in particular the Roper resonance N(1440) deservespecial attention at the current moment from the theoretical as well as theexperimental point of view. The Roper resonance has been observed at Sat-urne as an excitation of a hydrogen target scattered by α-particles [1]. Thetransition form factors for the excitation of baryon resonances are going to besystematically measured at TJNAF in a near future. Related experimentalprograms are carried out in MAMI, ELSA and GRAAL. Theoretically, to pre-dict the precise position of the Roper in the nucleon resonance spectrum andthe correct values (including signs) of its strong and electromagnetic transitionamplitudes has been a challenge for many years.Though the simplest image of the Roper corresponds to a radial excitationof the 3q nucleon state, due to the difficulty in describing its phenomenology,’more completed’ or alternative descriptions have been proposed. Among thefirst ones we can mention the relativized versions of the constituent quarkmodel [2, 3] and the light front constituent quark model [4, 5]. Among thesecond ones the breathing bag model [6, 7] and the hybrid state descriptions[8, 9, 10, 11] which confer to the Roper an exotic nature. Nonetheless a unifieddescription of the Roper phenomenology has not been achieved and its natureis still under discussion.In the last few years very precise fits of the low-lying baryon spectrum havebeen obtained making use of potential models [12, 13, 14]. One of the models[12] has also been succesfully applied to predict the pionic decay widths ofthe low-lying nucleon and delta resonances [15]. Concerning the Roper, theconsideration, via a 3P0 model, of the qq¯ pion structure, becomes essential tocorrectly reproduce the strong pionic decay data. The purpose of this paperis to apply the same model to the calculation of photo and electroproductionamplitudes of the Roper, the aim being to check whether it is possible or not,from a non-relativistic constituent quark model, to get a unified description ofthe Roper phenomenology and to learn about its elusive nature.1 Photo and electroproduction amplitudesSome years ago, Gavela et al. [16] got a reasonable description at that timeof the Roper photoproduction amplitude N(1440)→ Nγ by combining vectormeson dominance (VMD) with a 3P0 model for the meson production. Asimple and immediate generalization to Q2 6= 0 is obtained by introducing anintermediate propagator for the meson (a more fundamental, but at the sametime more complicated technical approach would be to couple the photon tothe qq¯ structure of the meson). The transition matrix element for a generalprocess B → B′γ reads:2〈B′γ|H|B〉 =∑V〈B′V |H3P0|B〉1m2V +Q2〈γ|HV γ|V 〉 (1)with V = ρ, ω and [17]:〈γ|HV γ|V 〉 = em2VfV(2)The matrix element in the 3P0 model is given by [18]:〈B′V |H3P0|B〉 = −3γ∑m(110|m −m)∫d~p1 d~p2 d~p3 d~p4 d~p5 Y1m(~p4 − ~p5)δ(~p4 + ~p5)Φ−mPair[ΨB′(~p1, ~p2, ~p4)ΦB′ ]∗ [ΨV (~p3, ~p5)ΦV ]∗ [ΨB(~p1, ~p2, ~p3)ΦB](3)where γ is the pair creation coupling constant, Y1m a solid harmonic, Ψ standsfor the momentum space wave function and Φ for the spin-isospin wave func-tion.By using eqs. (2) and (3) we can write after some algebra the transitionmatrix element as:〈B′ γ|H|B〉 = −3eγ(mAωγ)1/2δ(3)(~PB − ~PB ′ − ~q )(− 