Electroexcitation of the Roper resonance in the relativistic models

I.G. Aznauryan1, 2 1 Thomas Jefferson 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 assuming that the P11(1440) is the first radial excitation of the 3q 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 unified definition of helicity amplitudes consistent with the definition used in the extraction of the helicity amplitudes from experimental data in one- electroproduction. The results of relativistic quark models are qualitatively in good agreement with each other and differ 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 final 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 differ 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 π+ 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 final results 11 2 in the LF relativistic quark model and to find 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 different definitions 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 differences of defini- 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 first HELICITY AMPLITUDES radial excitation of the 3q nucleon state led to the diffi- 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 defined 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 → πN: 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 , 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 = −√ 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 Γ a = 2π π /C, (3) 11]. Within this framework, one can set up an impulse K m βπN approximation and avoid difficulties caused by different r 1 C = − for γ∗p → pπ0, (4) 3 r 2 Notice: Authored by The Southeastern Universities Research As- C = − for γ∗p → nπ+, (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 Γ and M are the total width and mass of the res- U.S. Government purposes. onance, βπN is its branching ratio to πN channel, m is 2

2 2 the mass of the nucleon, K = (M − m )/2M, qπ is the hadrons in the infinite momentum frame, where the center of mass momentum of the pion at the resonance initial moves along the z-axis with momentum position. P → ∞, and the momentum of the photon is k = ³ 2 2 2 2 2 2 ´ The helicity amplitudes defined 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 NNπ RNπ 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 → πN. Ref. [19], where the model of Ref. [18] was used to de- 2 ∗ Let us define 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 |Jµ|N >= eu¯(P )Γµu(P ), (6) dial part of the final state nucleon wave function by that ¡ ¢ 02 02 0 Γ = k2γ − (kγ)k F ∗(Q2) + iσ kν F ∗(Q2).(7) of the P11(1440): ΦN (M0 ) → ΦR(M0 ), where M0 is the µ µ µ 1 µν 2 invariant mass of quarks in the final state hadron. Under Using the definitions (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 ΦR(M ) = N(β − M )ΦN (M ). (14) £ ¤ 0 0 0 2 ∗ 2 ∗ 2 A1/2 = −c Q F1p(Q ) − (M + m)F2p(Q ) , (8) The parameters N and β are determined by the condi- k∗ £ ¤ √cms ∗ 2 ∗ 2 tions: S1/2 = −c F2p(Q ) + (M + m)F1p(Q ) , (9) 2 q R R Mk∗2 2 2 2 2 cms 4πα ΦR(M0 )ΦN (M0 )dΓ = 0, ΦR(M0 )dΓ = 1. (15) c = −ξR mK (M+m)2+Q2 , (10) P k2 +m2 where ξ is the relative sign between g and g Here dΓ is the phase space volume, M 2 = i⊥ q , R NNπ RNπ 0 i xi 2 coupling constants, and α = e /4π = 1/137. quark momenta are parametrized by ki = xiP + ki⊥, In quark model calculations another definition 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]: Φ(M0 ) ∼ exp(−M0 /6αHO). The only parameters of the approach are the quark mass (mq = 0.22 GeV ) and the q ∗+ ∗ 1 1 harmonic-oscillator parameter (α = 0.38 GeV ) which A = b < N ,S = |J²|p, 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∗ = |J²|p, S = >,(12) properties of the nucleon. 1/2 Q z 2 z 2 We have found the relative sign between NNπ and · ¸1/2 P (1440)Nπ amplitudes (ξ ) by relating these ampli- 2πα 11 R b = . (13) tudes to the matrix elements of the axial-vector current K (¯uγµγ5u) 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 definition (14) for the radial part of the ∗ ∗ 2 2 P11(1440) wave function, we have obtained ξR = 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 −ξR: A1/2 = −ξRA1/2, q tained in Refs. [12–14]. In accordance with the discus- S1/2 = −ξRS1/2. sion in Sec. II, we took into account difference of the ∗ q In some papers (for example, in Ref. [12]), the N,N signs of the amplitude A1/2 obtained via Eq. (11) and helicities are used in formulas (11,12) instead of the using the corresponding formula from Ref. [12]. We have projections. This leads to the opposite sign for the am- q corrected inaccuracies in Eqs. (18) of Ref. [12] and recal- plitude A1/2. culated the helicity amplitudes using the wave functions from that work. In Ref. [12], we did not find information on the sign ξR. The results from Ref. [12] presented in ∗ III. THE γ p → P11(1440) AMPLITUDES IN THE Fig. 1 correspond to the sign obtained in this work and RELATIVISTIC QUARK MODEL in Ref. [13]. The quark mass mq = 0.22 GeV coincides with that ∗ The calculations of the γ N → P11(1440) ampli- from Refs. [13, 14] and agrees with the light-quark mass tudes we have performed in the relativistic quark model obtained from the description of baryon masses in Ref. of Ref. [18], constructed for radiative transitions of [22]. The wave functions in Refs. [12, 13] are taken also 3 in the Gaussian form. Our