Review of Polarization Observables in Incoherent Pion Photoproduction on the Deuteron

Transcription

1 App. Math. Inf. Sci. 9, o. 2, (25) 527 Appied Mathematics & Information Sciences An Internationa Journa Review of Poarization Observabes in Incoherent Pion Photoproduction on the Deuteron Eed M. Darwish,2, Physics Department, Facuty of Science, Sohag University, Sohag 82524, Egypt 2 Physics Department, Facuty of Science, Taibah University, A-Madinah A-Munawarah, B.O. Box 32, Saudi Arabia Received: 2 May 24, Revised: 3 Aug. 24, Accepted: 5 Aug. 24 Pubished onine: Mar. 25 Abstract: Poarization observabes in incoherent pion photoproduction from the deuteron are investigated in the energy region from -threshod up to the (232)-resonance with incusion of a eading effects. Forma expressions for poarization observabes are derived and described by various beam, target and beam-target asymmetries for poarized photons and/or poarized deuterons. For the eementary pion photoproduction operator on the free nuceon, a reaistic effective Lagrangian approach is used which incudes seven nuceon resonances, in addition to Born and vector-meson exchange terms. The interactions in the fina two-body subsystems are taken from separabe representations of reaistic potentias. Resuts are given for the unpoarized cross sections, the douby poarized cross sections for parae and antiparae heicity states, the inear photon asymmetry, the doube poarization E-asymmetry and, the vector and tensor deuteron asymmetries for the d pp, d + nn, and d np channes. The contributions to the spin asymmetry and the Gerasimov-Dre-Hearn (GDH) integra from separate channes are evauated by expicit integration up to a photon ab-energy of 35 MeV. Effects of fina-state interaction are investigated and their roe in these observabes are found to be significant, speciay for production. The extracted resuts are compared with avaiabe experimenta data and predictions of other works, and a satisfactory agreement is obtained. The sensitivity of the d resuts to the eementary operator is aso investigated and a considerabe dependence is found. This indicates that it can serve as a fiter for different eementary operators. We expect that the resuts presented here may be usefu to interpret the recent measurements from the high-intensity and high duty-factor eectron acceerators MAMI, ELSA, Jefferson Lab, LEGS, and MAX-Lab. Keywords: Meson production, Photoproduction reactions, Few-Body Systems, Poarized beams, Poarized targets, Spin observabes, Sum rues, Fina-state interactions Introduction The study of pseudoscaar meson production in eectromagnetic reactions on ight nucei has become a very active fied of research in medium-energy nucear physics with respect to the study of hadron structure. For the foowing reasons the deuteron pays an outstanding roe besides the free nuceon. The first one is that the deuteron is the simpest nuceus on whose structure we have abundant information and a reiabe theoretica understanding, i.e., the structure of the deuteron is very we understood in comparison to heavier nucei. Furthermore, the sma binding energy of nuceons in the deuteron, which from the kinematica point of view provides the case of a neary free neutron target, aows one to compare the contributions of its constituents to the eectromagnetic and hadronic reactions to those from free nuceons in order to estimate interaction effects. Meson photo- and eectroproduction on ight nucei is primariy motivated by the foowing possibiities: (i) study of the eementary neutron ampitude in the absence of a free neutron target, (ii) investigation of medium effects, i.e., possibe changes of the production operator in the presence of other nuceons, (iii) it provides an interesting means to study nucear structure, and (iv) it gives information on pion production on off-she nuceon, as we as on the very important -interaction in a nucear medium. The major reason for studying poarization phenomena ies in the fact that ony the use of poarization degrees of freedom aows one to obtain compete information on a possibe reaction matrix Corresponding author e-mai: eeddarwish@yahoo.com c 25 SP atura Sciences Pubishing Cor.

2 528 E. M. Darwish: Review of Poarization Observabes in Incoherent Pion... eements. Without poarization, the cross section is given by the incoherent sum of squares of the reaction matrix eements ony. Thus, sma ampitudes are masked by the dominant ones. On the other hand, sma ampitudes very often contain interesting information on subte dynamica effects. This is the pace where poarization observabes enter, because such observabes in genera contain interference terms of the various matrix eements in different ways. Thus, a sma ampitude may be consideraby ampified by the interference with a dominant matrix eement. The eariest cacuations for pion photoproduction from the deuteron were performed using the impuse approximation (IA) [, 2]. Approximate treatments of fina-state interaction (FSI) effects within a diagrammatic approach have been reported in Refs. [3, 4, 5]. Photoproduction of pions from the deuteron has been investigated with the spectator nuceon mode [6] ignoring a kinds of FSI and two-body processes. The -FSI is considered in Ref. [7] and a good agreement with experiment was obtained. The infuence of - and -rescattering on the unpoarized cross sections is investigated in Refs. [8, 9], whereas the roe of the -FSI in pion photoproduction from the deuteron is investigated in []. It has been shown that fu cacuations with the off-she ampitudes of - and -FSI are necessary to obtain a quantitative description of the cross sections. Most of cacuations have treated ony unpoarized observabes, ike the differentia and tota cross sections. These cross sections provide information ony regarding the sum of the absoute squares of the ampitudes, whereas the poarization observabes aow extraction of more information. Observabes with a poarized photon beam and/or poarized deuteron target have not been throughy investigated. Recent years have witnessed an increasing interest in theoretica research on meson eectromagnetic production incuding poarization observabes. This interest is party due to the new generation of high-intensity and high duty-cyce eectron acceerators, such as MAMI at Mainz and ELSA at Bonn (Germany), JLab at ewport ews (USA), and MAX-Lab at Lund (Sweden) as we as aser backscattering faciities such as GRAAL at Grenobe (France) and LEGS at Brookhaven (USA) - for an experimenta overview see Refs. [, 2, 3, 4]. With the deveopment of these new faciities, it is now possibe to obtain accurate data for meson eectromagnetic production, incuding singe and doube spin-dependent observabes. Poarization observabes for -photoproduction on the deuteron via the reaction d(, )pp have been studied within a diagrammatic approach [5]. In [8, 9], the energy dependence of the three charge states of the pion for incoherent pion photoproduction on the deuteron in the (232)-resonance region has been investigated. Resuts for differentia and tota cross sections and the beam-target spin asymmetry which determines the Gerasimov-Dre-Hearn (GDH) sum rue were given. In Refs. [6, 7, 8], we have investigated various singeand doube-poarization asymmetries in incoherent pion photoproduction from the deuteron without any kind of FSI effects. We found that interference of Born terms and the (232)-contribution pays a significant roe in the cacuations. The vector target asymmetry T has been found to be very sensitive to this interference. Furthermore, our resuts for the inear photon asymmetry [8] have been compared with the preiminary experimenta data from the LEGS Spin Coaboration [4] and major discrepancies were evident. Since a strong infuence of the FSI on the unpoarized cross sections was found in [8,9], one might expect that the LEGS data can be understood in terms of the FSI and possiby two-body effects. Therefore, we have investigated the infuence of -FSI on the heicity structure (Ref. [9]) and the inear photon asymmetry (Ref. [2]) of the d pp reaction. The differentia poarized cross-section difference for the parae and antiparae heicity states has been predicted and compared with recent experimenta data from MAMI (Mainz/Pavia) [2]. It has been shown that the effect of -rescattering is much ess important in the poarized differentia cross-section difference than in the unpoarized one. It has been aso found in [2], that -FSI is quite important and eads to a better agreement with existing experimenta data. In addition, we have investigated in Ref. [2] the roe of the -rescattering effect on severa singe- and doube-poarization observabes of photon and deuteron target in -photoproduction from the deuteron with poarized photon beam and/or oriented deuteron target. The understanding of this mechanism is of great importance to understand the basic interaction. Based on the effective Lagrangian mode [6], the heicity dependence of the reaction channes is investigated in Ref. [22] and compared with recent experimenta data from the GDH Coaboration, and a good agreement was obtained. The work in [9] was improved by Arenhöve et a. [23,24] and by Levchuk et a. [25], where a better eementary production operators provided by MAID [26] and SAID [27], respectivey, were empoyed and FSI effects were considered. Expicit evauation of the deuteron spin asymmetry and the associated GDH sum rue were presented in said work. The A2 and GDH Coaborations at MAMI have undertaken a joint effort to verify experimentay the GDH sum rue [28], measuring the difference between the heicity components in both the tota [σ P σ A ] and the differentia [d(σ P σ A )/dω] photoabsorption cross sections, where σ P,A stands for the parae and antiparae spin orientations of the photon and target deuteron. In these experiments, circuary poarized photons are scattered on ongitudinay poarized deuterons to determine the doube poarization asymmetry in a arge kinematica range [2]. These heicity dependent cross sections provide vauabe c 25 SP atura Sciences Pubishing Cor.

