We present an extraction of the E2/M1 ratio of the Δ(1232) from experimental data applying an effective Lagrangian model. We compare the result obtained with different nucleonic models and we reconcile the experimental results with the lattice QCD calculations.Gobierno de España UAC2002-0009Ministerio de Educación y Ciencia FIS2005-00640 BFM2003-04147-C02-0

Fernández Ramírez, C.

Moya de Guerra, E.

Udías, J. M.

English

idUS. Depósito de Investigación Universidad de Sevilla

arXiv:nucl-th/0611062v1  16 Nov 2006EPJ manuscript No.(will be inserted by the editor)Analysis of the quadrupole deformation of ∆(1232) within aneffective Lagrangian model for pion photoproduction from thenucleonC. Fernández-Ramı́rez1,2 a, E. Moya de Guerra1,3, and J.M. Ud́ıas31 Instituto de Estructura de la Materia, CSIC, Serrano 123, E-28006 Madrid, Spain2 Departamento de F́ısica Atómica, Molecular y Nuclear, Universidad de Sevilla, Apdo. 1065, E-41080 Sevilla, Spain3 Departamento de F́ısica Atómica, Molecular y Nuclear, Facultad de Ciencias F́ısicas, Universidad Complutense de Madrid,Avda. Complutense s/n, E-28040 Madrid, SpainReceived: date / Revised version: dateAbstract. We present an extraction of the E2/M1 ratio of the ∆(1232) from experimental data applyingan effective Lagrangian model. We compare the result obtained with different nucleonic models and wereconcile the experimental results with the Lattice QCD calculations.PACS. 14.20.Gk Baryon resonances with S = 0 – 25.20.Lj Photoproduction reactions – 13.60.Le MesonproductionThe deformation of the nucleon and its first excita-tion, the ∆(1232), is a topic that has focused the atten-tion of many researchers in the last years from both theexperimental and the theoretical sides [1]. The possibilityof such deformation has been studied using the E2/M1ratio (EMR) of the γN → ∆(1232) transition [2]. Theemission (absorption) of a photon by a spin-3/2 particleinvolves a magnetic dipole (M1) multipolarity and an elec-tric quadrupole (E2) multipolarity. From experiments it isfound that E2 is small but not zero, which evokes a de-formed nucleon picture. A deviation from zero of the EMRis a clear indication of the existence of such deformationand allows to quantify it. This ratio is mainly obtained intwo different ways, from nucleonic models such as quarkmodels or Lattice QCD, and from experimental data. Rec-onciliation of both extractions is significant in order to un-derstand the structure of the nucleon. This work is con-cerned with the extraction of the intrinsic E2/M1 ratioof the ∆(1232) from experimental data using a reactionmodel.In [3,4] we have developed a pion photoproductionmodel up to 1 GeV of photon energy based upon effectiveLagrangians. The model follows closely the work of Gar-cilazo and Moya de Guerra [5] and has similarities withthe work of Sato and Lee [6] as well as with other works[7,8,9] based on the seminal work of Peccei [10]. The re-action model allows us to isolate the contribution of the∆(1232) – taking into account the high energy behaviourof the tail of the resonance –, to calculate its EMR, andto compare it with the values provided by nucleonic mod-a Corresponding author; e-mail: cesar@nuc2.fis.ucm.esels. In addition to Born terms (those which involve onlyphotons, nucleons, and pions) and vector meson exchangeterms (ρ and ω exchanges), the model includes all thefour star resonances in Particle Data Group (PDG) [19]up to 1.7 GeV mass and up to spin-3/2:∆(1232), N(1440),N(1520), ∆(1620), N(1650), and ∆(1700).The model displays chiral symmetry, gauge invariance,and crossing symmetry, as well as a consistent treatmentof the interaction with spin-3/2 particles that avoids well-known pathologies present in previous models [3,4]. Thedressing of the resonances is considered by means of a phe-nomenological width which takes into account decays intoone π, one η, and two π. The width fulfills crossing sym-metry and contributes to both direct and crossed channelsof the resonances.We assume that the final state interactions (FSI) fac-torize (πN rescattering) and can be included through thedistortion of the πN final state wave function. The calcula-tion of the distortion requires one to calculate higher orderpion loops or to