Helicity dependence and contribution to the Gerasimov-Drell-Hearn sum rule of the γ d → π NN reaction channels in the energy region from threshold up to the (1232) resonance

PHYSICAL REVIEW C 76, 044005 (2007) Helicity dependence and contribution to the Gerasimov-Drell-Hearn sum rule of the  γ d→πNN reaction channels in the energy region from threshold up to the (1232) resonance E. M. Darwish,1,*C. Fern´ andez-Ram´ ırez,2,3,†E. Moya de Guerra,2,4and J. M. Ud´ ıas4 1Physics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt 2Instituto de Estructura de la Materia, CSIC, Serrano 123, E-28006 Madrid, Spain 3Departamento de F´ ısica At´ omica, Molecular y Nuclear, Universidad de Sevilla, Apdo. 1065, E-41080 Sevilla, Spain 4Grupo de F´ ısica Nuclear; Departamento de F´ ısica At´ omica, Molecular y Nuclear, Facultad de Ciencias F´ ısicas, Universidad Complutense de Madrid, Avda. Complutense s/n, E-28040 Madrid, Spain (Received 18 January 2007; published 15 October 2007) The helicity dependence of the γ d→π−pp, γ d→π+nn,and γ d→π0np reaction channels is studied for incident photon energies from threshold up to the (1232) resonance with inclusion of leading πNN effects. The doubly polarized total and differential cross sections for parallel and antiparallel helicity states are predicted. Then the contribution of various channels to the deuteron spin asymmetry and the double polarization Easymmetry is calculated. In addition, the contribution to the Gerasimov-Drell-Hearn (GDH) integral from separate channels is evaluated by explicit integration up to a photon lab energy of 350 MeV. Sizeable effects from final-state interactions, specially for π0production, are found. The sensitivity of the results to the elementary pion photoproduction operator is also investigated. Considerable dependence of the results on the elementary amplitude is found. We expect that these results may be useful to interpret the recent measurements from LEGS@BNL, A2, and GDH@MAMI Collaborations. DOI: 10.1103/PhysRevC.76.044005 PACS number(s): 13.60.Le, 25.20.Lj, 14.20.−c, 29.27.Hj I. INTRODUCTION Recent years have witnessed an increasing interest in theoretical research on quasifree pion production from the deuteron including polarization observables—see recent works in Refs. [1–4] where also references to earlier research can be found. This interest is partly due to the new generation of high-intensity and high duty-cycle electron accelerators, such as MAMI at Mainz and ELSA at Bonn (Germany), JLab at Newport News (USA), and MAX-Lab at Lund (Sweden) as well as laser backscattering facilities such as GRAAL at Grenoble (France) and LEGS at Brookhaven (USA)—for an experimental overview see Refs. [5–8]. With the development of these new facilities, and thanks to the improvement in polarized beam and polarized target techniques, it is now possible to obtain accurate data for meson electromagnetic production, including single and double spin-dependent observables. These measurements allow us to obtain more information on the reaction mechanism than unpolarized measurements do. The latter only provide information on the sum of the absolute squares of the reaction amplitudes. A series of measurements of double polarization observables in both charged and neutral pion production reactions from the deuteron have been carried out or planned at different laboratories such as LEGS and MAMI. *[email protected]; Corresponding author. Fax:+20 93 4601159. †Present address: Grupo de F´ ısica Nuclear, Departamento de F´ ısica At´ omica, Molecular y Nuclear, Facultad de Ciencias F´ ısicas, Universidad Complutense de Madrid, Avda. Complutense s/n, E-28040 Madrid, Spain. In particular, the A2 and GDH Collaborations at MAMI have undertaken a joint effort to verify experimentally the Gerasimov-Drell-Hearn (GDH) sum rule [9], measuring the difference between the helicity components in both the total [σP−σA] and the differential [d(σP−σA)/d] photoabsorption cross sections, where σP,A stands for the parallel and antiparallel spin orientations of the photon and target deuteron. In these experiments, circularly polarized photons are scattered on longitudinally polarized deuterons to determine the double polarization asymmetry in a large kinematical range [6]. These helicity-dependent cross sections provide valuable information on the nucleon spin structure and allow us to extract information on the neutron. The knowledge of these cross sections is also required to test the validity of the GDH sum rule on the deuteron and the neutron as well as to explore which are the dominant contributions to the GDH integral. These new measurements will make possible