3 App. Math. Inf. Sci. 9, o. 2, (25) / information on the nuceon spin structure and aow to extract information on the neutron. The knowedge of these cross sections is aso required to test the vaidity of the GDH sum rue on the deuteron and the neutron as we as to expore which are the dominant contributions to the GDH integra. These new measurements wi make possibe to test experimentay the behavior of the GDH integra. However, in order to expore the contributions of the resonances, the tota photoabsorption cross section does not provide enough information. Fixing degrees of freedom by seection of particuar partia reaction channes aows to obtain more detaied information. This is possibe with doube poarization observabes of the differentia cross section. The present focus of the LEGS Coaboration is the measurement of doube poarization spin observabes in photoreactions on the proton, the neutron, and the deuteron. For instance, the beam-target doube poarization E-asymmetry of the d np reaction at photon ab-energy E = 349±5 MeV, has been recenty measured [4]. This asymmetry wi certainy provide an important test, which, indirecty, aows to check our knowedge on the process of pion production from the neutron. To the best of our knowedge, there is no cacuation in the iterature for the doube poarization E-asymmetry of the d reaction channes. Theoretica predictions for the unpoarized cross sections and poarization observabes of the processes d pp, d + nn, and d np, in the energy range from -threshod up to the (232)-resonance, were performed in Refs. [9,2,2] and extended in Refs. [29, 3] using as eementary reaction a reaistic effective Lagrangian approach (ELA) from [3]. This eementary reaction mode provides a reiabe description of the -threshod region. umerica resuts for unpoarized cross sections and various poarization observabes have been presented [29, 3] where in addition to the pure IA and -FSI, the -rescattering is aso incuded. Extension of our previous works [7, 8, 9, 2, 2, 22] to the -threshod region was given in order to understand the dynamics of the pion photoproduction ampitude near -threshod. It was found in [29, 3] that the resuts for unpoarized cross sections and spin asymmetries are strongy dependent on the eementary operator. The cacuations presented in this review are of high interest in view of the extensive recent poarization measurements from A2 and GDH@MAMI [2] and LEGS@BL [4] Coaborations. A theoretica understanding of these data on the deuteron target wi provide information about the pion photoproduction from the neutron reaction, which is not we known yet, but is required for a compete understanding of -excitations in the pion photoproduction process. This review is structured as foows: in the next section we briefy outine the eectromagnetic and hadronic two-body eementary reactions which we incude in our treatment of pion photoproduction from the deuteron. In section 3, a brief review of the framework for the reaction d(, ) in which the transition matrix eements are cacuated [8] is presented. The main resuts for the d pp, d + nn, and d np reactions together with a comparison to experimenta data and other theoretica predictions are presented in section 4. In addition, the sensitivity of our resuts to the eementary pion photoproduction operator is discussed here. Finay, we cose with some concusions and an outook in section 5. Throughout this work we use natura units h=c=. 2 Eementary eectromagnetic and hadronic reactions In this section, we wi briefy describe the necessary ingredients for the various eementary reactions which govern the process of incoherent pion photoproduction from the deuteron. These are the pion photoproduction from free nuceons, which pays the centra part in the reaction, and hadronic two-body scattering reactions, namey nuceon-nuceon ( ) and pion-nuceon ( ) scattering, which constitute the rescattering effects in the fina state. The eementary reactions are cassified as two-body reactions. The genera form of these reactions is given by a(p a )+b(p b ) c(p c )+d(p d ), () where p i = (E i, p i ) denotes the four-momentum of partice i with i {a,b,c,d}. Partices a and b stand for a photon and a nuceon in the case of pion photoproduction from free nuceons, a pair of baryons in the case of scattering, or a meson and a baryon in the case of scattering. Corresponding assignments stand for the fina partices c and d. Foowing the conventions of Bjorken and Dre [32] the genera form for the differentia cross section of a two-partice reaction in the center of mass (c.m.) system is given by dσ = E a E b E c E d dω c 4 2 W 2 F a F b F c F d µ d µ c µ b µ a M f i p c p a s µ d µ c µ b µ a (p d,p c,p b,p a ) with M f i µ d µ c µ b µ a the reaction matrix, µ i denotes the spin projection of partice i, and F i is a factor arising from the covariant normaization of the states and its form depends on whether the partice is a boson ( F i = 2E i ) or a fermion ( F i = E i /m i ), where E i and m i are its energy and mass, respectivey. The factor s = (2s a + )(2s b + ) takes into account the averaging over the initia spin states, where s a and s b denote the spins of the incoming partices a and b, respectivey. If partice a is a rea photon, then s A = /2 2 c 25 SP atura Sciences Pubishing Cor.

4 53 E. M. Darwish: Review of Poarization Observabes in Incoherent Pion... because the rea photon is a boson with two poarization states. A momenta are functions of the invariant mass of the two-body system W, i.e. p i = p i (W). For the scattering processes, it is more convenient to use non-covariant normaization of the states and to switch to a couped spin representation repacing the M -matrix by the T -matrix via M f i µ d µ c µ b µ a (p d,p c,p b,p a )=(2) 3 F a F b F c F d C s cs d S as b S S µ S Sµ µ c µ d µ S Cs µ a µ b µ S T f i S µ S Sµ (p d,p c,p b,p a ), S S with C s cs d S µ c µ d µ S the appropriate Cebsch-Gordan coefficient. The partia wave representation of the T -matrix reads T f i S µ S Sµ S (W,p,p)= J where we have introduced (2) X J S µ S Sµ S ( ˆp, ˆp)T f i J (W, p, p), X J S µ S Sµ ( ˆp, ˆp)= S C S J µ µ µ µ µ S µ CSJ J µ µ S µ J J Y µ ( ˆp )Y µ ( ˆp). Here and J denote, respectivey, the orbita and tota anguar momenta of the system. Y µ ( ˆp) is the spherica harmonic, p and p are the fina and initia reative momenta, respectivey, and ˆp=(θ p,φ p ). The partia wave T-matrix is obtained as a soution of the Lippmann-Schwinger (LS) equation [33] T f i J (W, p, p)= V f i J (p, p) + n 2m n (W, p d p n (p n) 2 V f n J (p, p n)t ni J q 2 n (p n) 2 + iε n, p), where V f i J (p, p) is the interaction potentia between the partices and n abes possibe intermediate two-partice configurations with tota mass M n, reduced mass m n, and with reative momentum q n = 2m n (W M n ) in the c.m. system. ow we wi briefy review the different eementary processes in some detai. (3) (4) (5) T = (D) (A) (E) ρ,ω (B) * (F) (C) (G) Fig. : Feynman diagrams for pion photoproduction from free nuceons. Born terms: (A) direct nuceon poe or s-channe, (B) crossed nuceon poe or u-channe, (C) pion in fight or t-channe, and (D) Kro-Rudermann contact term; (E) vector meson exchange; resonance excitations: (F) direct or s-channe and (G) crossed or u-channe. approach in the energy region of the mass of the nuceon. It incudes Born terms (diagrams (A)-(D) in Fig. ()), vector-meson exchanges (ρ and ω, diagram (E) in Fig. ()), and a the four star resonances in Partice Data Group (PDG) [34] up to.7 GeV and up to spin-3/2: (232), (44), (52), (535), (62), (65), and (7) (diagrams (F) and (G) in Fig. ()). In the pion photoproduction mode from free nuceons [3] it was assumed that fina-state interaction (FSI) factorize and can be incuded through the distortion of the fina-state wave function (pion-nuceon rescattering). -FSI was incuded by adding a phase δ FSI to the eectromagnetic mutipoes. This phase is set so that the tota phase of the mutipoe matches the tota phase of the energy dependent soution of SAID [27]. In this way it was possibe to isoate the contribution of the bare diagrams to the physica observabes. The parameters of the resonances were extracted from data fitting the eectromagnetic mutipoes from the energy independent soution of SAID [27] appying a modern optimization technique based upon genetic agorithms combined with gradient based routines [3, 35] which provides reiabe vaues for the parameters of the nuceon resonances. Once the bare properties of the nuceon resonances have been extracted from data, their contribution to more compex probems can be cacuated. * 2. Pion photoproduction from free nuceons To study the d processes we first need a mode for the eementary reaction. The mode we use for this eementary process is the one eaborated in Ref. [3], which has been appied successfuy from -threshod up to GeV of photon energy in the aboratory reference system. The mode is based upon an effective Lagrangian approach (ELA) which from a theoretica point of view is a very appeaing, reiabe, and formay we-estabished 2.2 uceon-nuceon scattering For a given potentia V, the T -matrix of the -scattering is obtained from the LS equation which in partia wave decomposition reads T (p, p;e)= V (p, p) + M dkk 2 V (p,k) M E k 2 + iε T (k, p;e) (6) c 25 SP atura Sciences Pubishing Cor.