develop a phenomenological potential FSImodel. Both approaches are far of the one we apply. Thefirst one is overwhelmingly complex and the second wouldintroduce additional model-dependencies, which are to beavoided in the present analysis, because we are mainlyinterested in the bare properties of the resonances. Werather include FSI in a phenomenological way by addinga phase δFSI to the electromagnetic multipoles. We de-termine this phase so that the total phase of the electro-magnetic multipole is identical to the one of the energydependent solution of SAID [11]. In this way we are ableto isolate the electromagnetic vertex and remove the FSIeffects.2 C. Fernández-Ramı́rez et al.: Analysis of the quadrupole deformation of ∆(1232). . .-20-1001020304050600.2 0.3 0.4 0.5 0.6M1+3/2   (mF)Eγ (GeV)Real partImaginary part -4-20240.2 0.3 0.4 0.5 0.6E1+3/2   (mF)Eγ (GeV)Fig. 1. M3/21+ and E3/21+ electromagnetic multipoles. Curve conventions: solid, real part of the multipole; dashed, imaginary partof the multipole. Data taken from [11].In order to assess the parameters of the model we hadto minimize the function χ2 defined byχ2 =m∑j=1[Mexpj −Mthj (λ1, . . . , λn)∆Mexpj]2, (1)where Mexp stands for the current energy independentextraction of the multipole analysis of SAID up to 1 GeVfor E0+, M1−, E1+, M1+, E2−, and M2− multipoles inthe three isospin channels I = 32 , p, n for the γp → π0pprocess [11]. ∆Mexp is the error and Mth is the multi-pole given by the model which depends on the parametersλ1, . . . , λn, which stand for the electromagnetic couplingconstans of the resonances and the cutoff Λ which regu-larizes the high energy behaviour of the Born terms. Themasses and the widths of the resonances have been takenfrom the multichannel analysis of Vrana, Dytman, andLee [12] and from a speed plot calculation [4]. The EMRwe present has been obtained as an average of the resultsobtained with both sets of masses and widths. Electro-magnetic multipoles are complex quantities and we havetaken into account 763 data for the real part of the multi-poles and the same amount for the imaginary part. Thus,m = 1526 data points have been used in the fits.In order to fit the data and determine the best pa-rameters of the resonances we have written a genetic al-gorithm combined with the E04FCF routine from NAGlibraries [13]. Although genetic algorithms are computa-tionally more expensive than other algorithms, in a min-imisation problem it is much less likely for them to getstuck at local minima than for other methods, namelygradient based minimisation methods. Thus, in a mul-tiparameter minimisation like the one we face here it isprobably the best possibility to search for the minimum[3,14]. In Fig. 1 we show the fits to M3/21+ and E3/21+ elec-tromagnetic multipoles.Different definitions of the EMR have been employedin the literature. We should distinguish between the in-trinsic or bare EMR of the ∆(1232) and the directly mea-sured value in experiments which is often called physi-cal or dressed EMR value [15,8,6]. The physical EMRis obtained as the ratio between the imaginary parts ofE3/21+ and M3/21+ electromagnetic multipoles at the invari-ant mass (photon energy) at which Re[M3/21+]= 0 =Re[E3/21+]. Since all the reaction models are fitted to theexperimental electromagnetic multipoles, they generallyreproduce the physical EMR value within the error bars(see table 1). We obtainEMRphysical =Im[E3/21+]Im[M3/21+] × 100% = (−3.9± 1.1)%. (2)However, this measured EMR value is not directly avail-able from theoretical models of the nucleon and its reso-nances. Instead, if we want to compare to models of nu-cleonic structure, it is necessary to extract the bare EMRvalue of ∆(1232) which is defined asEMRbare =G∆(1232)EG∆(1232)M× 100% = (−1.30± 0.52)%, (3)The EMR defined in this way depends only on theintrinsic characteristics of the ∆(1232) and can thus becompared directly to predictions from nucleonic models. Itis not, however, directly measurable but must be inferred(in a model dependent way) from reaction models.The intrinsic quadrupole deformation of the ∆(1232)is found to be EMR= (−1.30± 0.52)%, indicative of asmall oblate deformation. In Tables 1 and 2 we compareour EMR values (bare and physical) to the ones extractedby other authors using other models for pion photopro-duction, as well as to predictions of nucleonic models.C. Fernández-Ramı́rez et al.: Analysis of the quadrupole deformation of ∆(1232). . . 