1√4π12√23)[1fωm2ωm2ω +Q2〈ηB′ |ηB〉+ 1fρm2ρm2ρ +Q2〈ηB′ |τ (3)0 |ηB〉]〈χB′|∫d~pξ1 d~pξ2 Ψ∗B′(~pξ1 , ~pξ2 +√23~q )Ψ∗V (−√23~pξ2 +~PB3− ~q2)[(~p3 − ~q )~ǫ ∗ + i~ǫ ∗(~σ(3) × (~p3 − ~q ))]ΨB(~pξ1 , ~pξ2)|χB〉 (4)where mA is an average ρ and ω mass, ωγ (~q ) is the photon energy (trimomen-tum), η stands for the isospin wave function, χ for the spin one, (~ξ1, ~ξ2) arethe Jacobi coordinates and (~pξ1 , ~pξ2) its associated momenta operators. Thepolarization of the vector meson has been denoted by ~ǫ. ~σ’s (~τ ’s) as usual de-note spin (isospin) matrices. The superindex in the spin and isospin operatorsindicates the quark on which they act.All the dependence on a specific quark model for the baryon is containedin the baryon wave functions ΨB, ΨB′ . For ΨV we shall take a gaussian form[18]:3ΨV (~q ) =2R3/2Aπ1/4exp(−R2A2~q 2)Y00(qˆ) (5)The value chosen for RA is 2.82 GeV−1 [16], quite close to the one obtainedfrom the study of the leptonic decay of ρ0 in the quark model. If we assumethe SU(3) constraint, fω = 3fρ, the only free parameter is the ratioγfρ.1.1 Nucleon form factors.As a first application of expression (4) it is interesting to analyze the formfactors of the nucleon for which we use the Breit system of reference. Toget the magnetic form factor we take a transverse polarization vector ~ǫ± =∓ 1√2(1,±i, 0) and compare expression (4) with the corresponding matrix ele-ment in terms of the form factors of the nucleon. Thus we get:Gp,nM (Q2) = −3γ(mA)1/2(− 1√4π12√23)(2MN(2π)3/2)[1fωm2ωm2ω +Q2〈ηB′(12,±12)|ηB(12,±12)〉+1fρm2ρm2ρ +Q2〈ηB′(12,±12)|τ (3)0 |ηB(12,±12)〉]〈χB′(12,−12)|∫d~pξ1 d~pξ2 Ψ∗B′(~pξ1 , ~pξ2 +√23~q )Ψ∗V (−√23~pξ2 −~q3)1q[−qσ(3)− + p3zσ(3)− − p3−σ(3)z − p3−]ΨB(~pξ1, ~pξ2)|χB(12,12)〉(6)MN being the nucleon mass.The electric form factor is given in terms of the time component of thecurrent (GE(Q2) = 〈j0〉) that can be related to the spatial one j3 by gaugeinvariance (j0 =qωj3, ~q ≡ (0, 0, q)). Then:Gp,nE (Q2) = −3γ(mA)1/2(− 1√4π12√23)(−√2(2π)3/2)[1mωfωm2ωm2ω +Q2〈ηB′(12,±12)|ηB(12,±12)〉+1mρfρm2ρm2ρ +Q2〈ηB′(12,±12)|τ (3)0 |ηB(12,±12)〉]4〈χB′ |∫d~pξ1 d~pξ2 Ψ∗B′(~pξ1, ~pξ2 +√23~q )Ψ∗V (−√23~pξ2 −~q3)q[p3z − q + σ(3)+ p3− − σ(3)− p3+]ΨB(~pξ1 , ~pξ2)|χB〉 (7)Symbolically 〈χB′ |σ(3)− |χB〉GE(Q2) ∝ Q2GM(Q2). Let us note that thefactor q multiplying the last bracket makes the electric form factor to vanishat Q2 = 0, what can be interpreted as an indication that the total charge ofthe mesonic cloud (ρ, ω) in the nucleon is zero. Therefore, in order to get thebaryon charge for GpE(Q2 = 0), VMD can not be the only mechanism for thephoton-baryon interaction.We propose instead an extended vector meson dominance model (EVMD)as schematically pictured in fig. 1. The first diagram on the r.h.s. correspondsto VMD as calculated above. The second diagram on the r.h.s. corresponds toa direct coupling to the baryon for which we shall use a modified elementaryemission model [19] where a convergent (p/E) instead of a (p/m) expansion hasbeen applied [20]. Quite similar assumptions underlie the so called two-phasemodels [21] where the electromagnetic properties of the nucleon come fromtwo sources, the vector meson cloud and the core of quarks respectively corre-sponding to the two diagrams we consider in fig. 1. Hence we can summarizethe