5 App. Math. Inf. Sci. 9, o. 2, (25) / 53 for singe channes. For the couped channes it reads T (p, p;e)= V (p, p) + M dkk 2 V (p,k) M E k 2 + iε T (k, p;e), (7) where p, k and p are the reative three-momenta of the two interacting nuceons in the initia, intermediate and fina state, respectivey. E denotes the energy of the two interacting nuceons in the c. m. frame and is given by E = p 2 /M, where p is the c. m. on-she momentum. Using the identity x x + iε = P iδ(x x ), (8) x x where P denotes the principa vaue integra, we get the T -matrix eements in partia wave decomposition T (p, p;e)= V (p, p) +P M dkk 2 V (p,k) p 2 T (k, p;e) k2 2 im p V (p, p )T (p, p;e) for singe channes and T (p, p;e)= V (p, p) + P =J± M dkk 2 V (p,k) p 2 T k2 (k, p;e) 2 im p V (p, p )T (p, p;e) (9) () for couped channes. These one-dimensiona integra equations can be soved numericay by means of a matrix inversion agorithm which is expained in Ref. [36]. For the interaction in the -subsystem, the separabe representations of the reaistic Paris potentia from [37] are used in this work. These separabe interactions represent a good approximation of the on-she as we as off-she properties of the origina Paris potentia and show a good fit to the modern data base. Thus, the use of such a reaistic potentia is good enough for our purpose. 2.3 Pion-nuceon scattering Anaogous to the case of -scattering, the T -matrix for scattering is obtained from the LS equation T(p, p;e)= V(p, p) + dkk 2 V(p,k) G (k,e) T(k, p;e), () with the propagator G (k,e)= E E (k) E (k)+iε, (2) where p, k and p are the reative momentum in the initia, intermediate and fina state, respectivey; and p p, k k and p p. E = M 2 + p2 + m 2 + p2 denotes the tota coision energy with on-she momentum p in the c.m. frame. For our cacuations, reativistic kinematics for both pion and nuceon are used, thus E (k)= m 2 + k2, E (k)= M 2 + k2. (3) The partia wave T -matrix for scattering is then given by where T(p, p;e)= V(p, p) + dkk 2 V(p G(k),k) p 2 k2 + iε T(k, p;e), (4) G(k)= [E (p )+E (k)][e (p )+E (k)] [E (p )+E (k)+e (p )+E (k)]. (5) To transform this equation into a principa vaue integra equation using identity (8) to get the matrix eements in partia wave decomposition as T(p, p;e)= V(p, p) +P dkk 2 V(p,k) G(k) p 2 T(k, p;e) k2 2 i p G(p )V(p, p )T(p, p;e). (6) This one-dimensiona integra equation can be soved numericay for a given potentia mode V(p, p) by using the matrix inversion method [36]. For the potentia, the reaistic separabe representation of interaction of ozawa et a. [38] is used. This mode is consistent with the existing unitary description of the system and treats the interaction dynamicay, with a S-, P- and D-wave phase shifts being we reproduced beow 5 MeV. 3 Theoretica treatment of the d reaction 3. Kinematics and cross section As a starting point, we wi first consider the formaism for incoherent singe pion photoproduction from the deuteron (k,ε µ )+d(d) (q)+ (p )+ 2 (p 2 ), (7) c 25 SP atura Sciences Pubishing Cor.

6 532 E. M. Darwish: Review of Poarization Observabes in Incoherent Pion... y scattering pane d 2 θ nuceon pane Fig. 2: The kinematics of incoherent pion photoproduction from the deuteron in the aboratory frame. where the four-momenta of the participating partices are indicated in the parentheses. The poarization vector of the photon is denoted by ε µ (µ =±). The genera expression of the fivefod differentia cross section is given by [32] d 5 σ = δ 4 (k+d q p p 2 )M 2 d3 qd 3 p d 3 p 2 96(2) 5 v v d E E d E E 2 ω q sm µm d T sm µmd (q,p,p 2,d,k) 2, z (8) d d () t () t (a) (c) d T Ν () t 2 2 G () t + d T + ΝΝ 2 (b) + Fig. 3: Diagrammatic representation of the d ampitude incuding rescattering contributions in the two-body subsystems and negecting a contributions of two-body meson-exchange currents. Diagram (a): impuse approximation (IA); (b), (c) and (d) driving terms from - and -rescattering, respectivey. Diagrams when the eementary pion photoproduction operator acts on nuceon 2 are not shown in the figure but are incuded in the cacuations. In the cacuations, each diagram shown in the figure goes accompanied by the diagram obtained by the exchange 2. d ΝΝ t () 2 = (d) T Ν 2 where the definition of a kinematica variabes and quantum numbers is given in Ref. [8]. For the cacuation of the cross sections, the deuteron rest frame is utiized. We have chosen a right-handed coordinate system where the z-axis is defined by the photon momentum k and the y-axis by k q, where q is the pion momentum. The kinematica situation is shown in Fig. (2). The scattering pane is defined by the momenta of photon k and pion q whereas the momenta of the two outgoing nuceons p and p 2 define the nuceon pane. As independent variabes for the characterization of the fina state, the outgoing pion momentum q = (q,θ,φ ) is chosen. Furthermore, the spherica anges Ω p = (θ p,φ p ) of the reative momentum p = (p p 2 )/2 = (p,ω p ) of the two outgoing nuceons. The semi-incusive differentia cross section, where ony the fina pion is detected without anayzing its energy, is obtained from (8) d 2 σ qmax = dq dω dω p ρ s 6 sm µm d Tsm µmd (q, p, k) 2, (9) where the maximum pion momentum q max is determined by the kinematics and ρ s is a phase space factor. 3.2 The photoproduction ampitude A observabes are determined by the photoproduction ampitude T sm µmd of the eectromagnetic pion production current J between the initia deuteron and the fina states. In a genera frame, it is given by T sm µmd (q, Ω, Ω p )= ( ) q, p; sm ε µ J () d; m d. (2) Introducing a partia wave decomposition, one finds for the scattering matrix the reation T sm µmd (Ω, Ω p )=e i(µ+m d m)φ t sm µmd (θ, θ p, φ p ), (2) where φ p = φ p φ. The sma t-matrix depends ony on θ, θ p, and the reative azimutha ange φ p. The eements of this t-matrix are the basic quantities which determine unpoarized differentia cross section and spin asymmetries. If parity is conserved, the foowing symmetry reation hods t s m µ md (θ, θ p, φ p )= ( ) s+m+µ+m d t sm µmd (θ, θ p, φ p ) (22) which eads to a corresponding reation for the T sm µmd - matrix T s m µ md (θ p, φ p, θ, φ )= ( ) s+m+µ+m d T sm µmd (θ p, φ p, θ, φ ). (23) Figure (3) shows the diagrammatic representation of the scattering matrix, where the eementary pion photoproduction operator acts on nuceon ony. A possibe diagrams when the production operator acts on c 25 SP atura Sciences Pubishing Cor.