to test experimentally the behavior of the GDH integral. However, to explore the contributions of the resonances, the total photoabsorption cross section does not provide enough information. Fixing degrees of freedom by selection of particular partial reaction channels allows us to obtain more detailed information. This is possible with double polarization observables of the differential cross section. The present focus of the LEGS Collaboration is the measurement of double polarization spin observables in photoreactions on the proton, neutron, and deuteron. For instance, the beam-target double polarization Easymmetry of the γ d→π0np reaction at photon lab energy Eγ=349 ±5 MeV has been recently measured [8]—see Eq. (7) for the definition of Easymmetry. This asymmetry will certainly provide an important test that, indirectly, allows us to check our knowledge on the process of pion production from the neutron. To the best of our 0556-2813/2007/76(4)/044005(13) 044005-1 ©2007 The American Physical Society E. M. DARWISH et al. PHYSICAL REVIEW C 76, 044005 (2007) T T G T d tγπ N (a) N 2 (1) 1 π γ d tγπ N (1) 2 π N1 ΝΝ γ (b) d tγπ (1) γ (d) π N1 N2 πΝ = d tγπ N N (1) 1 2 π γ πΝΝ ++ + 2 d tγπ N (1) γ πΝ π N1 (c) FIG. 1. Diagrammatic representation of the γd →πNN amplitude including rescattering contributions in the two-body subsystems and neglecting all contributions of two-body meson-exchange currents. (a) Impulse approximation (IA); (b, c, and d) “driving terms” from NN and πN rescattering, respectively. Diagrams when the elementary pion photoproduction operator acts on nucleon 2 are not shown in the figure but are included in the calculations. In the calculations, each diagram shown in the figure goes accompanied by the diagram obtained by the exchange N1↔N2. knowledge, there is no calculation in the literature for the double polarization Easymmetry of the γ d→πNN reaction channels. The helicity structure of quasifree pion photoproduction from the deuteron was studied in Refs. [10–12]. In Ref. [10], special emphasis was given to the beam-target spin asymmetry and the corresponding GDH sum rule for the deuteron, including complete hadronic rescattering in the NN and πN subsystems. The helicity dependence of the  d(γ,π−)pp reaction channel was investigated in Ref. [11] with just the inclusion of NN rescattering in the final state. In that work, the differential polarized cross-section difference for parallel and antiparallel helicity states was predicted and the contribution of the  d(γ,π−)pp channel to the deuteron spin asymmetry was evaluated. It was found that the effects of final state interactions (FSI) are larger in the asymmetry than in the unpolarized cross sections, which leads to a sizeable reduction of the deuteron spin asymmetry in the position of the peak. Both approaches [10,11] were restricted to the (1232)-resonance region in view of a relatively simple elementary pion photoproduction operator based on an effective Lagrangian approach from Ref. [13] that included only Born terms and the (1232) resonance. Later on, the work in Ref. [10] was improved by Arenh¨ ovel, Fix, and Schwamb in Ref. [12], where a better elementary production operator provided by MAID model [14]was employed and FSI effects were considered. Explicit evaluation of the deuteron spin asymmetry and the associated GDH sum rule were presented in said work. In this article we present the first calculation for the d(γ,π)NN reaction channels with polarized photon beam and polarized deuteron target in the near-threshold region. The ultimate goal of the present article is to extend our approach, recently presented in Ref. [1], to make theoretical predictions for the helicity dependence in total and differential photoabsorption cross sections of the processes γ d→π−pp, γ d→π+nn, and γ d→π0np, in the energy range from threshold up to the (1232) resonance, using as elementary γN →πN reaction a realistic effective Lagrangian approach (ELA) from Refs. [15,16]. This elementary reaction model displays chiral symmetry, gauge invariance, and crossing symmetry, as well as a consistent treatment of the interaction with spin-3/2 particles. It also provides a reliable description of the threshold region. In this article we provide a theoretical overview of the double polarization Easymmetry for different pion photoproduction channels and we explicitly evaluate the contribution from separate channels to the GDH sum rule for the deuteron. It was found in Ref. [1] that the results for single-spin asymmetries are strongly dependent on the elementary pion photoproduction operator. Hence, we also investigate the sensitivity of the