EVMD scenario in the equations:Gp,nE (Q2) = [Gp,nE (Q2)]eem + [Gp,nE (Q2)]vmd (8)Gp,nM (Q2) = c[Gp,nM (Q2)]eem + [Gp,nM (Q2)]vmd (9)where the relative weight parameter of the two diagrams c plays the samerole as an anomalous magnetic moment of the quarks. Note that althoughthe value of the quark mass used is mq ≈ MN3 , due to the (p/E) expansionin the EEM, the magnetic moment is not saturated by [Gp,nM (Q2 = 0)]eem,the remaining contribution having to be provided by [Gp,nM (Q2 = 0)]vmd. Forthe electric form factor the relative weight is 1 due to the charge conservation(recall that [Gp,nE (Q2 = 0)]vmd = 0).To fix the two parameters γfρand c, we can use the magnetic moment µpand the square mean radius of the proton (〈r2p〉) for instance. One shouldbe aware however that important contributions to 〈r2p〉, as the one from theDarwin-Foldy term ≈ ( 23m2q), have not been considered. Actually, a value of〈r2p〉 between 0.5 and 0.6 fm2 seems to be a better choice to get a very goodfit of most electromagnetic transitions [22].In order to extract some results we shall make use of two potential modelsVI, VII, extensively detailed elsewhere [15]. For the sake of completeness weonly quote here their expressions (values of the parameters as in ref. [15]):5VI =∑i<j12[rija2− κrij+κmimjexp(−rij/r0)r20rij~σi~σj −D](10)VII =∑i<j12[rija2− κrij+κ6mimjexp(−rij/r0)r20rij~σi~σj −D]+∑i6=j 6=k 6=i12V0mimjmke−m0rijm0rije−m0rikm0rik(11)VI contains the ’minimal’ one gluon exchange kind of potential and providesa reasonable average fit for the meson and baryon spectra (not fitting the Roperin the spectrum) [23] whereas VII, that incorporates a phenomenological three–quark interaction, fits very precisely the low lying baryon spectrum [12] andmost of its pionic strong decay widths, in particular the Roper resonanceN(1440) mass and decay widths [20].By fixing µp = 2.79 and 〈r2p〉 = 0.54 fm2 one gets c = 0.49(0.17) andγfρ= −0.38(−0.71) for VI (VII). Then for the neutron we get µn = −1.82(-1.84) and 〈r2n〉 = −0.003 (-0.002) for VI (VII). The separate contributionsfrom the two diagrams are shown in table 1.Let us emphasize that we do not do a completely consistent constructionof the transition operator according to the mass operator. We assume that thewave functions we obtain by solving the Schro¨dinger equation for VI of VII rep-resent, through the effective parameters and interactions of the potential a sortof averaged (p/E) quark core wave function. One should realize however thatthe corrections introduced by the (p/E) expansion in the transition operatorget correlated to the model wave functions through the weight parameter.It is ilustrative to draw the magnetic form factor. Looking at fig. 2 itis clear that the data lie between the predictions of the two models. Thisemphasizes the major role of the form of the transition operator (as it was thecase in the strong decays of the Roper where the qq¯ structure of the pion wasessential) leaving a fine tunning effect to the specific quark potential modelemployed. In particular, the 3P0 structure is mostly responsible to get thetendency of experimental data.1.2 Roper transition amplitudes.The Roper transition amplitudes calculated in the EVMD model are shownin figs. 3.a and 3.b. Though a clarification