7 App. Math. Inf. Sci. 9, o. 2, (25) / nuceon 2 are not shown in this figure but are incuded in the cacuations. In addition, a contributions of two-body currents are not dispayed in Fig. (3). In principe, the fu treatment of a interaction effects requires a fu unitary three-body cacuation. In the present work, however, we wi restrict ourseves to the incusion of compete rescattering in the various two-body subsystems of the fina state (diagrams (b), (c), and (d) in Fig. (3)). For the cacuation of the T sm µmd -matrix we start from the IA (diagram (a) in Fig. (3)) to which the contributions from - and -rescattering (diagrams (b), (c), and (d), respectivey) are added. Then, the T sm µmd -matrix is given by the sum T sm µmd = T IA sm µm d + T sm µm d + T sm µm d. (24) In the foowing we briefy review these terms in some detai The impuse approximation For the IA contribution (diagram (a) in Fig. (3)), which describes the production on one nuceon whie the other acts as a spectator, one has Tsm IA µm d = q,p, sm [t () + t (2) ] m d = [ sm p t () (W ) p 2 φ m m d (p 2 ) m m ( 2)], (25) where t () and t (2) denote the eementary pion photoproduction operator from nuceon and 2, respectivey, W /2 the invariant energy of the /2 system, p /2 = (k q)/2 ± p, and φ mmd (p) is reated to the interna deuteron wave function in momentum space by φ mmd (p)= L=,2 m L i L (Lm L m m d )u L (p)y LmL ( ˆp), (26) where u (p) and u 2 (p) are the S and D components of the deuteron wave function, for which the reaistic Paris potentia mode [39] is used The -rescattering The transition matrix eement (see diagram (b) in Fig. (3)) has the form Tsmµm d (k,q,p,p 2 )= d 3 p E E 2 m E E 2,tµ R smm (W,p,p ) M (27) p 2 p 2 + iε Tsm IA µm d (k,q,p,p 2 ). R,tµ The conventiona -scattering matrix smm is introduced with respect to noncovarianty normaized states. It is expanded in terms of the partia wave contributions T,tµ Js as foows R,tµ smm (W,p,p )= J F,Js mm ( ˆp, ˆp,tµ )TJs (W, p, p ), (28) where the purey anguar function F,Js mm ( ˆp, ˆp ) is defined by F,Js mm ( ˆp, ˆp )= M m m C sj m mm C sj m m M Y m ( ˆp )Y m ( ˆp ). (29) The necessary haf-off-she -scattering matrix T,tµ Js was obtained from separabe representation of a reaistic -interaction [37]. Expicity, a S, P, and D waves were incuded in the -scattering matrix The -rescattering For the contribution of -rescattering, two terms are considered. The diagrammatic representation of these terms are given by diagrams (c) and (d) in Fig. (3). The transition matrix eement of diagram (c) has the form T (c) smµm d (k,q,p,p 2 )= 2 d 3 p ωq E 2 2 α ω q E 2 [ J G (+) m 2 m 2 µ 2 µ 2 t µ t µ Cαα (m 2,m 2, µ 2, µ 2 ) F Jm 2 m ( ˆp, ˆp, t µ )T 2 J (W (p 2 ), p, p ) (E d,q,p,p 2 ] ( ) s+t (p p 2 ), with and )T IA s m µ m d (k,q,p,p 2 ) t µ Cαα (m 2,m 2, µ 2, µ 2 )=C 2 t µµ 2 µ C 2 t µ µ 2 µ 2 C 2 s m m m 2 mc 2 2 s m m C 2 2 t 2 m µ F Jm 2 m 2( ˆp, ˆp )= µ µ 2 µ C 2 2 t µ µ 2, µ C 2 J m m M m 2 m M C 2 J m 2 m M (3) (3) Y m ( ˆp )Y m ( ˆp ), (32) c 25 SP atura Sciences Pubishing Cor.

8 534 E. M. Darwish: Review of Poarization Observabes in Incoherent Pion... where α =(smt µ) and t denotes the tota isospin of the interacting pion and nuceon. The reative momentum of the fina (initia) -subsystem is given by p = M q m p 2 M + m, p = M q m p 2 M + m, (33) where q = q+p 2 p 2. Anaogous to diagram (c), the matrix eement of diagram (d) is given by T (d) smµm d (k,q,p,p 2 )= 2 d 3 p ωq E t µ α ω q E Cαα (m,m, µ, µ ) m m µ µ t µ [ F Jm m ˆ p (, ˆ p ) J, t µ TJ (W (p ), p, p ) G (+) (E d,q,p 2,p ) ] Ts IA m µ m d (k,q,p 2,p ) ( ) s+t (p p 2 ), (34) where the reative momentum of the fina (initia) -subsystem is given in this case by p = M q m p, p M + m = M q m p, (35) M + m with q = k p 2 p. Then the contribution from -rescattering of both diagrams (c) and (d) in Fig. (3) is given by T smµm d = T (c) smµm d + T (d) smµm d. (36), t µ The haf-off-she partia wave ampitudes TJ in Eqs. (3) and (34) were found for the S, P, and D partia waves by numerica soution of the LS-equation. In the present cacuations, these ampitudes are obtained from the separabe energy-dependent potentia presented in [38]. 3.3 Definition of poarization observabes The cross section for arbitrary poarized photons and initia deuterons can be computed for a given t-matrix by appying the density matrix formaism simiar to that given for deuteron photodisintegration [4]. The most genera expression for a possibe poarization observabes is given by O = αα dω p ρ s t s m µ m d Ω s m sm t sm µmd ρ µµ ρ d m d m, (37) d where ρ µµ and ρ d m d m denote the density matrices of d initia photon poarization and deuteron orientation, respectivey, Ω s m sm is an operator associated with the observabe, which acts in the two-nuceon spin space and ρ s is a phase space factor. For further detais we refer to Refs. [4,4]. As shown in Refs. [6, 7, 8, 23] a possibe poarization observabes for the pion photoproduction reaction from the deuteron can be expressed in terms of the quantities V µ 2I+ IM (q, θ ) = d Ω p ρ s (E,q,Ω,Ω p ) 3 ( ) ( ) m d I m m d m d m d M d t smµm θ sm d(q,, θ p, φ p ) t smµmd (q, θ, θ p, φ p ), (38) and 2I+ W IM (q, θ ) = d Ω p ρ s (E,q,Ω,Ω p ) 3 ( ) ( ) m d I m m d m d m d M d tsm m θ sm d(q,, θ p, φ p ) t smmd (q, θ, θ p, φ p ), (39) where we use the convention of Edmonds [42] for the Wigner 3 j-symbos. The unpoarized cross section is given by d 3 σ dqdω = V (q, θ ). (4) The photon asymmetry for ineary poarized photons is given by d3 σ Σ = W. (4) dqdω The vector and tensor target asymmetries are given by T IM d 3 σ dqdω = (2 δ M )Re[i I V IM (q, θ )]. (42) The photon and target doube poarization asymmetries are given by (i) Circuar asymmetries: T c IM d 3 σ dqdω = (2 δ M )Im[i I V IM(q, θ )]. (43) (ii) Longitudina asymmetries: T IM d 3 σ dqdω = i I W IM (q, θ ). (44) c 25 SP atura Sciences Pubishing Cor.