harmonic-oscillator parameter The amplitudes are presented along with the results ob- αHO = 0.38 GeV is close to that from Ref. [13]. The tained in close approaches in Refs. [12–14] and with non- parameters used in Ref. [12] are different; by this reason relativistic amplitudes. All results are presented within above 1 GeV 2, the slope of the amplitudes corresponding unified definition of helicity amplitudes consistent with to Ref. [12] is larger. the definition used in the extraction of the helicity am- The wave functions in Ref. [14] (see also Ref. [23]) dif- plitudes from experimental data in one-pion electropro- fer from simple harmonic-oscillator wave functions due duction. The results of relativistic quark models are to the effects connected with the one-gluon-exchange in- qualitatively in good agreement with each other showing teraction which is taken into account in the form used in nontrivial behaviour of the transverse A1/2 amplitude. Ref. [22]. In Refs. [14, 23], the effects connected with the Being negative at Q2 = 0, this amplitude changes sign form factors and anomalous magnetic moments of quarks at Q2 ' 0.1 GeV 2 and becomes large and positive at are also taken into account. As it is shown in Ref. [23], Q2 ' 1 GeV 2; then it slowly falls with increasing Q2. these effects lead to the suppression of the absolute values The longitudinal amplitude S1/2 is large and positive at 2 2 of the amplitudes A1/2 and S1/2. Q = 0; with increasing Q it decreases. In order to demonstrate how important is the role of At Q2 < 0.6 GeV 2 , the results of different relativistic the relativistic effects, in Fig. 1 we have presented the quark models for the transverse amplitude A1/2 are con- results obtained according to the standard nonrelativistic sistent, but significantly higher than experimental data. calculations [24, 25] using the definitions (11,12). Our From our point of view, this is an indication on the ∗ parameters and the sign ξR obtained in this work have large pion cloud contribution to the γ N → P11(1440) been used. Let us note that the sign of our nonrelativistic transition which is expected to be significant at small amplitudes differs from the sign of the amplitudes in Refs. Q2. When final results on the helicity amplitudes at 2 2 [24, 25]. In Refs. [24, 25], the sign ξR was found not 1.7 < Q < 4.2 GeV will become available, complete within the quark model, but was fixed using experimental simultaneous description of the nucleon form factors and 2 ∗ data on the amplitude A1/2 at Q = 0. Our sign of the γ N → P11(1440) amplitudes will be neccessary. the nonrelativistic amplitudes is in agreement with that Such investigation will be important to find and specify obtained in Ref. [13]. the size of the pion cloud contribution, to investigate the From Fig. 1 it is seen that the relativistic effects possible mixing between N and P11(1440), as well to un- strongly change the quark model predictions for the derstand the role of other effects such as the quark form ∗ γ p → P11(1440) transition. The results obtained in factors and anomalous magnetic moments, and probably the relativistic approaches show nontrivial behavior of the quark mass dependence on the quark virtualities and 2 the transverse amplitude A1/2, which changes sign at therefore on Q . Q2 ' 0.1 GeV 2. Such behaviour of this amplitude is caused by the fact that due to the correct treatment of the ∗ 2 relativistic effects, the form factor F2 (Q ) changes the V. ACKNOWLEDGMENTS sign at Q2 ' 0.15 GeV 2 from positive at Q2 < 0.15 GeV 2 to negative. In the nonrelativistic calculations this form I am grateful to the participants of the JLab Theory 2 ∗ 2 factor is negative at all Q . The form factor F1 (Q ) is Group seminar for interest to this problem which stim- positive at all Q2 in both relativistic and nonrelativistic ulated this work, and to V.Burkert, H.Lee, A.Thomas approaches. for encouraging interest. I am thankful to B.Julia-Diaz and S.Stepanyan for their help in numerical calcula- tions, to T.Sato for discussions of the definitions of the IV. SUMMARY ∗ γ N → P11(1440) helicity amplitudes, to H.Matevosyan for discussion of the problems related to the pion cloud ∗ In this work we have calculated the γ p → P11(1440) contribution, and to M.Paris for valuable remarks while transition amplitudes in the LF relativistic quark model. reading the draft of this paper.

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100 80 1/2 1/2 S A 80 70

60 60 40 50 20 40 0 30 -20 -40 20 -60 10 -80 0 0 1 2 3 4 0 1 2 3 4 Q2 (GeV2) Q2 (GeV2)

∗ −3 −1/2 FIG. 1: Helicity amplitudes for the γ p → P11(1440) transition (in 10 GeV units). Full boxes and circles are the results obtained in the analysis of π electroproduction data in Refs. [4, 5]. Full triangle at Q2 = 0 is the PDG estimate [7]. Thick lines correspond to the LF relativistic quark models. Thick solid lines are the results obtained in this paper. Dashed and dash-dotted lines are the results of Refs. [13, 14], respectively. Dotted lines correpond to the results of Ref. [12] recalculated according to the discussion in Sec. III. Thin solid lines are the results of the nonrelativistic calculations [24, 25] obtained using our parameters and the sign ξR between gNNπ and gRNπ coupling constants found in this work.

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