Sara Parsaei and Ali Akbar Rajabi?

1Faculty of Physics,Shahrood University of Technology,P.O.Box 3619995161-316,Shahrood,Iran

2Shams Institute of Higher Education,Golestan Province,Gonbad,Iran

1 Introduction

The structure of the nucleon and its excited states has been one of the most extensively studied subjects in nuclear and particle physics.Various constituent quark models(CQMs)and the nonrelativistic constituent quark model have been proposed for the internal structure of baryons.The Constituent Quark Model(CQM)and the nonrelativistic constituent quark model,[1?3](NRCQM)which has existed for nearly 25 years,[4]have been extensively applied to the description of baryon properties[5?7]and most attention has been devoted to the spectrum.[8]Note that,a common characteristic is that,although the models use different ingredients,they are able to give a satisfactory description of the baryon spectrum and,in general,of the nucleon static properties.[9]The study of hadron spectroscopy is not sufficient to distinguish among the various models of quark,but the electromagnetic transition between the nucleon and excited baryons has been shown to be a very important probe to the structure of nucleon and baryon resonances and is an important test for the various models of quark. The electromagnetic transition allows us to understand important aspects of the underlying theory of the strong interaction,QCD,in the con finement regime where solutions are very difficult to obtain[10]also the excitation of nucleon resonances in electromagnetic interactions has long been recognized as a sensitive source of information on the long-and shortrange structure of the nucleon and its excited states in the domain of quark con finement.[11]An important advantage of electromagnetic experiments is the ability to extract the matrix elements forγN→N?,commonly called the photon coupling amplitudes that these amplitudes are primarily sensitive to the quark wave function used.[12]Elastic electron scattering experiments provide information on the ground state of the nucleon,while studying theQ2evolution of the transition amplitudes for the nucleon ground state in the excited states provides insight into the internal structure of the excited nucleon[10]therefore the question of how quarks contribute to radiative transitions between hadrons has been investigated for many years.The basic assumption is that a single quark absorbs the photon and leads to excitation of the system,where the photon is a clean probe in that it couples to the spin and flavor of the constituent quarks and reveals the correlation among the flavors and spin inside the target.[4]The total transverse amplitude and the virtual photon polarization depend on the invariant momentum transfer to the resonance(Q2),which theQ2behavior is more sensitive to the quark wavefunctions.All the information about the electromagnetic structure of the baryon is contained in structure functions and form factors that for a spin 1/2 resonance,there is one transverse amplitude(A1/2),one longitudinal amplitude(S1/2)and for a spin 3/2 resonance there is one transverse amplitude(A3/2)therefore the measurement of all three electromagnetic form factors(transverse,longitudinal and scalar)could provide stimulating tests of QCD-inspired models of baryon structure.[13]The CLAS detector at Jefferson Lab is a first large acceptance instrument designed for the comprehensive investigation of exclusive electroproduction of mesons with the goal to study the electroexcitation of nucleon resonances in detail.[5]In this paper,in order to perform a systematic study of amplitudes according to a hypercentral approach,we study the electromagnetic excitation of nucleon resonances in the inelastic scattering of high-energy electron beams and obtain the amplitudes.Since these amplitudes are depended to the quark wave function using the non-relativistic threebody quark model and the hyper-central potential(Cornell potential)to obtain the nucleon wave function and its resonances by the Nikiforov–Uvarov method.Our results compared with the experimental data[14?17]and calculation of light-cone distribution amplitudes.[18]The paper is organized as follows.In Sec.2.the electromagnetic transition form factors are evaluated.In Sec.3,we brie fly describe the non-relativistic three-body quark model.In Sec.4.the Nikiforov–Uvarov method described and wave functions obtained then in Subsec.4.2 the results are presented and compared with the light-cone model predictions in Ref.[18].In Sec.5,a summary of the discussion is presented.