9 App. Math. Inf. Sci. 9, o. 2, (25) / Resuts and discussion The discussion of our resuts is divided into four parts as foows: First, we wi discuss various poarization observabes for incoherent pion photoproduction from the deuteron in the (232)-resonance region. umerica resuts for the three isospin channes of the d(,) reaction wi be given. The d scattering ampitude is given as a inear combination of the on-she matrix eements of pion photoproduction on the two nuceons (diagram (a) in Fig. (3)). The resuts of this part are obtained with the deuteron wave function of the Paris potentia [39] and the eementary pion production operator on the free nuceon of Schmidt et a. [6] which describes we the reaction channes (see Ref. [22] in which we have presented resuts for the heicity dependence of the reaction channes in comparison with the recent experimenta data from the GDH Coaboration [43].). In the second part, the infuence of fina-state interaction (diagram (b) in Fig. (3)) in various poarization observabes for the reaction d pp wi be studied. The understanding of this mechanism is of great importance to understand the basic interaction. The second point of interest is to anayze the preiminary experimenta data for the differentia poarized cross-section difference for parae and antiparae heicity states from [2] and for the inear photon asymmetry from LEGS [4] in order to keep up with the deveopment of the experimenta data. For nuceon-nuceon scattering in the -subsystem, we use a specific cass of separabe potentias [37] which historicay have payed and sti pay a major roe in the deveopment of few-body physics and aso fit the phase shift data for -scattering. This separabe mode is most widey used in the case of the system (see, for exampe, Ref. [44] and references therein). In the third part, we wi present and discuss resuts for tota cross section and various poarization observabes when in addition to the pure IA (diagram (a) in Fig. (3)) and -FSI (diagram (b) in Fig. (3)), the -rescattering (diagrams (c) and (d) in Fig. (3)) is aso incuded. Resuts for the three isospin channes of the d(,) reaction wi be given in the energy region from -threshod up to the (232)-resonance. Extension of our previous works [7,8,9,2,2] to the -threshod region is presented in order to understand the dynamics of the pion photoproduction ampitude near -threshod. For this purpose, a more reaistic eementary ampitude from Ref. [3] is used which aows to provide a more reaistic description of the -threshod region. For both - and -scattering in the two-body subsystems, we use separabe representations of reaistic potentias from T 2 T 2 T 2 Refs. [37,?]. The tota cross section, the douby poarized cross sections for the parae and antiparae heicity states, the spin response of the deuteron, i.e., the asymmetry of the tota photoabsorption cross section with respect to parae and antiparae spins of photon and deuteron, the inear photon asymmetry, target asymmetries, the beam-target doube poarization E-asymmetry, and the deuteron GDH integra wi be discussed in this part. The ast part is devoted to study the sensitivity of the resuts for tota cross section and poarization observabes to the eementary pion photoproduction operator on the free nuceon pp + nn np.3.4 ω =285 MeV ω =33 MeV T 2 T ω =285 MeV ω =33 MeV T 2 T ω =285 MeV ω =33 MeV ω =45 MeV ω =45 MeV ω =45 MeV.3. T 2 -. Fig. 4: The doube poarization asymmetry T2 of d(,) channes as a function of pion ange θ at various photon abenergies. The soid curves show the resuts of the IA-cacuation whie the dotted curves represent the resuts when ony the (232)-resonance is taken into account. The eft, midde and right panes represent the resuts for d pp, + nn and np, respectivey. 4. The impuse approximation We start the discussion with a sampe of resuts for the ongitudina doube poarization asymmetry T2 which are T c 25 SP atura Sciences Pubishing Cor.

10 536 E. M. Darwish: Review of Poarization Observabes in Incoherent Pion... potted in Fig. (4) as a function of emission pion ange θ at various photon ab-energies for a the three charge states of the pion for the reaction d (see aso Ref. [7]). For neutra pion production (see the right panes of Fig. (4)), we see that T2 has aways positive vaues for a photon ab-energies and emission pion anges. For energies beow and above the (232)-resonance, the doube poarization asymmetry T2 shows sensitivity on the Born terms, in particuar for anges between 6 and 2. Furthermore, it is apparent that at extreme forward and backward emission pion anges the asymmetry T2 is very sma in comparison to other anges. At 45 MeV the dominant contribution comes from the resonance term. For the cacuations of charged pion production channes (see the eft and midde panes of Fig. (4)), we see that T2 has negative vaues at forward pion anges which is not the case at backward anges. It is aso noticeabe that the contributions of Born terms are arge, in particuar at energies above the resonance region. At extreme backward anges, we see that T2 has sma positive vaues. T ω =25 MeV ω =36 MeV ω =5 MeV T T Fig. 5: Tensor target asymmetry T 2 of d(,) channes as a function of emission pion ange at various photon ab-energies. The soid, dashed, and dotted curves correspond to d pp, + nn, and np, respectivey. In Fig. (5) we present the resuts for the tensor target asymmetry T 2 as a function of θ [8]. Predictions for the three isospin channes of d(, ) at various photon ab-energies are given. For the reaction d at forward and backward emission pion anges, the asymmetry T 2 aows one to draw specific concusions about detais of the reaction mechanism. Comparing with the resuts for inear photon and vector target asymmetries [7, 8] we found that for charged pion production channes the asymmetry T 2 has reativey arge positive vaues at pion forward anges (at θ < 3 ) whie sma negative ones are found when θ changes from 3 to 8. For neutra pion production channe, we see that T 2 has negative vaues at forward anges and positive ones at backward anges. Ony at energies above the -region we observe sma negative vaues at extreme backward anges. The ongitudina doube-poarization asymmetry T 2+2 is potted in Fig. (6) as a function of pion ange for a the T 2+2 T 2+2 T ω =25 MeV ω =29 MeV ω =3 MeV T 2+2 T T ω =34 MeV ω =36 MeV ω =39 MeV T 2+2 T ω =42 MeV ω =47 MeV ω =5 MeV Fig. 6: The doube-poarization asymmetry T2+2 of d(,) at various photon ab-energies. otation as in Fig. (5). Σ ω =27 MeV T 2+2 ω =33 MeV Fig. 7: Photon beam asymmetry Σ for d(, )pp reaction in comparison with the preiminary experimenta data from LEGS@BL Coaboration [4]. three charge states of the pion for the reaction d at various photon ab-energies (see aso Ref. [8]). The soid, dashed, and dotted curves correspond to d pp, + nn, and np, respectivey. In genera, one readiy notes that the ongitudina asymmetry T2+2 has negative vaues. For neutra and charged pion production channes, it is apparent that the asymmetry T2+2 has quaitativey the same behavior. We see aso that the vaues of T2+2 for channe are greater (in absoute vaue smaer) than its vaues for ± channes, in particuar at θ =. Furthermore, we noticed that the c 25 SP atura Sciences Pubishing Cor.