of experimental data seems to benecessary, it is worth to emphasize that we reproduce the experimental sign ofthe transverse amplitude at Q2 = 0 (contrary to what happens with an EEMcalculation) for what the VMD as provided by the 3P0 model becomes essential6(see table 1). The smallness of the EEM contribution is related to the use ofa (p/E) expansion instead of a (p/m) one, combined with the presence of anode in the Roper wave function (note the higher sensitivity of this processas compared to the magnetic nucleon form factor case though the expressionfor the amplitude is similar). Furthermore, we predict a change of sign atQ2 ≈ 0.5−1 GeV2, a feature suggested by the present data. The behaviour ofour amplitude resembles the one obtained with a light-front formalism [4, 5],though we do not have a significant configuration mixing as it was the case inref. [4].In order to have a deeper understanding of the nature of the Roper atest based on the calculation of the longitudinal amplitudes was proposed[11]. In figs. 3.c and 3.d we show our results for S1/2. The interpretation ofexperimental points [26] in terms of the longitudinal amplitude is controversial(see for instance refs. [4] and [11]) about the sign of this amplitude. Thenno definitive conclusion shoud be extracted before more confident data areavailable and agreement about its interpretation is reached. Meantime wemay conclude that a 3q structure of the Roper cannot be discarded once thetransition operator incorporates the physically relevant ingredients.The resulting picture of the Roper is that of a small quark core dressedby a mesonic cloud of total charge zero. The contribution of the cloud tothe baryon masses can be taken into account quite approximately, throughthe values of the effective parameters or the potential, in a 3q state. Forthe transitions, the cloud contribution has to be considered explicitely (as anexplicit |qqq qq¯〉 component or in the transition operator as we have done).No exotic explanation of the Roper nature seems to be needed, reinforcing thesimilar conclusions obtained in different frameworks [4, 5].The analysis can be extended to other resonances [22]. In particular, forthe ∆(1600), the Roper of the ∆, the Q2 behaviour of the amplitude is depictedin fig. 4, showing a much more pronounced quark model dependence. In ourmodel it also corresponds to a radial excitation so the comparison to datawhen available may be a more stringent test about our conclusions.We are grateful to S. Singh and S. Noguera for their suggestions and com-ments and to G. Salme` for his immediate answer and clarification to our ques-tions on the subject. This work has been partially supported by DGES undergrants PB95-1096, PB94-0080 and by EC-TMR network HaPHEEP undercontract ERBTMRX CT96-0008.References[1] H.P. Morsch et al., Phys. Rev. Lett. 69 (1992) 1336.7[2] S. Capstick, Phys. Rev. D46 (1992) 1965; Phys. Rev. D46 (1992) 2864.[3] F.E. Close and Z. Li, Phys. Rev. D42 (1990) 2194. Z. Li and F.E. ClosePhys. Rev. D42 (1990) 2207.[4] F. Cardarelli, E. Pace, G. Salme` and S. Simula, Phys. Lett. B397 (1997)13.[5] S. Capstick and B.D. Keister, Phys. Rev. D51 (1995) 3598.[6] G.E. Brown, J.W. Durso and M.B. Johnson, Nucl. Phys. A39 (1983) 447.