double-spin polarization observables to the elementary γN →πN operator. The calculations presented in this article are of high interest in view of the extensive recent polarization measurements from the A2 and GDH@MAMI [6] and LEGS@BNL [8] Collaborations. A theoretical understanding of these data on the deuteron target will provide information about the pion photoproduction from the neutron reaction, which is not well known yet, but is required for a complete understanding of excitations in the pion photoproduction process. In Sec. II we briefly review the general formalism for the d(γ,π)NN reaction [1,4]. Results for the γ d→ π−pp, γ d→π+nn, and γ d→π0np reactions are presented and discussed in Sec. III. Finally, we provide conclusions in Sec. IV. T γπ NN = NNN NNNN γ * NN ∆ N NN N πγ γππ γ γπ π * NN ∆ γπ N NN γ π π ρ,ω γπ (A) (D) (E) (B) (C) (F) (G) FIG. 2. Feynman diagrams for pion photoproduction from free nucleons. Born terms: (A) direct nucleon pole or schannel, (B) crossed nucleon pole or uchannel, (C) pion in flight or tchannel, and (D) Kroll-Rudermann contact term; (E) vector-meson exchange (ρand ω); resonance excitations contribution: (F) direct or schannel and (G) crossed or uchannel. 044005-2 HELICITY DEPENDENCE AND CONTRIBUTION TO THE . . . PHYSICAL REVIEW C 76, 044005 (2007) -0.08 -0.06 -0.04 -0.02 0 0 60 120 180 d(σP -σA )/dΩ [µb/sr] θπ [deg] θπ [deg] θπ [deg] d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ =150 MeV -6 -4 -2 0 0 60 120 180 θπ [deg] θπ [deg] θπ [deg] d(γ,π0 )np d(γ,π-d(γ,π+ )nn Eγ =150 MeV Eγ =150 MeV -6 -4 -2 0 0 60 120 180 θπ [deg] θπ [deg] θπ [deg] d(γ,π0 )np d(γ,π-d(γ,π+ )nn Eγ =150 MeV Eγ =150 MeV Eγ =150 MeV -10 -4 2 8 14 0 60 120 180 d(σP -σA )/dΩ [µb/sr] θπ [deg] θπ [deg] θπ [deg] d(γ,π0 )np d(γ,π-d(γ,π+ )nn Eγ =150 MeV Eγ =150 MeV Eγ =150 MeV Eγ =230 MeV -15 -10 -5 0 0 60 120 180 θπ [deg] θπ [deg] θπ [deg] d(γ,π0 )np d(γ,π-d(γ,π+ )nn Eγ =150 MeV Eγ =150 MeV Eγ =150 MeV Eγ =230 MeV Eγ =230 MeV -15 -10 -5 0 0 60 120 180 θπ [deg] θπ [deg] θπ [deg] d(γ,π0 )np d(γ,π-d(γ,π+ )nn Eγ =150 MeV Eγ =150 MeV Eγ =150 MeV Eγ =230 MeV Eγ =230 MeV Eγ =230 MeV -80 -40 0 40 80 120 0 60 120 180 d(σP -σA )/dΩ [µb/sr] θπ [deg] θπ [deg] θπ [deg] d(γ,π0 )np d(γ,π-d(γ,π+ )nn Eγ =150 MeV Eγ =150 MeV Eγ =150 MeV Eγ =230 MeV Eγ =230 MeV Eγ =230 MeV Eγ =350 MeV -40 -20 0 20 0 60 120 180 θπ [deg] θπ [deg] θπ [deg] d(γ,π0 )np d(γ,π-d(γ,π+ )nn Eγ =150 MeV Eγ =150 MeV Eγ =150 MeV Eγ =230 MeV Eγ =230 MeV Eγ =230 MeV Eγ =350 MeV Eγ =350 MeV -40 -20 0 20 0 60 120 180 θπ [deg] θπ [deg] θπ [deg] d(γ,π0 )np d(γ,π-d(γ,π+ )nn Eγ =150 MeV Eγ =150 MeV Eγ =150 MeV Eγ =230 MeV Eγ =230 MeV Eγ =230 MeV Eγ =350 MeV Eγ =350 MeV Eγ =350 MeV FIG. 3. (Color online) Circular photon polarization asymmetry for the separate channels of semiexclusive pion photoproduction on the deuteron as a function of the pion angle in the laboratory frame at various photon lab energies using the electromagnetic multipoles of set no. 2 from the effective Lagrangian approach in Refs. [15,16]. Curve conventions: (dotted) IA; (long-dashed) IA+πN; (short-dashed) IA+NN; (solid) IA+NN+πN. 0 200 400 600 800 1000 150 200 250 300 350 σP [µb] d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] 0 100 200 300 150 200 250 300 350 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] 0 100 200 300 150 200 250 300 350 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] 0 200 400 600 800 1000 150 200 250 300 350 σA [µb] d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] 0 100 200 300 150 200 250 300 350 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] 0 100 200 300 150 200 250 300 350 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] 0 200 400 600 800 150 200 250 300 350 σP - σA [µb] d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] -200 -100 0 100 150 200 250 300 350 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] -150 -50 50 150 150 200 250 300 350 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] FIG. 4. (Color online) The helicitydependent total photoabsorption cross sections for the separate channels of  d(γ,π)NN for circularly polarized photons on a longitudinally polarized deuteron with spin parallel σP (upper panel) and antiparallel σA(middle panel) to the photon spin in the energy region from threshold up to the (1232) resonance. The lower panel shows the difference σP−σA, i.e., the deuteron spin asymmetry of total photoabsorption cross section. Curve conventions as described in the caption to Fig. 3. 044005-3 E. M. DARWISH et al. PHYSICAL REVIEW C 76, 044005 (2007) 0 0.5 1 1.5 0 5 10 15 20 σP [µb] d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] 0 2 4 6 0 5 10 15 20 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] 0 2 4 6 0 5 10 15 20 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] 0 0.5 1 1.5 2 0 5 10 15 20 σA [µb] d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] 0 50 100 150 0 5 10 15 20 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] 0 50 100 150 0 5 10 15 20 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] -1 -0.5 0 0.5 0 5 10 15 20 σP - σA [µb] d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] -150 -100 -50 0 0 5 10 15 20 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] -150 -100 -50 0 0 5 10 15 20 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] FIG. 5. (Color online) Same as in Fig. 4but in the energy region near threshold. Eγ−Ethr γ denotes the excess energy above threshold in the laboratory system. 0.5 0.6 0.7 0.8 0.9 1 0 60 120 180 E-asymmetry θπ [deg] θπ [deg] θπ [deg] d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ =150 MeV 0.97 0.98 0.99 1 0 60 120 180 θπ [deg] θπ [deg] θπ [deg] Eγ =150 MeV Eγ =150 MeV 0.97 0.98 0.99 1 0 60 120 180 θπ [deg] θπ [deg] θπ [deg] Eγ =150 MeV Eγ =150 MeV Eγ =150 MeV -1 -0.5 0 0.5 1 0 60 120 180 E-asymmetry θπ [deg] θπ [deg] θπ [deg] Eγ =150 MeV Eγ =150 MeV Eγ =150 MeV Eγ =230 MeV 0 0.2 0.4 0.6 0.8 1 0 60 120 180 θπ [deg] θπ [deg] θπ [deg] Eγ =150 MeV Eγ =150 MeV Eγ =150 MeV Eγ =230 MeV Eγ =230 MeV 0 0.2 0.4 0.6 0.8 1 0 60 120 180 θπ [deg] θπ [deg] θπ [deg] Eγ =150 MeV Eγ =150 MeV Eγ =150 MeV Eγ =230 MeV Eγ =230 MeV Eγ =230 MeV -1 -0.5 0 0.5 1 0 60 120 180 E-asymmetry θπ [deg] θπ [deg] θπ [deg] Eγ =150 MeV Eγ =150 MeV Eγ =150 MeV Eγ =230 MeV Eγ =230 MeV Eγ =230 MeV Eγ =350 MeV -1 -0.5 0 0.5 1 0 60 120 180 θπ [deg] θπ [deg] θπ [deg] Eγ =150 MeV Eγ =150 MeV Eγ =150 MeV Eγ =230 MeV Eγ =230 MeV Eγ =230 MeV Eγ =350 MeV Eγ =350 MeV -1 -0.5 0 0.5 1 0 60 120 180 θπ [deg] θπ [deg] θπ [deg] Eγ =150 MeV Eγ =150 MeV Eγ =150 MeV Eγ =230 MeV Eγ =230 MeV Eγ =230 MeV Eγ =350 MeV Eγ =350 MeV Eγ =350 MeV FIG. 6. (Color online) The double polarization Easymmetry [see Eq. (7) for its definition] as a function of the pion angle in the laboratory frame at various photon lab energies for the separate channels of the reaction  d(γ,π)NN for circularly polarized photons on a longitudinally polarized deuteron. Curve conventions as in Fig. 3. 044005-4 HELICITY DEPENDENCE AND CONTRIBUTION TO THE . . . PHYSICAL REVIEW C 76, 044005 (2007) 0 40 80 120 160 150 200 250 300 350 IGDH [µb] d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] -80 -60 -40 -20 0 150 200 250 300 350 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] -60 -40 -20 0 150 200 250 300 350 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] FIG. 7. (Color online) The Gerasimov- Drell-Hearn integral [see Eq. (9)foritsdefinition] as a function of the upper integration limit for separate channels of the  d(γ,π)NN reaction. Curve conventions as in Fig. 3. II. FORMALISM We start with a brief description of the basic formal ingredients of quasifree pion photoproduction reaction from the deuteron γ(k, εµ)+d(d)→π(q)+N1(p1)+N2(p2),(1) where the four-momenta of the involved particles are indicated within parentheses. In Fig. 1we present the Feynman diagrams of the scattering matrix for reaction (1). Full treatment of all interaction effects requires, in principle, a full unitary πNN three-body calculation. In the present work, however, we restrict ourselves to the inclusion of complete rescattering in the various two-body subsystems of the final state [Figs. 1(b), 1(c), and 1(d)]. The treatment follows previous works [1,4], to which the reader is referred for formal details. The three contributions in Figs. 1(a)–1(d) present three different contributions to the reaction amplitude. We refer to them as (a) impulse approximation (IA); (b) the NN- rescattering contribution; and (c) πNc-rescattering and (d) the πNd-rescattering contributions. The total transition matrix -450 -250 -50 150 350 150 200 250 300 350 σP - σA [µb] Eγ [MeV] FIG. 8. (Color online) The helicity-dependent total photoabsorption cross section difference (σP−σA) for the semiexclusive channels γ d→π±NN. Curve conventions: (dashed) IA∗+NN+πNc using MAID [14]; (solid) IA+NN+πN using the bare electromagnetic multipoles of ELA [15,16,20]. Experimental data are taken from Ref. [7]. element reads Tsmµmd=TIA smµmd+TNN smµmd+TπN(c)+(d) smµmd.(2) The starting point of our formalism is the IA [Fig. 1(a)], which is completely determined by the elementary amplitude γN →πN and the deuteron wave function as follows TIA smµmd= m [sm| p1|tγπ(WγN 1)|− p2 ×φmmd(p2)|1m−(1 ↔2)],(3) where tγπ denotes the elementary pion photoproduction operator on the free nucleon; WγN 1the invariant energy of the γN 1system; p1=( k−q)/2+p, and p2=( k−q)/2−p. φmmd(p) is given by φmmd(p)= L=0,2 mL iLCL11 mLmm duL(p)YLmL(ˆ p)(4) and we compute the radial deuteron wave function uL(p)using the realistic Paris potential [17]. The contribution from NN rescattering [Fig. 1(b)]tothe transition matrix is TNN smµmd=q, p, sm |TNNGNNt(1) γπ(WγN 1) +t(2) γπ(WγN 2)|1md,(5) where TNN stands for the half-off-shell NN-scattering matrix and GNN for the corresponding free NN propagator. The former is obtained from separable representation of a realistic NN interaction [18], including all S,P, and Dpartial waves. Similarly to the NN matrix element, we obtain the πN- rescattering contribution [Figs. 1(c) and 1(d)] TπN(c)+(d) smµmd= j=c, d q, p, sm|TπNjGπNjt(1) γπ(WγN 1) +t(2) γπ(WγN 2)|1md,(6) where we calculate the half-off-shell πN-scattering matrix TπN from a separable energy-dependent πN potential [19], including all Sto Dwaves. The explicit formal expressions for the three contributions in Eqs. (3), (5), and (6) can be found in Refs. [1,4] and we refer to these article for details. For the evaluation of Eqs. (3), (5), and (6) one has also to specify the elementary pion photoproduction operator for 044005-5 E. M. DARWISH et al. PHYSICAL REVIEW C 76, 044005 (2007) -0.12 -0.08 -0.04 0 0 60 120 180 d(σP -σA )/dΩ [µb/sr] d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ =155 MeV Eγ =155 MeV Eγ =155 MeV Eγ =230 MeV Eγ =230 MeV Eγ =230 MeV Eγ =350 MeV Eγ =350 MeV Eγ =350 MeV θπ [deg] θπ [deg] θπ [deg] -15 -10 -5 0 0 60 120 180 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ =155 MeV Eγ =155 MeV Eγ =155 MeV Eγ =230 MeV Eγ =230 MeV Eγ =230 MeV Eγ =350 MeV Eγ =350 MeV Eγ =350 MeV θπ [deg] θπ [deg] θπ [deg] -15 -10 -5 0 0 60 120 180 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ =155 MeV Eγ =155 MeV Eγ =155 MeV Eγ =230 MeV Eγ =230 MeV Eγ =230 MeV Eγ =350 MeV Eγ =350 MeV Eγ =350 MeV θπ [deg] θπ [deg] θπ [deg] -5 0 5 10 0 60 120 180 d(σP -σA )/dΩ [µb/sr] d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ =155 MeV Eγ =155 MeV Eγ =155 MeV Eγ =230 MeV Eγ =230 MeV Eγ =230 MeV Eγ =350 MeV Eγ =350 MeV Eγ =350 MeV θπ [deg] θπ [deg] θπ [deg] -30 -20 -10 0 0 60 120 180 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ =155 MeV Eγ =155 MeV Eγ =155 MeV Eγ =230 MeV Eγ =230 MeV Eγ =230 MeV Eγ =350 MeV Eγ =350 MeV Eγ =350 MeV θπ [deg] θπ [deg] θπ [deg] -25 -15 -5 5 0 60 120 180 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ =155 MeV Eγ =155 MeV Eγ =155 MeV Eγ =230 MeV Eγ =230 MeV Eγ =230 MeV Eγ =350 MeV Eγ =350 MeV Eγ =350 MeV θπ [deg] θπ [deg] θπ [deg] -50 0 50 100 0 60 120 180 d(σP -σA )/dΩ [µb/sr] d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ =155 MeV Eγ =155 MeV Eγ =155 MeV Eγ =230 MeV Eγ =230 MeV Eγ =230 MeV Eγ =350 MeV Eγ =350 MeV Eγ =350 MeV θπ [deg] θπ [deg] θπ [deg] -50 -30 -10 10 30 0 60 120 180 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ =155 MeV Eγ =155 MeV Eγ =155 MeV Eγ =230 MeV Eγ =230 MeV Eγ =230 MeV Eγ =350 MeV Eγ =350 MeV Eγ =350 MeV θπ [deg] θπ [deg] θπ [deg] -50 -30 -10 10 30 0 60 120 180 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ =155 MeV Eγ =155 MeV Eγ =155 MeV Eγ =230 MeV Eγ =230 MeV Eγ =230 MeV Eγ =350 MeV Eγ =350 MeV Eγ =350 MeV θπ [deg] θπ [deg] θπ [deg] FIG. 9. (Color online) Circular photon polarization asymmetry for the separate channels of semiexclusive pion photoproduction on the deuteron as a function of the pion angle in the laboratory frame at various photon lab energies using different elementary pion photoproduction operators and including FSI effects. Curve conventions: (dashed) IA∗+NN+πNcusing MAID [14]; (dotted) IA∗+NN+πNcusing the dressed multipoles of ELA [15,16,20]; (solid) IA+NN+πN using the bare electromagnetic multipoles of ELA. the N(γ,π)Nreaction. As stated in the Introduction, we use a realistic elementary operator from Refs. [15,16,20]. This model has been successfully applied to the N(γ,π)Nprocess from threshold to 1 GeV of photon energy in the laboratory reference system. The model is based on an effective Lagrangian approach that, from a theoretical point of view, is an appealing, reliable, and formally well-established approach in the energy region of the mass of the nucleon. The model includes Born terms [Figs. 2(A)–2(D)], vector-meson exchanges [ρand ω, in Fig. 2(E)], and all the four star resonances in the Particle Data Group (PDG) [21] up to 1.7 GeV and up to spin-3/2: (1232), N(1440), N(1520), N(1535), (1620), N(1650), and (1700) [Fig. 2(F) and 2(G)]. In Ref. [15] the single and double polarization asymmetries for pion photoproduction from free nucleons were computed and compared to the available experimental database. The agreement with data was found to be fairly good. Recently, the model has also been applied successfully to ηphotoproduction from the proton [22]. The model displays chiral symmetry, gauge invariance, and crossing symmetry as well as a consistent treatment of the spin- 3/2 interaction that overcomes pathologies present in former analyses [23]. The dressing of the resonances is considered by means of a phenomenological width that contributes to both sand uchannels and takes into account decays into one π, one η, and two π. In the pion photoproduction model from free nucleons [15,16] it was assumed that FSI factorize and can be included through the distortion of the πN final state wave function (pion-nucleon rescattering). πN-FSI was included [15,16] by adding a phase δFSI to the electromagnetic multipoles. This phase is set so that the total phase of the multipole matches the total phase of the energy dependent solution of SAID [24]. In this way it was possible to isolate the contribution to the physical observables from the bare diagrams. The parameters of the resonances were extracted from data fitting the electromagnetic multipoles of the energy-independent solution of SAID [24], applying a modern optimization technique based on genetic algorithms combined with gradient-based routines [16,22,25] that provide reliable values for the parameters of the nucleon resonances. Once the bare properties of the nucleon resonances have been extracted from data, their contribution to other processes can be calculated. For further details on the model we refer the reader to Refs. [1,15,16,20]. III. RESULTS AND DISCUSSION We divide into four parts the discussion of the results obtained. First, we report results on the doubly polarized differential and total cross sections for the parallel and antiparallel helicity states of the γ d→π0np, γ d→π−pp, and γ d→π+nn processes in the energy region from threshold up to the (1232) resonance. The contribution of these processes to the spin response of the deuteron, i.e., the asymmetry 044005-6 HELICITY DEPENDENCE AND CONTRIBUTION TO THE . . . PHYSICAL REVIEW C 76, 044005 (2007) 0 200 400 600 800 150 200 250 300 350 σP [µb] d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] 0 100 200 300 400 150 200 250 300 350 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] 0 100 200 300 400 150 200 250 300 350 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] 0 200 400 600 800 150 200 250 300 350 σA [µb] d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] 0 100 200 300 400 150 200 250 300 350 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] 0 100 200 300 400 150 200 250 300 350 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] 0 200 400 600 150 200 250 300 350 σP - σA [µb] d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] -200 -100 0 100 200 150 200 250 300 350 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] -200 -100 0 100 200 150 200 250 300 350 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ [MeV] Eγ [MeV] Eγ [MeV] FIG. 10. (Color online) Total photoabsorption cross sections for the separate channels of  d(γ,π)NN for circularly polarized photons on a longitudinally polarized deuteron with spin parallel σP(upper part) and antiparallel σA (middle part) to the photon spin using different elementary operators and including FSI effects. The lower part shows the difference σP−σA. Curve conventions as described in the caption to Fig. 9. of the total photoabsorption cross section with respect to parallel and antiparallel spins of photon and deuteron, is explicitly evaluated in this part. The second part is devoted to the results for the beam-target double polarization E asymmetry, whereas in the third part, the contribution of the separate pion photoproduction channels to the deuteron GDH integral is evaluated by explicit integration up to a photon lab-energy of Eγ=350 MeV. In the last one,we discuss the sensitivity of our results to the elementary γN →πN amplitude. In the figures we compare calculations with different ingredients. We display separately the contributions from the bare and the final-state interactions. We call impulse approximation (IA) to the contribution to the observables shown in Fig. 1(a) using the bare electromagnetic multipoles together with the same diagram exchanging N1↔N2.In what follows, when we cite Figs. 1(a)–1(d) we implicitly mean the contributions of both the depicted diagram and the corresponding one obtained under N1↔N2exchange. As we stated in our preceding article [1], one has to be careful with what one calls IA. We would like to explain carefully what we call IA and how we compute it. A truly IA calculation [a calculation that includes only Fig. 1(a)] cannot employ directly the elementary amplitudes that fit the data on electromagnetic multipoles for the γN →πN process. This is due to the fact that πN rescattering is unavoidably included in the “elementary” amplitude in these fits to data. For example, if we use the multipoles provided by the ELA model that fits the experimental data, and afterwards we include πN rescattering between the pion and the spectator nucleon, we are not really including just Figs. 1(a)+1(c) but rather Figs. 1(a)+ 1(c)+1(d), with a contribution of πN rescattering of the pion with the knocked-out nucleon that comes from the photoproduction operator. For the same reason computations in Refs. [3,4,10,12] implicitly include to some extent Fig. 1(d), although it is not explicitly considered, and a certain content of such diagram is also present in the rest of the calculated rescattering terms. Therefore, if we wish to calculate the contribution coming just from Fig. 1(a), the bare IA contribution to the amplitude has to be extracted from the analysis of the γN → πN, where the final-state interaction has to be removed. This was done in Refs. [15,16]. We name the calculations where the πN rescattering is included in the elementary reaction IA∗.IA ∗is, therefore, somehow equivalent to a Fig. 1(a)+1(d) calculation. In Sec. III D we compare results for IA∗+rescattering and IA+rescattering results using MAID and ELA models and, thus, we survey the effects of πN rescattering and the influence of the basic pion production operator. We refer to the inclusion of Fig. 1(a) in a calculation as IA, to the inclusion of diagram (b) as NN, and to the inclusion of Figs. 1(c)+1(d),usingtheπN rescattering explained in this article, as πN. When only Fig. 1(c) is accounted for in the πN rescattering it is labeled πNc. 044005-7 E. M. DARWISH et al. PHYSICAL REVIEW C 76, 044005 (2007) 0 0.4 0.8 1.2 1.6 0 5 10 15 20 σP [µb] d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] 0 2 4 6 8 0 5 10 15 20 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] 0 2 4 6 0 5 10 15 20 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] 0 0.5 1 1.5 2 0 5 10 15 20 σA [µb] d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] 0 40 80 120 160 0 5 10 15 20 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] 0 40 80 120 0 5 10 15 20 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] -1 -0.8 -0.6 -0.4 -0.2 0 0 5 10 15 20 σP - σA [µb] d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] -160 -120 -80 -40 0 0 5 10 15 20 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] -120 -80 -40 0 0 5 10 15 20 d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] Eγ -Eγ thr [MeV] FIG. 11. (Color online) Same as in Fig. 10 but in the near-threshold region. The full consistent calculation with all the effects considered in this work is what we refer as IA+πN+NN. A. Helicity dependent cross sections First, we present results for the semiexclusive differential spin asymmetry d(σP−σA)/d with respect to circularly polarized photons, where P(A) stands for the deuteron spin oriented parallel (antiparallel) to the photon spin. In Fig. 3 we depict results for the pure IA calculation as well as with the inclusion of rescattering effects, for the individual pion photoproduction channels. We have chosen three laboratory photon energies covering the region from near threshold to the (1232) resonance. This spin asymmetry has been measured by the A2 and GDH Collaborations at MAMI [6]. The importance of rescattering in the neutral channel is clearly addressed when the full (IA +NN +πN) and the IA calculations are compared (respectively, solid and dotted curves in Fig. 3). We observe that the πN rescattering is not very important in both neutral and charged channels and that the NN rescattering plays a central role. Only at energies close to threshold, a small influence of πN-FSI is observed. This means, in particular, that the helicity asymmetry d(σP−σA)/d at photon energies near threshold is sensitive to the choice of the elementary reaction amplitude and the πN-rescattering. In charged channels, we find that FSI rescattering is quite small, and indeed almost negligible at pion backward angles and energies near the (1232) region. It is due to the fact that in charged-pion production the 3S1 contribution to the NN final state is forbidden. The results for the doubly polarized total cross sections in the case of the IA alone and with FSI effects are shown in Fig. 4(from threshold up to the (1232)-resonance region) and Fig. 5(near threshold region). Results are displayed in Figs. 4and 