11 App. Math. Inf. Sci. 9, o. 2, (25) / (dσ/dω ) P - (dσ/dω ) A [µb/sr] (dσ/dω ) P - (dσ/dω ) A [µb/sr] (dσ/dω ) P - (dσ/dω ) A [µb/sr] ab ω =34 MeV ab ω =42 MeV ab ω =48 MeV ab (dσ/dω ) P - (dσ/dω ) A [µb/sr] (dσ/dω ) P - (dσ/dω ) A [µb/sr] (dσ/dω ) P - (dσ/dω ) A [µb/sr] ab ω =364 MeV ab ω =435 MeV ab ω =54 MeV ab θ [deg] Fig. 8: The differentia poarized cross-section difference (dσ/dω ) P (dσ/dω ) A for d pp as a function of θ ab in comparison with experimenta data from Ref. [2] at different photon ab-energies. Dashed curves: IA; soid curves: IA+rescattering; dotted curves: predictions for n p reaction. Σ ω =27 MeV ω =33 MeV Fig. 9: Photon asymmetry Σ for the reaction d(, )pp at two different photon ab-energies in comparison with the preiminary data from LEGS [4]. otation of curves: dashed: IA; soid: IA+-rescattering. vaues of T 2+2 in the case of -production are insensitive to the photon energy and/or pion ange, which is not the case for ± -production channes. It is very interesting to examine these asymmetries experimentay. T θ =.2 θ =9.2 θ = ω [MeV] ω [MeV] ω [MeV] Fig. : The doube-poarization asymmetry T 2 for d(, )pp as a function of photon ab-energy at fixed vaues of θ. otation as in Fig. (9). The photon asymmetry Σ for ineary poarized photons is shown in Fig. (7) (see aso Ref. [8]) in comparison with experimenta data. In view of the fact that data for + and production channes are not avaiabe, we concentrate the discussion on production, for which we have taken the preiminary experimenta data from the LEGS@BL Spin Coaboration [4]. In agreement with these preiminary data, one can see that the predictions in the pure IA can hardy provide a reasonabe description of the data. Major discrepancies are evident which very ikey come from the negect of FSI effects. This means that the simpe spectator approach cannot describe the experimenta data. 4.2 Infuence of fina-state interaction Here, we discuss the infuence of the -FSI effect on poarization observabes for the reaction d pp ony. We begin the discussion by presenting the resuts for the differentia poarized cross-section difference for parae (dσ/dω ) P and antiparae (dσ/dω ) A heicity states in pure IA and with -rescattering, as shown in Fig. (8) as a function of the emission pion ange in the aboratory frame at various photon ab-energies (see aso Ref. [9]). It is readiy seen that -rescattering - the difference between the dashed and the soid curves - is quite sma, and indeed amost competey negigibe at pion backward anges. The reason for this stems from the fact that in charged-pion production, 3 S -contribution to the fina state is forbidden. By comparing the resuts for the difference (dσ/dω ) P (dσ/dω ) A in the case of d pp (soid curves in Fig. (8)) with those in the case of the free reaction n p (dotted curves in Fig. (8) and aso presented in Ref. [22] in which we have investigated the heicity dependence of the reaction channes), we see that a arge correction is needed to go from the bound deuteron to the free neutron case [9]. The difference between the two resuts decreases to a tiny effect at backward anges. Figure (8) aso gives a comparison of the resuts for the heicity difference with the experimenta data from the GDH Coaboration [2]. c 25 SP atura Sciences Pubishing Cor.

12 538 E. M. Darwish: Review of Poarization Observabes in Incoherent Pion... It is obvious that quite satisfactory agreement with experiment is achieved [9]. An experimenta check of the heicity difference at extreme forward and backward pion anges is needed. Aso, an independent check in the framework of effective fied theory woud be very interesting. Figure (9) shows a comparison of the numerica resuts for the inear photon asymmetry Σ in the pure IA (dashed curves) and with -rescattering (soid curves) [2] with the preiminary experimenta data from the LEGS Spin Coaboration [4]. We see that the genera feature of the data is reproduced. However, the discrepancy is rather significant in the region where the photon energy cose to the -resonance. In the same figure, we aso show the resuts from the IA ony (dashed curves). It is seen that the -rescattering yieds an about % effect in the region of the peak position. We found that this is mainy due to the interference between the IA ampitude and the -FSI ampitude. In agreement with our resuts in the pure IA [8], one notes that the pure IA (dashed curves in Fig. (9)) cannot describe the experimenta data. The incusion of -FSI eads at 27 MeV to a quite satisfactory description of the data, whereas at 33 MeV the -FSI effect is sma and therefore differences between theory and experiment are sti evident. It is appear that our mode is sti not capabe of describing the measured photon asymmetry, even if -FSI is incuded. The infuence of -rescattering on the ongitudina doube-spin asymmetry T2 was investigated in our previous work [2] as a function of photon ab-energy for fixed anges (see Fig. ()). We see that T2 has very sma vaues at forward pion anges around θ < 6. These vaues increase (in absoute vaue decrease) with increasing the photon energy. At extreme backward anges, we see that T2 has sma positive vaues. From the foregoing discussion for the poarized cross sections and spin asymmetries [9, 2, 2] it is apparent that the contribution of -FSI is much important in the energy region around the -resonance, especiay in the peak position and must be considered in the anaysis of the forthcoming experiments. One notices that the incusion of -FSI eads to an overestimation by about % of the pane wave resuts. For ower and higher energies, one can see that the -FSI effect is sma. This means, in particuar with respect to a test of theoretica modes for pion production ampitudes on the neutron, that one needs a reiabe description of the scattering process. Hopefuy, these predictions can be tested in the near future when the data from the on-going experiments become avaiabe. 4.3 Effects of and -rescattering The main goa of this section is to report on a theoretica prediction for the tota cross section and spin observabes for both the charged and neutra pion photoproduction channes of the reaction d in the energy region from -threshod up to the (232)-resonance [29, 3]. We aso disentange the different contributions to the observabes from the bare eementary reaction, the -rescattering, and the -rescattering. We are going to extend the work in the preceding papers [7,8,22,9, 2, 2] to the -threshod region in order to understand the dynamics of pion photoproduction ampitude near -threshod. The extension incudes the foowing important aspects: (i) A more reaistic eementary pion photoproduction operator from Ref. [3] is used. This mode based on an effective Lagrangian approach (ELA) incudes seven nuceon resonances ( (232), (44), (52), (535), (62), (65), and (7)), in addition to Born and vector-meson exchange terms. It incorporates a new theoretica treatment of spin-3/2 resonances avoiding pathoogies present in previous modes and aso aows to provide a more reiabe description of the -threshod region. (ii) The infuence of both - and -FSI effects on unpoarized cross sections and spin asymmetries is quantitativey studied. We consider, besides the pure impuse approximation (IA), compete rescattering in the fina two-body subsystems, i.e. in the - and -subsystems. (iii) Comparison with recent experimenta data and other theoretica modes is discussed. We compare in the figures cacuations with different ingredients, showing separatey the contribution from the bare interaction and the fina state interaction one. We ca impuse approximation (IA) to the bare contribution to the observabes shown in diagram (a) of Fig. (3) together with the same diagram exchanging 2. In what foows, when we cite any diagram (a)-(c) in Fig. (3) we impicity mean the contributions of both the depicted diagram and the corresponding one obtained under 2 exchange. We have to be carefu with what we ca IA. A truy IA cacuation (a cacuation that incudes ony diagram (a)) cannot empoy directy the eementary ampitudes which fit the eectromagnetic mutipoes of process, because the effect of -rescattering is incuded in those fits. For exampe, if we use the mutipoes provided by the ELA mode which fit the experimenta data, and afterwards we incude -rescattering, we are not reay incuding just diagram (a) + diagram (c), but rather diagram (a) + diagram (c) + diagram (c) cacuation. Therefore, if we wish to cacuate the contribution coming just from diagram (a), the bare contribution to the ampitude has to be extracted from the anaysis of the, where FSI has to be removed. This was done for instance in Ref. [3]. We name IA to the cacuations where the -rescattering is incuded in the eementary reaction. IA is therefore equivaent to a diagram (a) + diagram (c) cacuation. In section 4.4 we compare resuts for IA +rescattering and IA+rescattering resuts using MAID and ELA modes. We refer to the cacuation that incudes diagram (b) as and diagram c 25 SP atura Sciences Pubishing Cor.

13 App. Math. Inf. Sci. 9, o. 2, (25) / (c) using the rescattering, as. Therefore the fu cacuation, consistent and with a the effects considered in this work is what we refer as IA++. Resuts in Refs. [23, 24, 25] contain a doube-counting of the -rescattering because their starting point is MAID and SAID ampitudes as eementary reaction and afterwards they incude -rescattering (they perform what we ca an IA + + cacuation). σ tot [µb] σ tot [µb] E [MeV] thr E -E [MeV] Fig. 3: Same as in Fig. () but for d(, + )nn. Experimenta data are from Booth et a. [48]. The three-dashed curve in the right pane denotes the ChPT prediction of Lensky et a. [49] Tota cross sections E [MeV] thr E -E [MeV] Fig. : Tota cross section for d(, )np using the eectromagnetic mutipoes of set no. 2 from the effective Lagrangian approach [3]. Left pane: resuts from -threshod up to the (232)-resonance region in comparison with the experimenta data from Krusche et a. []. Right pane: resuts near -threshod as a function of the excess energy above - threshod E E thr. Curve conventions: (dotted) IA; (ongdashed) IA+; (short-dashed) IA+; (soid) IA+ +. σ tot [µb] E [MeV] thr E -E [MeV] Fig. 2: Same as in Fig. () but for d(, )pp. Experimenta data from Benz et a. [45] (soid circes), Chiefari et a. [46] (open circes) and Quraan et a. [47] (open square). In the eft panes of Figs. (), (2), and (3) we show the resuts for the tota cross section from -threshod up to the (232)-resonance region for the d np, d pp, and d + nn reactions, respectivey (see aso Ref. [3]). In the right panes of the same figures our predictions for the region near -threshod as function of the excess energy above -threshod, E E thr, in the aboratory system, are aso shown. From the comparison with the experimenta data it stems that our mode overestimates the tota cross section for the channe and sighty underestimates it for the + channe, however, on the overa, we obtain a very good agreement. Compared to the ChPT prediction from [49] for the + production channe near -threshod (right pane of Fig. (3)) a good agreement is obtained when FSI is incuded. The importance of rescattering in the neutra channe is ceary addressed when the fu cacuation is compared to the IA one. It is very cear from the right panes in Figs. (), (2), and (3) that the charged pion production cross-sections differ significanty in magnitude from the neutra pion ones due to the Kro-Rudermann term (diagram (D) in Fig. ()) which does not contribute to the neutra pion production reaction. We woud ike to emphasize that at photon energies cose to -threshod, amost a the contribution to the cross sections comes from Born terms. We observe aso that the -rescattering is not very important in both neutra and charged channes in what regards to the tota cross section and that the -rescattering pays a centra roe in neutra pion production but not so in charged pion channes. The resut on charged pion production is compatibe with what is obtained for the eementary process where the charged-production observabes are aready we predicted without the incusion of fina state interactions [3]. Since the contribution from -rescattering is sma, the effect of mutipe scattering of the pion with c 25 SP atura Sciences Pubishing Cor.

14 54 E. M. Darwish: Review of Poarization Observabes in Incoherent Pion... two outgoing nuceons (see diagrams (d) and (e) in Fig. (3)) is expected to be sma too (see aso [5])..5 d(, )np d(, - )pp d(, + )nn 6 6 σ P - σ A [µb] σ A [µb] σ P [µb] d(, )np d(, - )pp d(, + )nn E [MeV] E [MeV] E [MeV] σ P - σ A [µb] σ A [µb] σ P [µb] thr E -E [MeV] thr E -E [MeV] thr E -E [MeV] Fig. 5: Same as in Fig. (4) but in the energy region near - threshod. E E thr denotes the excess energy above -threshod in the aboratory system. Fig. 4: The heicity dependent tota photoabsorption cross sections for the separate channes of d(, ) for circuary poarized photons on a ongitudinay poarized deuteron with spin parae σ P (upper pane) and antiparae σ A (midde pane) to the photon spin in the energy region from -threshod up to the (232)-resonance. The ower pane shows the difference σ P σ A, i.e., the deuteron spin asymmetry of tota photoabsorption cross section. Curve conventions as in Fig. () Heicity dependent cross sections The resuts for the douby poarized tota cross sections in the case of the IA aone and with FSI effects are shown in Fig. (4) (from -threshod up to the (232)-resonance region) and Fig. (5) (near -threshod region) (see aso Ref. [29]). Resuts are dispayed in Figs. (4) and (5) as foows: (upper pane) tota photoabsorption cross sections σ P for circuary poarized photons on a target with spin parae to the photon spin; (midde pane) σ A, the same for antiparae spins of photon and deuteron target; (ower pane) spin asymmetry σ P σ A for the individua contributions of the different pion charge states to the d reaction. Severa experiments to measure the deuteron spin asymmetry are presenty underway [2]. In these experiments, deuterium serves as a neutron target and quasifree kinematics is preferred in order to minimize possibe interaction effects. Left panes in Figs. (4) and (5) show the importance of rescattering effects in the neutra channe when the fu cacuation (soid curve) is compared to the IA one (dotted curve). The most important contribution to FSI comes from -rescattering, whist -rescattering is much smaer and is significative ony in σ A at photon energies cose to -threshod (see the eft pane in Fig. (5)). This is due to the non-orthogonaity of the fina state pane wave in IA to the deuteron bound state wave function. We aso find that FSI effects are equay important for both σ P and σ A. FSI effects ead to an overa strong reduction of the spin asymmetry in the energy region of the (232)-resonance [29]. σ P is arger than σ A because of the (232)-excitation. The cross sections σ P and σ A as we as the spin asymmetry σ P σ A present quaitative and quantitativey simiar behaviors for both charged pion production channes (see aso Ref. [29]). The FSI effects appear mainy in σ A at photon ab-energies cose to -threshod (Fig. (5)). We find that the spin asymmetry σ P σ A starts out negative due to the E + mutipoe, which is dominant in the -threshod region, and has a strong positive contribution at higher energies due to the M + mutipoe, which is dominant in the (232)-resonance region. The effect of FSI is important for both the neutra c 25 SP atura Sciences Pubishing Cor.

15 App. Math. Inf. Sci. 9, o. 2, (25) / 54 and charged pion production channes in the -threshod region E =264.7 MeV E =297.7 MeV Σ E =8 MeV -.4 E =25 MeV Σ -.4 Σ E =322.6 MeV -.2 Σ -.4 Σ E =27 MeV E =33 MeV Fig. 6: The beam asymmetry Σ for ineary poarized photons for semi-excusive photoproduction from the deuteron as a function of pion ange in the aboratory frame of the deuteron at different photon ab-energies using the eectromagnetic mutipoes of set no. 2 from the effective Lagrangian approach [3]. Curve conventions as in Fig. (). Experimenta data are from the LEGS Coaboration (LEGS-exp-L3b) [4] Beam asymmetry In Figs. (6) and (7) we show a sampe of our resuts for the photon beam asymmetry Σ for and channes, respectivey (see Ref. [29]). This asymmetry does not exhibit a strong effect of the rescattering contribution and it is an exceent test of any weaknesses in the underying eementary reaction mode. The first effect that we observe is that -rescattering contribution is sma for this observabe in the energy region under study. The infuence of FSI is noticeabe at the owest energy and forward pion anges. With increasing photon ab-energy the effect of FSI becomes smaer athough not negigibe, even at the highest energy. In Fig. (7) we show our resuts for production together with the Lee-Sato cacuation [5] which is equivaent to our IA cacuation but using the dynamica mode deveoped in [52]. The Sato and Lee mode provides a resut cose to the one from our mode Fig. 7: Same as in Fig. (6) but for the d pp channe. The three-dashed curves in the bottom panes denote the resuts of Lee and Sato [5] without FSI effects. Experimenta data are from the LEGS Coaboration (LEGS-exp-L3b) [4]. except at the sma anges region, where a sizeabe deviation at 3 degrees is observed due to rescattering contributions. The sma ange region is sensitive to mode dependencies and rescattering effects and the resuts are again more dependent on the -rescattering than on the -rescattering. The -rescattering tends to enarge the asymmetry in charged channes and to decrease its vaue sighty in the neutra channe. When compared to the recent mode by Levchuk et a. [25] our resuts seem to compare better to the data from LEGS (LEGS-exp-L3b) [4], Figs. (6) and (7), but we have to wait unti the fina data are avaiabe to make a fina statement on this comparison Target asymmetries The target asymmetries T 22, T 2, and T 2 for the neutra channe are potted in Fig. (8) (see aso Ref. [3]). These asymmetries show a arge effect on the rescattering contribution and the empoyed eementary reaction mode. Thus, they constitute an exceent set of observabes to test our mode and to compare it to other modes. Actuay, our predictions for these observabes are quite different from those by Levchuk et a. [25] and by Arenhöve and Fix [23]. Unfortunatey no data are currenty avaiabe for these observabes and therefore we cannot make a comparison to experimenta data. c 25 SP atura Sciences Pubishing Cor.

