Electromagnetic Coupling of Negative Parity Nucleon Resonances N (1535) Based on Nonrelativistic Constituent Quark Model
Parsaei Sara1, *, Akbar Rajabi Ali1, 2, †
Faculty of Physics, Shahrood University of Technology, P.O. Box 3619995161-316, Shahrood, Iran
Shams Institute of Higher Education, Golestan Province, Gonbad, Iran

 

† Corresponding author. E-mail: sarahparsaei@shahroodut.ac.ir a.a.rajabi@shahroodut.ac.ir

Abstract
Abstract

The electromagnetic transition between the nucleon and excited baryons has long been recognized as an important source of information for understanding strong interactions in the domain of quark confinement. We study the electromagnetic properties of the excitation of the negative parity the N*(1535) resonances in the nonrelativistic constituent quark model at large momentum transfers and have performed a calculation the longitudinal and transverse helicity amplitudes. Since the helicity amplitudes depend strongly on the quark wave function in this paper, we consider the baryon as a simple, non-relativistically three-body quark model and also consider a hypercentral potential scheme for the internal baryon structure, which makes three-body forces among three quarks. Since the hyper central potential depends only on the hyper radius, therefore, the Cornell potential which is a combination of the Coulombic-like term plus a linear confining term is considered as the potential for interaction between quarks. In our work, in solving the Schrodinger equation with the Cornell potential, the Nikiforov–Uvarov method employed, and the analytic eigen-energies and eigen-functions obtained. By using the obtained eigen-functions, the transition amplitudes calculated. We show that our results in the range lead to an overall better agreement with the experimental data in comparison with the other three non-relativistic quark models.

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,[13] (NRCQM) which has existed for nearly 25 years,[4] have been extensively applied to the description of baryon properties[57] 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 confinement 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 short-range structure of the nucleon and its excited states in the domain of quark confinement.[11] An important advantage of electromagnetic experiments is the ability to extract the matrix elements for , 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 the evolution 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 (), which the behavior 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 (), one longitudinal amplitude () and for a spin 3/2 resonance there is one transverse amplitude () 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 three-body 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[1417] 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 briefly 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 transitions 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 defined by

where is a unit vector of polarization and k is the photon momentum. It is sufficient to consider photons with right-handed polarization (photons with helicity +1) and k along the z axis. 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]

where , are the charge operators of the i-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

where m, e, s and () denote the mass, charge, spin, and the proton magnetic moment of 3-th quark , and p are 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 defined in terms of helicity states by

The concept of helicity is illustrated in Fig. 1 for and [6] where x is excited baryon states, , n and is the final state helicity, the initial state helicity, and is the virtual photon helicity. We have the helicity for and the helicity for .

Fig. 1 Graphical representation of helicity amplitudes A.

Therefore, in general, there is a pair of amplitudes and associated 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 momentum and integrating over the center-of-mass coordinate. The EM interaction Hamiltonian can be shown as[5]

The operators and act only on the spatial part of the baryon wave function and the transverse couplings can be obtained from the following formula:

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 the operator is the operator in Eq. (5) (with m = 0). Therefore the longitudinal helicity amplitudes, is calculated by

where electromagnetic transition operators and act 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.

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 confining 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[56,23] 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 confinement interaction, are spin and flavor independent and therefore SU(6) invariant, while the second violates the SU(6) symmetry.[2426] 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) configurations for the combination of the three quarks is beneficial.[9] In Appendix A of Ref. [9], there is the explicit form of the SU(6)-configurations 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 confining potential.[27] After removal of the center of mass coordinate, the internal quark motion is usually described by means of Jacobi coordinates, ρ and λ.[9,28]

In order to describe three-quark dynamics it is convenient to introduce the hyperspherical coordinates the hyper-radius x and the hyper-angle defined by[9]

In this way one can use the hyperspherical harmonic formalism.[29] The hyper central Schrodinger equation for the 3q-state with a potential can be written as:

where

The potential is assumed to depend on x only, that is to be hyper-central.[3031] 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] is the hyper-radial part of Schrodinger equation eigenfunction and is the quadratic Casimir operator and eigen functions are the hyperspherical harmonics, ,

is the grand angular quantum number given by , and & being the angular momenta associated with the and variables.[29] denotes the number of nodes of the three-quark wave function, where the potential is assumed to depend on x only, on the hyper central Constituent Quark Model it is assumed to be given by the hyper central potential.[30,32] m is 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 briefly outlined and more detailed description of the method can be obtained in Refs. [3233]. In this method the one-dimensional Schrödinger equation can be reduced to a generalized equation of hyper-geometric type:

where and are polynomials, at most, of second-degree, and is a most first-degree polynomial. In order to find a particular solution for Eq. (14), we introduce as follows:

where Eq. (14) reduces into an equation of a hyper-geometric type as:

where

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

And

To find the value of t, 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] where is a constant defined in the form

Furthermore, the other part of the wave function Eq. (15) can be expressed in terms of Rodrigues relation

where are the normalization constant and the weight function is the solution of the differential below equation

In addition, can be solved through the NU method.[34]

4.1 Wave Functions

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

We solve the Eq. (23) exactly. For this purpose, we suggest this change variable , so we have

Now setting where , a is the length representing the surface thickness, using approximation similar to Pekeris.[35] Therefore, this changes variable, , around y = 0 can be expanded into a series of powers so we get

And so Schrödinger equation turns into

where

Equation (26) can be solved by the NU method for this purpose, we compare it with Eq. (16). We have

Therefore the parameters are found as

And

We solve and find , the energy eigenvalues as follows:

where

Unknown coefficients are obtained by fitting the observed baryon spectrum. Let us now find the corresponding eigen functions for this model. Using Eqs. (21) and (22), we find

And further, substituting Eqs. (33) into Eq. (15), by setting x = 1/s we get

where is the normalization constant determined by arguing that 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 equal-mass quarks thus the state function can be written as where the subscripts S and A indicate 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 flavors u, d, and s quarks that the three flavors imply an approximate flavor SU(3), which requires the baryons made of these quarks belong to the multiplets on the , where the subscripts indicate antisymmetric, mixed-symmetry, or symmetric states under interchange of any two quarks[3637] and for the spin part, the composite system of two spin 1/2 particles may have spin J = 1 or 0. By combining a third spin 1/2 particle imply an approximate SU(2), we have 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 (, ) and the baryons belong to the multiplets on the where , , and . Accordingly the notation for the spin-flavor part of the baryons wave function used is where S is the total spin and the superscript (2S + 1) gives the net spin S of the quarks for each particle in the SU(3) multiplet, dim (SU(n)) is the dimension of the SU(n) representation, J, L and P are the spin, orbital angular momentum and parity of the resonance and X denotes the type of baryon resonance[38] the introduction of SU(6) configurations for the combination of the three quarks is beneficial.[39] In Appendix A of Ref. [1] there is the explicit form of the SU(6)-configurations describing the various baryon states.

4.2 The Helicity Amplitudes for (1535) and Results

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

where i and f represent the spatial part of initial and final states of the baryon, where denotes the helicity, the coefficients contain the contribution of the spin-flavor matrix element of Clebsch–Gordan coefficients.[5] Here A and B represent the orbit and spin-flip spatial amplitudes (radial integrals), in the following form

We evaluate the amplitudes in the Breit frame, using the relation

where M is the nucleon mass, W is the mass of the resonance and is the magnitude of the four-momentum transfer,[31] Now, the matrix elements of the operator U and T of Eqs. (36) to be evaluated are of the type:

where

which depend strongly on the details of the wave functions, the baryon states i and f. 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) and is the hyperspherical harmonics, defined in the following form:

where is the Jacobi polynomial function. We use the notation and define the amplitudes for as below:

Therefore, for the , the matrix elements A1/2 of the operator (4) between nucleon states with λ = −1/2 and resonance states S11 (1535) with λ = 1/2 are given by

where and we have

According Eq. (42) and Eq. (43) we can obtain

And the longitudinal helicity amplitudes get

where is Clebsch–Gordan coefficients that contain the contribution of the spin-flavor matrix element. According to Eq. (43) and Eq. (45) we have

At the end, the matrix elements of Eq. (44) and Eq. (46) can be evaluated in terms of two basic integrals:

The Laplace transform of the ordinary Bessel functions

For example, in Eq. (46) we have

With the wave functions , obtained from Eq. (34) Finally, the and for the electroexcitation of the S11(1535) from proton targets calculated and show in Figs. 2 and 3 for the range .