5as follows: (upper panel) total photoabsorption cross sections σPfor circularly polarized photons on a target with spin parallel to the photon spin; (middle panel) σA,the same for antiparallel spins of photon and deuteron target; (lower panel) spin asymmetry σP−σAfor the individual contributions of the different pion charge states to the γ d→πNN reaction. Several experiments to measure the deuteron spin asymmetry are presently underway [6]. In these experiments, deuterium serves as a neutron target and quasifree kinematics are preferred to minimize possible interaction effects. The left panels in Figs. 4and 5show the importance of rescattering effects in the neutral channel when the full calculation (solid curve) is compared to the IA one (dotted curve). The most important contribution to FSI comes from NN rescattering, whereas πN rescattering is much smaller and is significative only in σAat photon energies close to threshold in the neutral channel (see the left panel in Fig. 5). This is due to the nonorthogonality of the final-state plane wave in IA to the deuteron bound-state wave function. We 044005-8 HELICITY DEPENDENCE AND CONTRIBUTION TO THE . . . PHYSICAL REVIEW C 76, 044005 (2007) 0 0.2 0.4 0.6 0.8 1 0 60 120 180 E-asymmetry d(γ,π0 )np d(γ,π- )pp d(γ,π+ )nn Eγ =155 MeV Eγ =155 MeV Eγ =155 MeV Eγ =230 MeV Eγ =230 MeV Eγ =230 MeV Eγ =350 MeV Eγ =350 MeV Eγ =350 MeV θπ [deg] θπ [deg] θπ [deg] 0.94 0.96 0.98 1 0 60 120 180 ,, 0.94 0.96 0.98 1 0 60 120 180 ,, -1.4 -1 -0.6 -0.2 0.2 0.6 1 0 60 120 180 E-asymmetry ,, -0.4 0 0.4 0.8 0 60 120 180 ,, -0.4 0 0.4 0.8 0 60 120 180 ,, -1 -0.6 -0.2 0.2 0.6 1 0 60 120 180 E-asymmetry , -0.6 -0.2 0.2 0.6 1 0 60 120 180 , -0.6 -0.2 0.2 0.6 1 0 60 120 180 , FIG. 12. (Color online) The double polarization Easymmetry as a function of the pion angle in the laboratory frame at various photon lab energies for the separate channels of  d(γ,π)NN using different elementary pion photoproduction operators and including FSI effects. Curve conventions as in Fig. 9. also find that FSI effects are equally important for both σP and σA. FSI effects lead to an overall strong reduction of the spin asymmetry in the energy region of the (1232) resonance. σPis larger than σAbecause of the (1232) excitation. The cross sections σPand σAas well as the spin asymmetry σP−σApresent qualitative and quantitatively similar behaviors for both charged pion production channels. The FSI effects appear mainly in σAat photon lab energies close to threshold (Fig. 5). We find that the spin asymmetry σP−σA starts out negative due to the E0+multipole, which is dominant in the threshold region and has a strong positive contribution at higher energies due to the M1+multipole, which is dominant in the (1232)-resonance region. The effect of FSI is important for both the neutral and charged pion production channels in the threshold region. B. The double polarization Easymmetry As already mentioned in the Introduction, there is a great deal of interest in experiments [8] to determine the beamtarget double polarization Easymmetry for the γ d→πNN reaction channels. In connection with this study, we provide in Fig. 6a sample of Easymmetry as a function of the emission pion angle θπin the laboratory frame for the individual pion photoproduction channels. This asymmetry is given by E(θπ)=d(σA−σP)/d d(σA+σP)/d =d(σA−σP)/d 2(dσ0/d),(7) where dσP/d and dσA/d represent the spin parallel and antiparallel differential cross sections, respectively, and dσ0/d denotes the unpolarized differential cross section. The helicity dependent cross sections are well suited to verify the GDH sum rule, to do partial channel analysis, and to give contributions to the double polarization Easymmetry. This asymmetry appears as an interference between the amplitudes with different parity-exchange properties. The helicity Easymmetry (Fig. 6) has qualitatively a similar behavior for all pion photoproduction channels. The maximum value of Eequals unity at θπ=0◦and 180◦. The curves begin with unity and decrease as the pion angle increases until the minimum value is reached at θπ≃90◦. Then, it increases again to unity. The negative values in the Easymmetry come from higher positive contribution in dσP/d. Figure 6shows also that the helicity Easymmetry does not exhibit a strong effect of the rescattering contribution and, therefore, it is an excellent observable to test any weakness in the underlying elementary reaction model. We find that FSI are sizable in the neutral channel, whereas in charged channels they are noticeable only at the lowest energy. The E asymmetry proves to be sensitive to the choice of the input 044005-9