[7] P.A.M. Guichon, Phys. Lett. B164 (1985) 361.[8] T. Barnes and F.E. Close, Phys. Lett. B123 (1983) 89.[9] F.E. Close, Rep. Prog. Phys. 51 (1988) 833. F.E. Close, Nucl. Phys. A497(1989) 109c.[10] C.E. Carlson and N.C. Mukhopadhyay, Phys. Rev. Lett. 67 (1991) 3745.[11] Z. Li, V. Burkert and Z. Li, Phys. Rev. D46 (1992) 70.[12] B. Desplanques, C. Gignoux, B. Silvestre–Brac, P. Gonza´lez, J. Navarroand S. Noguera, Z. Phys. A343 (1992) 331. A. Valcarce, F. Ferna´ndez, P.Gonza´lez and V. Vento, Phys. Lett. B367 (1996) 35.[13] Y.B. Dong, J.C. Su and S.S. Wu, J. Phys. G20 (1994) 73.[14] L.Ya. Glozman and D.O. Riska, Phys. Rep. 268 (1996) 263.[15] F. Cano, P. Gonza´lez, S. Noguera and B. Desplanques, Nucl. Phys. A603(1996) 257.[16] M.B. Gavela, A. Le Yaouanc, L. Oliver, O. Pe`ne, J.-C. Raynal and S.Sood, Phys. Rev. D21 (1980) 182.[17] J.J. Sakurai, Currents and Mesons (The University of Chicago Press,1969).[18] A. Le Yaouanc, L. Oliver, O. Pe`ne and J.-C. Raynal, Phys. Rev. D8 (1973)2223; Phys. Rev. D11 (1975) 1272.[19] A. Le Yaouanc, Ll. Oliver, O. Pe`ne and J.-C. Raynal, Hadron Transitionsin the Quark Model (Gordon and Breach Science Publishers, 1988).[20] F. Cano, P. Gonza´lez, B. Desplanques and S. Noguera, Z. Phys. A359(1997) 315.8[21] G.E. Brown, M. Rho and W. Weise, Nucl. Phys. A454 (1986) 669.[22] F. Cano, P. Gonza´lez, B. Desplanques and S. Noguera, work in prepara-tion.[23] B. Silvestre–Brac and C. Gignoux, Phys. Rev. D32 (1985) 743.[24] W. Bartel et al., Nucl. Phys. B58 (1973) 429. R.G. Arnold et al., Phys.Rev. Lett. 57 (1986) 174. P.E. Bosted et al., Phys. Rev. Lett. 68 (1992)3841.[25] Particle Data Group, Phys. Rev. D54 (1996) 1.[26] Ch. Gerhardt, Z. Phys C4 (1980) 311.9Table captionTable 1 Charge radii (in fm2) and magnetic moments (in nuclear magnetons)for the nucleon in the two models VI and VII. The contribution from eachdiagram in fig. 2 is shown. The photoproduction amplitudes Ap,n1/2, Sp,n1/2(in units of 10−3 GeV−1/2) correspond to the process N(1440) → Nγ.The EEM contribution includes the relative weight c.Figure captionsFigure 1 Extended vector meson dominance picture.Figure 2. Magnetic form factor for the proton calculated in the EVMDmodel for the two potentials VI (solid line), VII (dashed line). For exp.data see for instance [24]Figure 3. Transverse (a,b) and longitudinal (c,d) electroproduction ampli-tudes for N(1440). The solid line corresponds to VI and the dashed lineto VII. Experimental points at Q2 = 0 in a) and b) are taken from [25].For other experimental points see [11].Figure 4. Transverse electroproduction amplitude for the process∆+(1600)→ pγ. Notation as in fig. 3.10VI VIIEEM VMD Total EEM VMD Total Exp.〈r2p〉 0.238 0.302 0.54 0.133 0.407 0.54 0.74± 0.02〈r2n〉 −0.02 −0.012 −0.032 −0.005 −0.01 −0.015 −0.119± 0.004µp 0.93 1.86 2.79 0.27 2.52 2.79 2.79µn −0.61 −1.21 −1.82 −0.18 −1.66 −1.84 −1.91Ap1/2 0.63 −82.1 −81.5 −1.9 −164 −166 −65± 4An1/2 −.26 52.3 52.0 1.3 108 110 40± 10−Sp1/2 −16.5 −30.3 −46.8 −4.7 −61.4 −66.1 –−Sn1/2 2.1 3.7 5.9 0.25 2.6 2.8 –Table 111VB’ Bγ γBγVΣ +B’B’=Bfig. 1fig. 212fig. 3a fig. 3bfig. 3c fig. 3d13fig. 414

A consistent explanation of the Roper phenomenology

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Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura