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Electroexcitation of the Roper Resonance in the Relativistic Quark Models

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Electroexcitation of the Roper Resonance in the Relativistic Quark Models Electroexcitation of the Roper resonance in the relativistic quark models I.G. Aznauryan1, 2 1 Thomas Je®erson National Accelerator Facility, Newport News, Virginia 23606, USA 2 Yerevan Physics Institute, 375036 Yerevan, Armenia ¤ The amplitudes of the transition γ N ! P11(1440) are calculated within light-front relativistic quark model assuming that the P11(1440) is the ¯rst radial excitation of the 3q nucleon state. The results are compared with those obtained in close approaches by other authors and with standard nonrelativistic results. One of the reasons for this study was to present all these results within uni¯ed de¯nition of helicity amplitudes consistent with the de¯nition used in the extraction of the helicity amplitudes from experimental data in one-pion electroproduction. The results of relativistic quark models are qualitatively in good agreement with each other and di®er strongly from nonrelativistic 2 calculations. At small Q , these results for the transverse amplitude A1=2 are consistent, but fail to reproduce experimental data. The most probable explanation of this discrepancy is the absence of pion cloud contribution in the approaches under consideration. PACS numbers: 12.39.Ki, 13.40.Gp, 13.40.Hq, 14.20.Gk I. INTRODUCTION momenta of initial and ¯nal hadrons in the γ¤N ! N ¤ transition. For the P11(1440) such approach has been realized in Refs. [12{14]. However, quantitatively the The amplitudes of the electroexcitation of the Roper predicted results di®er from each other and there are in- resonance on proton are expected to be obtained from consistencies in the signs of the amplitudes obtained in the CLAS ¼+ electroproduction data at 1:7 < Q2 < 2 Refs. [12{14]. For this reason, we found it important to 4:2 GeV . There are already results [1] extracted perform calculations of the γ¤N ! P (1440) amplitudes from the preliminary data [2, 3], and the ¯nal results 11 2 in the LF relativistic quark model and to ¯nd sources of will be available soon. The results at 1:7 < Q < disagreement of the results obtained in Refs. [12{14]. We 4:2 GeV 2 combined with the previous CLAS data at 2 2 will also specify and compare di®erent de¯nitions of the Q = 0:4; 0:65 GeV [4{6] and with the information at γ¤N ! P (1440) helicity amplitudes in order to elim- Q2 = 0 [7] will give us knowledge of the Q2 evolution of 11 2 inate inconsistencies caused by the di®erences of de¯ni- the P11(1440) electroexcitation in wide Q region. This tion. information can be very important for understanding of the nature of the Roper resonance which has been the subject of discussions since its discovery [8], because the ¤ II. DEFINITIONS OF THE γ N ! P11(1440) simplest and most natural assumption that this is a ¯rst HELICITY AMPLITUDES radial excitation of the 3q nucleon state led to the di±- culties in the description of the mass of the resonance. ¤ The γ p ! P11(1440) helicity amplitudes extracted In order to utilize the information on the the electroex- in Refs. [1{3] from electroproduction data are de¯ned citation of the Roper resonance, it is important to have an through the P11(1440) contribution to the multipole am- understanding of which are the quark model predictions plitudes of the reaction γ¤p ! ¼N: M R ;SR . The 2 ¤ 1¡ 1¡ for the Q evolution of the transition γ N ! P11(1440). commonly used formulas presented in Refs. [15, 16] were 2 It is known that with increasing Q , when the mo- used: mentum transfer becomes larger than the masses of the constituent quarks, a relativistic treatment of the elec- R A1=2 = aImM1¡(W = M); (1) tromagnetic excitations, which is important already at a 2 R Q = 0, becomes crucial. The consistent way to realize S1=2 = ¡p ImS1¡(W = M); (2) 2 the relativistic treatment of the γ¤N ! N ¤ transitions s is to consider them in the light-front (LF) dynamics [9{ q M ¡ a = 2¼ ¼ =C; (3) 11]. Within this framework, one can set up an impulse K m ¯¼N approximation and avoid di±culties caused by di®erent r 1 C = ¡ for γ¤p ! p¼0; (4) 3 r 2 Notice: Authored by The Southeastern Universities Research As- C = ¡ for γ¤p ! n¼+; (5) sociation, Inc. under U.S. DOE Contract No. DE-AC05-84150. 3 The U.S. Government retains a non-exclusive, paid-up, irrevoca- ble, world-wide license to publish or reproduce this manuscript for where ¡ and M are the total width and mass of the res- U.S. Government purposes. onance, ¯¼N is its branching ratio to ¼N channel, m is 2 2 2 the mass of the nucleon, K = (M ¡ m )=2M, q¼ is the hadrons in the in¯nite momentum frame, where the center of mass momentum of the pion at the resonance initial hadron moves along the z-axis with momentum position. P ! 1, and the momentum of the photon is k = ³ 2 2 2 2 2 2 ´ The helicity amplitudes de¯ned through Eqs. (1-5) M ¡m ¡k? M ¡m ¡k? k?; ¡ ; . Such approach is anal- include the relative sign between g and g cou- 4P 4P NN¼ RN¼ ogous to the LF calculations [9{14]. The formulas for the pling constants which determines the relative contribu- + + transition γ¤N( 1 ) ! N( 1 ) are presented in detail in tions of the Born term and the P11(1440) to the reaction 2 2 γ¤N ! ¼N. Ref. [19], where the model of Ref. [18] was used to de- 2 ¤ Let us de¯ne the γ¤N ! N ¤ transition current matrix scribe the Q evolution of the γ N ! N form factors. element in terms of the form factors F ¤(Q2) and F ¤(Q2): These formulas can be directly used for the evaluation of 1 2 ¤ the γ p ! P11(1440) amplitudes if we will replace the ra- ¤ ¤ < N jJ¹jN >= eu¹(P )¡¹u(P ); (6) dial part of the ¯nal state nucleon wave function by that ¡ ¢ 02 02 0 ¡ = k2γ ¡ (kγ)k F ¤(Q2) + iσ kº F ¤(Q2):(7) of the P11(1440): ©N (M0 ) ! ©R(M0 ), where M0 is the ¹ ¹ ¹ 1 ¹º 2 invariant mass of quarks in the ¯nal state hadron. Under Using the de¯nitions (1-5) one can express the helicity the assumption that P11(1440) is a radial excitation of ¤ 2 amplitudes A1=2, S1=2 through the form factors F1p(Q ) the 3-quark nucleon state, we have ¤ 2 ¤ and F2p(Q ) of the γ p ! P11(1440) transition: 2 2 2 2 ©R(M ) = N(¯ ¡ M )©N (M ): (14) £ ¤ 0 0 0 2 ¤ 2 ¤ 2 A1=2 = ¡c Q F1p(Q ) ¡ (M + m)F2p(Q ) ; (8) The parameters N and ¯ are determined by the condi- k¤ £ ¤ pcms ¤ 2 ¤ 2 tions: S1=2 = ¡c F2p(Q ) + (M + m)F1p(Q ) ; (9) 2 q R R Mk¤2 2 2 2 2 cms 4¼® ©R(M0 )©N (M0 )d¡ = 0; ©R(M0 )d¡ = 1: (15) c = ¡»R mK (M+m)2+Q2 ; (10) P k2 +m2 where » is the relative sign between g and g Here d¡ is the phase space volume, M 2 = i? q , R NN¼ RN¼ 0 i xi 2 coupling constants, and ® = e =4¼ = 1=137. quark momenta are parametrized by ki = xiP + ki?, In quark model calculations another de¯nition of i = 1; 2; 3 denotes the quark, and mq is the quark mass. ¤ the γ N ! P11(1440) helicity amplitudes through the As in Refs. [18, 19], we have taken the radial part γ¤N ! N ¤ transition current matrix elements commonly of the nucleon wave function in the Gaussian form: 2 2 2 is used [12, 17]: ©(M0 ) » exp(¡M0 =6®HO). The only parameters of the approach are the quark mass (mq = 0:22 GeV ) and the q ¤+ ¤ 1 1 harmonic-oscillator parameter (® = 0:38 GeV ) which A = b < N ;S = jJ²jp; Sz = ¡ >; (11) HO 1=2 z 2 2 were found in Ref. [18] from the description of the static k¤ 1 1 Sq = b cms < N ¤+;S¤ = jJ²jp; S = >;(12) properties of the nucleon. 1=2 Q z 2 z 2 We have found the relative sign between NN¼ and · ¸1=2 P (1440)N¼ amplitudes (» ) by relating these ampli- 2¼® 11 R b = : (13) tudes to the matrix elements of the axial-vector current K (¹uγ¹γ5u) using PCAC. The formulas for these matrix Here it is supposed that the virtual photon moves along elements were taken from Refs. [18, 19] (see also Ref. the z-axis in the N ¤ rest frame and its 3-momentum is [20]). With the de¯nition (14) for the radial part of the ¤ ¤ 2 2 P11(1440) wave function, we have obtained »R = 1. The kcms, P = P + k, Q ´ ¡k . 3 In order to compare the quark model predictions ob- same sign was obtained in Ref. [13] using P0 model of Ref. [21]. tained using Eqs. (11-13) with the results extracted from ¤ q Our results for the γ p ! P11(1440) helicity ampli- experimental data via Eqs. (1-5), the amplitudes A1=2, q q tudes are presented in Fig. 1 along with the results ob- S1=2 should be multiplied by ¡»R: A1=2 = ¡»RA1=2, q tained in Refs. [12{14]. In accordance with the discus- S1=2 = ¡»RS1=2. sion in Sec. II, we took into account di®erence of the ¤ q In some papers (for example, in Ref. [12]), the N; N signs of the amplitude A1=2 obtained via Eq.
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