2 The Electromagnetic Interaction and Calculation of Amplitudes

Most calculations of electromagnetic properties in the constituent quark model(CQM)have been performed in the so-called impulse approximation,which assumes that the total electromagnetic current of the quarks is given by a sum of free quark currents.[19]Evaluation of the strength of electromagnetic transitionsγN→X?between the nucleons(N)and excited baryon states(X)involves finding matrix elements of an EM transition Hamiltonian between the baryon states.[20]The EM interaction Hamiltonian can be found from a non-relativistic reduction of the electromagnetic field of a photon(A)with the quark current(J),where the electromagnetic field of a photon is de fined by

where?is a unit vector of polarization andkis the photon momentum.It is sufficient to consider photons with right-handed polarization(photons with helicity+1)andkalong thezaxis.The transverse coupling is obtained by interacting the radiation field of a right-handed photon with the quark current(J),considering(J)as follows:[9]

whereei,σare the charge operators of thei-th quark and the Pauli matrices.The basic assumption is that a single quark absorbs the photon and leads to excitation of the system.[21]Hence,the EM interaction Hamiltonian is given by

wherem,e,sandμ=eg/2m=0.13 GeV?1(g=2μm/e)denote the mass,charge,spin,and the proton magnetic moment of 3-th quark(k0,k),andpare the virtual photon four-momentum and the momenta of the quark.In Eq.(3),the first term is magnetic interaction with a quark and flips the quarks spin projection,and the second term is the interaction of the field with the quark convection current.[22]Amplitudes are de fined in terms of helicity states by

The concept of helicity is illustrated in Fig.1 forλn=?1/2 andλn=1/2[6]wherexis excited baryon states,N=p,nandλxis the final state helicity,λNthe initial state helicity,andλνis the virtual photon helicity.We have the helicityA3/2(Q2)forλn=?1/2 and the helicityA1/2(Q2)forλn=1/2.

Therefore,in general,there is a pair of amplitudesA1/2andA3/2associated with photoproduction from each target,which correspond to the two possibilities for aligning the spin of the photon and initial baryon in the center of momentum(c.m.)frame[20]which in order to calculate them we need to reduce the EM interaction Hamiltonian to an operator which acts between a ground state wave function nucleon and exicted state therefore,consider a right handed photon with momentumk=k?zand integrating over the center-of-mass coordinate.The EM interaction Hamiltonian can be shown as[5]

In addition to transverse couplings,one can also consider longitudinal and scalar couplings.The longitudinal coupling is obtained by inserting the radiation field for the absorption of a longitudinally polarized virtual photon in the EM interaction Hamiltonian as follows,

where electromagnetic transition operatorsHtandHlact both on the spin- flavor part and the space part of the baryon wave function.[5]This wave function is obtained in the non-relativistic three-body quark model in Sec.4 and the results are shown in Fig.3 and Fig.4.

Fig.1 Graphical representation of helicity amplitudes A.

Fig.2 The curve is the Nonrelativistic Constituent Quark Model calculation of the helicity amplitudesfor the electroproduction of the N(1535)resonance in our work and the dashed curve is the LCSR calculation for the helicity amplitudes for the electroproduction of the N(1535)resonance obtained using the central values of the lattice parameters.[18]

Fig.3 The curve is the Nonrelativistic Constituent Quark Model calculation of the helicity amplitudes for the electroproduction of the N(1535)resonance in our work and the dashed curve is the LCSR calculation for the helicity amplitudesfor the electroproduction of the N(1535)resonance obtained using the central values of the lattice parameters.[17]

3 The Non-relativistic Three-body Quark Model

The traditional theoretical approach is to describe the nucleon and its excitations using wave functions from the non-relativistic potential models,which describe baryons as being made up of “constituent” quarks moving in a con fining potential.[22]All the established baryons are apparently three-quark(qqq)states.The non-relativistic constituent quark model[2,6,21](NRCQM)has existed for nearly 25 years.[4]The Constituent Quark Model(CQM)has been extensively applied to the description of baryon properties[5?6,23]and most attention has been devoted to the spectrum.[4,24]Different versions of Constituent-Quark Models(CQM)have been proposed in the last decades in order to describe the baryon properties.What they have in common is the fact that the three-quark interaction can be separated into two parts:the first one,containing the con finement interaction,are spin and flavor independent and therefore SU(6)invariant,while the second violates the SU(6)symmetry.[24?26]The complete three-quark wave function can be factorized in four parts,that is the color,spin,flavor and space factors.The introduction of SU(6)con figurations for the combination of the three quarks is bene ficial.[9]In Appendix A of Ref.[9],there is the explicit form of the SU(6)-con figurations describing the various baryon states.The Constituent quark models(CQM)have been developed that relate electromagnetic resonance transition form factors to fundamental quantities,such as the quark con fining potential.[27]After removal of the center of mass coordinate,the internal quark motion is usually described by means of Jacobi coordinates,ρandλ.[9,28]

In order to describe three-quark dynamics it is convenient to introduce the hyperspherical coordinates the hyper-radiusxand the hyper-angleξde fined by[9]

The potentialV(x)is assumed to depend onxonly,that is to be hyper-central.[30?31]This potential is provided by the interquark “glue”,which is taken to be in its adiabatic ground state.The quarks interact at short distance via one-gluon exchange.[22]ψνγis the hyper-radial part of Schrodinger equation eigenfunction andL2(?ρ,?λ,ξ)is the quadratic Casimir operator and eigen functions are the hyperspherical harmonics,Y[γ],lρ,lλ(?ρ,?λ,ξ),

γis the grand angular quantum number given byγ=2n+lρ+lλ,n=0,1,...andlρ&lλbeing the angular momenta associated with theρandλvariables.[29]νdenotes the number of nodes of the three-quark wave function,where the potentialV(x)is assumed to depend onxonly,on the hyper central Constituent Quark Model it is assumed to be given by the hyper central potential.[30,32]mis the quark mass,considering a quark mass about one third of the nucleon mass,then we have performed a calculation of the wave functions based on the hyper central potential Eq.(14)and NU method.

4 The Nikiforov–Uvarov Method

In this section,N-U method is brie fly outlined and more detailed description of the method can be obtained in Refs.[32–33]. In this method the one-dimensional Schr¨odinger equation can be reduced to a generalized equation of hyper-geometric type:

whereσ(s)and?σ(s)are polynomials,at most,of seconddegree,and?τ(s)is a most first-degree polynomial.In order to find a particular solution for Eq.(14),we introduceψn(s)as follows:

The prime factors show the differentials at first-degree be negative,where

To find the value oft,the expression under the square root of Eq.(18)must be the square of the polynomial of degree at most one.This is possible only if its discriminant is zero,[34]whereλnis a constant de fined in the form

In addition,ψn(s)can be solved through the NU method.[34]

4.1 Wave Functions

Here,we want to solve the hyper-radial Schr¨odinger equation for the Cornell potential,the Schr¨odinger equation reduces to the form:

Now settingy=s?ξwhereξ=1/a,ais the length representing the surface thickness,using approximation similar to Pekeris.[35]Therefore,this changes variable,s=1/x,aroundy=0 can be expanded into a series of powers so we get

Equation(26)can be solved by the NU method for this purpose,we compare it with Eq.(16).We have(2?2)?2=(3?2)⊕(1?2)=4⊕2⊕2 that is,three spin particles 1/2 group together into a quartet states of spin 3/2 and two doublets states of spin 1/2.[36]Therefore,for the ordinary baryons,flavor and spin may be combined with an approximate flavor-spin SU(6)in which the six basic states are d↑,d↓,...,s↓(↑,↓=spin up,down)and thebaryonsbelongtothemultipletson the 6?6?6=56s⊕70M⊕70M⊕20Awhere 56=410⊕28,70=210⊕48⊕28⊕21,and 20=41⊕28.Accordingly the notation for the spin- flavor part of the baryons wave function used is|2s+1dim(SU(3))J,[dim(SU(6)),Lp],X>whereSis the total spin and the superscript(2S+1)gives the net spinSof the quarks for each particle in the SU(3)multiplet,dim(SU(n))is the dimension of the SU(n)representation,J,LandPare the spin,orbital angular momentum and parity of the resonance andXdenotes the type of baryon resonance[38]the introduction of SU(6)con figurations for the combination of the three quarks is bene ficial.[39]In Appendix A of Ref.[1]there is the explicit form of the SU(6)-con figurations describing the various baryon states.

4.2 The Helicity Amplitudes for γ?p→ S11(1535)and Results

In general the transverse helicity amplitudesAμcan be obtained from Eq.(35)

whereNnγis the normalization constant determined by arguing that∫∞0|ψnγ(x)|2dx=1 and L is the Laguerre polynomial function.The complete baryon states can be factorized in four parts,that is the color,spin,flavor and space factors and since the quarks are fermions,the state function for any baryon must be antisymmetric under interchange of any two equalmass quarks thus the state function can be written as|qqq>A=|color>A×|space,spin,flavor>swhere the subscriptsSandAindicate symmetry or antisymmetry under interchange of any two of the equal-mass quarks.[36]For the flavor and spin part of the baryons wave function,considered that the ordinary baryons are made up of the three flavorsu,d,andsquarks that the three flavors imply an approximate flavor SU(3),which requires the baryons made of these quarks belong to the multiplets on theqqq≡3?3?3=1A⊕8M⊕8M⊕10S,where the subscripts indicate antisymmetric,mixed-symmetry,or symmetric states under interchange of any two quarks[36?37]and for the spin part,the composite system of two spin 1/2 particles may have spinJ=1 or 0.By combining a third spin 1/2 particle imply an approximate SU(2),we have

whereiandfrepresent the spatial part of initial and final states of the baryon,where(μ=1/2,μ=3/2)denotes the helicity,the coefficients containαμ,βμthe contribution of the spin- flavor matrix element of Clebsch–Gordan coefficients.[5]HereAandBrepresent the orbit and spinflip spatial amplitudes(radial integrals),in the following form

whereMis the nucleon mass,Wis the mass of the resonance andQ2=k2?k20is the magnitude of the fourmomentum transfer,[31]Now,the matrix elements of the operatorUandTof Eqs.(36)to be evaluated are of the type:

which depend strongly on the details of the wave functions,the baryon statesiandf.ψnγ(x)Y[γ]lρlλ(?ρ,?λ,ξ)is the spatial part of the baryon wave function,in this work,this wave function obtained with the use of Nikiforov–Uvarov

method(Sec.4)andY[γ]lρlλ(?ρ,?λ,ξ)is the hyperspherical harmonics,de fined in the following form:

With the wave functionsψ00,ψ10obtained from Eq.(34)Finally,theA1/2andAlfor the electroexcitation of the S11(1535)from proton targets calculated and show in Figs.2 and 3 for the range 1≤Q2≤11.

5 Summary

We have calculated the helicity amplitudes for electromagnetic excitation of the negative parity resonance of the nucleon using the non relativistic constituent quark models.Since the helicity amplitudes depend strongly on the quark wave function,we employ the NU method to obtain the wave functions of the nucleon and the excited nucleon.From the analysis of our results,one sees that Cornell potential and NU method are able to give a reasonable description of the helicity amplitude data,especially at large values of the momentum transferQ2,that is 2–1(GeV2).These improvements in the reproduction of amplitudes obtained by using a suitable form of confinement potential and exact analytical solution of the ra-dial Schr¨odinger equation for our proposed potential(The Cornell potential).We have observed that the Cornell potential still has problems for low Q2-values that this can be an indication that further degrees of freedom,as pairs,should be considered in the CQM in a more explicit way.[9]is vanishing because the Spin- flavor coefficients ofHin transverse,Eq.(35),longitudinal,Eq.(45),helicity amplitudes for nucleon resonances(proton target coupling)is zero.

[1]F.E.Close,Introduction to Quarks and Partons,Academic Press,New York(1978).

[2]L.A.Copley,G.Karl,and E.Obryk,Nucl.Phys.B 13(1969)303.