3Table 1. Comparison of EMRphysical values from experimentscompared to the values obtained with reaction models.EMRphysical Ref.ExperimentsLEGS Collaboration (−3.07± 0.26 ± 0.24)% [16]A1 Collaboration (−2.28± 0.29 ± 0.20)% [17]A2 Collaboration (−2.74± 0.03 ± 0.30)% [18]Particle Data Group (−2.5± 0.5)% [19]Reaction modelsFernández-Ramı́rez et al. (−3.9± 1.1)% [15]Pascalutsa and Tjon (−2.4± 0.1)% [8]Sato and Lee −2.7% [6]Fuda and Alharbi −2.09% [20]Table 2. Comparison of EMRbare values extracted from ex-periments through reaction models compared to the values ob-tained with nucleonic models.EMRbare Ref.Reaction modelsFernández-Ramı́rez et al. (−1.30± 0.52)% [15]Pascalutsa and Tjon (3.8± 1.6)% [8]Sato and Lee −1.3% [6]Davidson et al. −1.45% [7]Garcilazo and Moya de Guerra −1.42% [5]Vanderhaeghen et al. −1.43% [9]Nucleonic modelsNon-relativistic quark model 0% [21]Constituent quark model −3.5% [22]Skyrme model (−3.5± 1.5)% [23]Lattice QCD (Leinweber et al.) (3± 8)% [24]Lattice QCD (Alexandrou et al.) [25](Q2 = 0.1 GeV2, mπ = 0) (−1.93± 0.94)%(Q2 = 0.1 GeV2, mπ = 370 MeV) (−1.40± 0.60)%Reconciliation of the experimental value of the E2/M1γN → ∆(1232) transition ratio (EMRphysical) with theone obtained using Lattice QCD (EMRbare ) within a con-sistent and sound framework (besides our analysis only in[8] a consistent treatment of the spin-3/2 interaction isperformed). is one of the goals of this work [15]. Our re-sults also indicate that quark models need improvementsin order to reproduce the value obtained from experiment.C.F.-R. work has been developed under Spanish Governmentgrant UAC2002-0009. This work has been supported in partunder contracts of Ministerio de Educación y Ciencia (Spain)FIS2005-00640 and BFM2003-04147-C02-01.References1. B. Krusche and S. Schadmand, Prog. Part. Nucl. Phys. 51(2003) 399.2. A.M. Bernstein, Eur. Phys. J. A 17 (2003) 349.3. C. Fernández-Ramı́rez, Electromagnetic production of lightmesons. PhD dissertation. Universidad Complutense deMadrid (2006).4. C. Fernández-Ramı́rez, E. Moya de Guerra, and J.M. Ud́ıas,Ann. Phys. (N.Y.) 321 (2006) 1408.5. H. Garcilazo and E. Moya de Guerra, Nucl. Phys. A 562(1993) 521.6. T. Sato and T.-S.H. Lee, Phys. Rev. C 63 (2001) 0552017. R.M. Davidson, N.C. Mukhopadhyay, and R.S. Wittman,(1991) Phys. Rev. D 43 (1991) 71.8. V. Pascalutsa and J.A. Tjon, Phys. Rev. C 70 (2004)035209.9. M. Vanderhaeghen, K. Heyde, J. Ryckebusch, and M. Waro-quier, Nucl. Phys. A 595 (1995) 219.10. R.D. Peccei, Phys. Rev. 181, 1902 (1969).11. R.A. Arndt, W.J. Briscoe, I.I. Strakovsky, and R.L. Work-man, Phys. Rev. C 66 (2002) 055213, SAID database,http://gwdac.phys.gwu.edu12. T.P. Vrana, S.A. Dytman, and T.-S.H. Lee, Phys. Rep.328 (2000) 181.13. Numerical Algorithms Group Ltd., WilkinsonHouse, Jordan Hill Road, Oxford OX2-8DR, UK.http://www.nag.co.uk/14. D.G. Ireland, S. Janssen, and J. Ryckebusch, Nucl. Phys.A 740 (2004) 147.15. C. Fernández-Ramı́rez, E. Moya de Guerra, and J.M.Ud́ıas, Phys. Rev. C 73 (2006) 042201(R).16. G. Blanpied et al., Phys. Rev. C 64 (2001) 025203.17. S. Stave et al., arXiv:nucl-ex/060401318. J. Ahrens et al., Eur. Phys. J. A 21 (2004) 323.19. W.-M. Yao et al., J. Phys. G 33 (2006) 1,http://pdg.lbl.gov20. M.G. Fuda and H. Alharbi, Phys. Rev. C 68 (2003) 064002.21. C.M. Becchi and G. Morpurgo, Phys. Lett. 17 (1965) 352.22. A.J. Buchmann, E. Hernández, and A. Faessler, Phys. Rev.C 55 (1997) 448.23. A. Wirzba and W. Weise, Phys. Lett. B 188 (1987) 6.24. D.B. Leinweber, T. Draper, and R.M. Woloshyn, Phys.Rev. D 48 (1993) 2230.25. C. Alexandrou, Ph. de Forcrand, H. Neff, J.W. Negele,W. Schroers, and A. Tsapalis, Phys. Rev. Lett. 94 (2005)021601.

Analysis of the quadrupole deformation of Δ(1232) within an effective Lagrangian model for pion photoproduction from the nucleon

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Analysis of the quadrupole deformation of Δ(1232) within an effective Lagrangian model for pion photoproduction from the nucleon

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