Fig. 2 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 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 amplitudes for the electroproduction of the N (1535) resonance obtained using the central values of the lattice parameters.[17]
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 transfer Q2, 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 radial Schrödinger 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 of H in transverse, Eq. (35), longitudinal, Eq. (45), helicity amplitudes for nucleon resonances (proton target coupling) is zero.

Reference
[1] Close F. E. Introduction to Quarks and Partons Academic Press New York 1978
[2] Copley L. A. Karl G. Obryk E. Nucl. Phys. B 13 1969 303
[3] Feynman R. P. Kislinger M. Ravndal F. Phys. Rev. D 3 1971 2706
[4] Close F. E. Li Z. Phys. Rev. D 42 1990 2194
[5] Bijker R. Iachello F. Leviatan A. Ann. Phys. 236 1994 69
[6] Aiello M. et al. Phys. Lett. B 387 1996 215
[7] Ferraris M. Giannini M. M. Pizzo M. Santopinto E. Tiator L. Phys. Lett. B 364 1995 231
[8] Giannini M. M. Rep. Prog. Phys. 54 1991 453
[9] Aiello M. Gianniniy M. M. Santopinto E. J. Phys. G: Nucl. Part. Phys. 24 1998 753
[10] Aznauryan I. G. Burkert V. D. Prog. Part. Nucl. Phys. 1 2012 1
[11] Aznauryan I. G. et al. Phys. Rev. C 78 2008 045209
[12] The CLAS Collaboration Thompson R. et al. Phys. Rev. Lett. 86 2001 1702
[13] Kelly J. J. et al. Phys. Rev. C 75 2007 025201
[14] Gothe R. W. Mokeev V. I. et al. Nucleon Resonance Studies With CLAS12 in JLab Experiment E12-09-003
[15] Denizli H. et al. Phys. Rev. C 76 2007 015204
[16] Stoler P. Phys. Rep. 226 1993 103
[17] Tiator L. Drechsel D. Kamalov S. Giannini M. M. Eur. Phys. J. A 19 2004 55
[18] Braun V. M. Gockeler M. Horsley R. Kaltenbrunner T. et al. Phys. Rev. Lett. 103 2009 072001
[19] Buchmann A. J. Hernandez E. Faessler A. Phys. Rev. C 55 1997 448
[20] Capstick S. Roberts W. Prog. Part. Nucl. Phys. 45 2000 241
[21] Copley L. A. Karl G. Obryk E. Nucl. Phys. B 13 1969 303
[22] Capstick S. Keister B. D. Phys. Rev. D 51 1995 3598
[23] Dziembowski Z. Fabre de la Ripelle M. Miller G. A. Phys. Rev. C 53 1996 2038
[24] Aznauryan I. G. Burkert V. D. Phys. Rev. C 80 2009 055203
[25] Bali G. S. et al. Phys. Rev. D 62 2000 054503
[26] Bali G. S. Phys. Rep. 343 2001 1
[27] Alexandrou C. de Forcrand P. Jahn O. Nucl. Phys. Proc. Suppl. 119 2003 667
[28] Ballot J. de la Ripelle M. F. Ann. Phys. 127 1980 62
[29] Fabre de la Ripelle M. Navarro J. Ann. Phys. 123 1979 185
[30] Badalyan A. M. Phys. Lett. B 199 1987 267
[31] Santopinto E. Iachello F. Giannini M. M. Eur. Phys. J. A 1 1998 307
[32] Ikhdair Sameer M. Int. J. Mod. Phys. C 20 2009 1563
[33] Berkdemir C. Berkdemir A. Sever R. Phys. Rev. C 74 2006 039902
[34] Ikhdair S. M. Sever R. Int. J. Theor. Phys. 46 2007 1643
[35] Pekeris C. L. Phys. Rev. 45 1934 98
[36] Barnett R. M. et al. Phys. Rev. D 54 1996 174
[37] Halzen F. Martin A. D. Quarks and Leptons John Wiley and Sons New York 1984
[38] Giannini M. M. Santopinto E. Vassallo A. Eur. Phys. J. A 25 2005 241
[39] Salehi N. Rajabi A. A. Mod. Phys. Lett. A 24 2009 2631