[3]R.P.Feynman,M.Kislinger,and F.Ravndal,Phys.Rev.D 3(1971)2706.

[4]F.E.Close and Z.Li,Phys.Rev.D 42(1990)2194.

[5]R.Bijker,F.Iachello,and A.Leviatan,Ann.Phys.236(1994)69.

[6]M.Aiello,et al.,Phys.Lett.B 387(1996)215.

[7]M.Ferraris,M.M.Giannini,M.Pizzo,E.Santopinto,and L.Tiator,Phys.Lett.B 364(1995)231.

[8]M.M.Giannini,Rep.Prog.Phys.54(1991)453.

[9]M,Aiello,M.M.Gianniniy,and E.Santopinto,J.Phys.G:Nucl.Part.Phys.24(1998)753.

[10]I.G.Aznauryan and V.D.Burkert,Prog.Part.Nucl.Phys.1(2012)1.

[11]I.G.Aznauryan,et al.,Phys.Rev.C 78(2008)045209.

[12]The CLAS Collaboration,R.Thompson,et al.,Phys.Rev.Lett.86(2001)1702.

[13]J.J.Kelly,et al.,Phys.Rev.C 75(2007)025201.

[14]R.W.Gothe,V.I.Mokeev,et al.,“Nucleon Resonance Studies With CLAS12”,in JLab Experiment E12-09-003.

[15]H.Denizli,et al.,Phys.Rev.C 76(2007)015204.

[16]P.Stoler,Phys.Rep.226(1993)103.

[17]L.Tiator,D.Drechsel,S.Kamalov,and M.M.Giannini,Eur.Phys.J.A 19(2004)55.

[18]V.M.Braun,M.Gockeler,R.Horsley,T.Kaltenbrunner,et al.,Phys.Rev.Lett.103(2009)072001.

[19]A.J.Buchmann,E.Hernandez,and A.Faessler,Phys.Rev.C 55(1997)448.

[20]S.Capstick and W.Roberts,Prog.Part.Nucl.Phys.45(2000)241.

[21]L.A.Copley,G.Karl,and E.Obryk,Nucl.Phys.B 13(1969)303.

[22]S.Capstick and B.D.Keister,Phys.Rev.D 51(1995)3598.

[23]Z.Dziembowski,M.Fabre de la Ripelle,G.A.Miller,Phys.Rev.C 53(1996)2038.

[24]I.G.Aznauryan and V.D.Burkert,Phys.Rev.C 80(2009)055203.

[25]G.S.Bali,et al.,Phys.Rev.D 62(2000)054503.

[26]G.S.Bali,Phys.Rep.343(2001)1.

[27]C.Alexandrou,P.de Forcrand,and O.Jahn,Nucl.Phys.Proc.Suppl.119(2003)667.

[28]J.Ballot and M.F de la Ripelle,Ann.Phys.127(1980)62.

[29]M.Fabre de la Ripelle and J.Navarro,Ann.Phys.123(1979)185.

[30]A.M.Badalyan,Phys.Lett.B 199(1987)267.

[31]E.Santopinto,F.Iachello,and M.M.Giannini,Eur.Phys.J.A 1(1998)307.

[32]Sameer M.Ikhdair,Int.J.Mod.Phys.C 20(2009)1563.

[33]C.Berkdemir,A.Berkdemir,and R.Sever,Phys.Rev.C.74(2006)039902.

[34]S.M.Ikhdair and R.Sever,Int.J.Theor.Phys.46(2007)1643.

[35]C.L.Pekeris,Phys.Rev.45(1934)98.

[36]R.M.Barnett,et al.,Phys.Rev.D 54(1996)174.

[37]F.Halzen and A.D.Martin,Quarks and Leptons,John Wiley and Sons,New York(1984).

[38]M.M.Giannini,E.Santopinto,and A.Vassallo,Eur.Phys.J.A 25(2005)241.

[39]N.Salehi and A.A.Rajabi,Mod.Phys.Lett.A 24(2009)2631.