16 542 E. M. Darwish: Review of Poarization Observabes in Incoherent Pion T -.5 T 22. T.4 T T T T 2.3. T Fig. 8: Target asymmetries T, T 22, T 2, and T 2 for semiexcusive photoproduction from the deuteron at photon ab-energy E = 265 MeV using the eectromagnetic mutipoes of set no. 2 from the effective Lagrangian approach [3]. Curve conventions as in Fig. (). In charged channes (see Fig. (9)) the structure of the target asymmetries T 22, T 2, and T 2 is very simpe. They a present a peak at a certain ange (T, T 22, and T 2 approximatey between 25 and 6 degrees, and T 2 at zero degrees) with a decreasing asymmetry towards zero for 8 degrees except for the T 2 asymmetry and ow-energy T. We find that -rescattering effects are negigibe. Our resuts for the channe are simiar to the ones of Levchuk et a. [25] probaby due to the fact that fina state interactions are not so important in charged channes. However, the T 2 asymmetry we cacuate behaves in a different way than those of Refs. [23,25] in the arge ange region and seems to have some mode dependency (see Figs. (9) in [23] and (2) in [25]) The doube poarization E-asymmetry There is a great dea of interest in experiments [4] to determine the beam-target doube poarization E-asymmetry for the d reaction channes. In connection with this study, we provide in Fig. (2) a sampe of E-asymmetry as a function of the emission pion ange θ in the aboratory frame for the individua pion photoproduction channes (see aso Ref. [29]). This Fig. 9: Same as in Fig. (8) but for d pp at photon abenergy E = 27 MeV. asymmetry is given by E(θ ) = d(σ A σ P )/dω d(σ A + σ P )/dω = d(σ A σ P )/dω, (45) 2(dσ /dω) where dσ P /dω and dσ A /dω represent the spin parae and antiparae differentia cross sections, respectivey, and dσ /dω denotes the unpoarized differentia cross section. The heicity dependent cross sections are we suited to verify the GDH sum rue, to do partia channe anaysis, and to give contributions to the doube poarization E-asymmetry. This asymmetry appears as an interference between the ampitudes with different parity-exchange properties. The heicity E-asymmetry (Fig. (2)) has quaitativey a simiar behavior for a pion photoproduction channes. The maximum vaue of E equas unity at θ = and 8. The curves begin with unity and decrease as the pion ange increases unti the minimum vaue is reached at θ 9. Then, it increases again to unity. The negative vaues in the E-asymmetry come mainy from higher positive contribution in dσ P /dω [29]. Figure (2) shows aso that the heicity E-asymmetry does not exhibit a strong effect of the rescattering contribution and, therefore, it is an exceent observabe to test any weakness in the underying eementary reaction mode. We find that FSI are sizeabe in the neutra channe, whereas in charged channes they are noticeabe ony at the owest energy. The E-asymmetry proves to be sensitive to the choice of the input eementary c 25 SP atura Sciences Pubishing Cor.

17 App. Math. Inf. Sci. 9, o. 2, (25) / E-asymmetry E-asymmetry E-asymmetry d(, )np d(, - )pp d(, + )nn E =5 MeV E =5 MeV E =5 MeV E =35 MeV -.5 E =35 MeV E =23 MeV.8 E =23 MeV.8 E =23 MeV E =35 MeV Fig. 2: The doube poarization E-asymmetry (see Eq. (45) for its definition) as a function of the pion ange in the aboratory frame at various photon ab-energies for the separate channes of the reaction d(, ) for circuary poarized photons on a ongitudinay poarized deuteron. Curve conventions as in Fig. (). ampitude and to the doube-counting of the -rescattering [29]. We woud ike to point out that very preiminary data for the heicity E-asymmetry of the d np reaction at photon ab-energy E = 349±5 MeV have been measured by the LEGS@BL Coaboration [4]. However, the data anaysis has not yet been competed. I GDH [µb] d(, )np d(, - )pp d(, + )nn E [MeV] E [MeV] E [MeV] Fig. 2: The Gerasimov-Dre-Hearn integra (see Eq. (46) for its definition) as a function of the upper integration imit for separate channes of the d(, ) reaction. Curve conventions as in Fig. () The Gerasimov-Dre-Hearn integra ext, we discuss the GDH sum rue [28] which inks the anomaous magnetic moment of a partice to the energy weighted integra of the spin asymmetry of the photo-absorption cross section. For a partice of mass M, charge eq, anomaous magnetic moment κ, and spin S it reads I GDH = de E ( σ P (E ) σ A (E )) = 4 2 κ 2 e2 M 2 S, (46) where σ P(A) denote the tota absorption cross sections for circuary poarized photons on a target with spin parae (P) and antiparae (A) to the photon spin. This sum rue provides a very interesting reation between a ground state property (κ) of a partice and its whoe excitation spectrum. Apart from the genera assumption that the integra in Eq. (46) converges, its derivation is based soey on first principes ike Lorentz and gauge invariances, unitarity, crossing symmetry, and causaity of the Compton scattering ampitude of a partice. Consequenty, from the experimenta and theoretica points of view, a test for various targets becomes very important. To present the resuts in a direct way, we introduce the finite GDH integra as defined by I GDH (E )= E de ( σ P (E ) σ A (E ) ). (47) E In Fig. (2) we depict our resuts for the evauation of the GDH integra for the individua contributions from the different charge states of the pion for the d reaction as a function of the upper integration imit. The contributions of various pion photoproduction channes to the finite GDH integra (Eq. (47)) up to 35 MeV and their sum are summarized in Tabe () (see aso Ref. [29]). We find that a arge positive contribution to the GDH integra comes from the -production channe, whereas the charged pion channes give a negative but - in absoute vaue - smaer contribution to the GDH integra. This negative contribution comes from the isovector M transition to the S state, which can ony be accessed when the spins of the photon and the deuteron are antiparae. The incusion of -rescattering reduces significanty (more than a haf, see Tabe ()) the vaue of the integra. The integra is sighty reduced by the -rescattering. Contrary with what we have found in other observabes, we find that FSI effects are sizeabe in the GDH integra for both neutra and charged pion production channes [29]. The measurement of spin asymmetry for the various pion photoproduction channes on the deuteron represents a stringent test of our present theoretica understanding of the d reaction. Hence, the experimenta programs at faciities ike MAMI at Mainz and ELSA at Bonn, concerning the GDH sum c 25 SP atura Sciences Pubishing Cor.

18 544 E. M. Darwish: Review of Poarization Observabes in Incoherent Pion... Tabe : Contributions of various channes for quasifree pion photoproduction from the deuteron to the finite GDH integra using the eementary reaction from [3] and expicity integrated up to a photon ab-energy of 35 MeV in µb. Contribution pp + nn np IA IA IA rue on the deuteron are of great importance for further progress in the fied. σ P - σ A [µb] E [MeV] Fig. 22: The heicity-dependent tota photoabsorption cross section difference (σ P σ A ) for the semiexcusive channes d ±. Curve conventions: dashed, IA ++ c using MAID [26]; soid, IA+ + using the bare eectromagnetic mutipoes of ELA [3]. Experimenta data taken from [3]. 4.4 Sensitivity to the eementary ampitude In what foows, we discuss the infuence of different choices for the input eementary pion photoproduction operator on the resuts presented above for the d reaction channes (see Refs. [29,3] for more resuts). We compare resuts for the heicity tota and differentia cross sections, spin asymmetries, and the GDH sum rue for the deuteron in the energy region from near -threshod to the (232)-resonance, using as eementary reaction ampitudes the ones provided by the ELA mode from Ref. [3] and those obtained using MAID mode [26]. The resuts of this comparison are coected in Figs. (22), (23), (24), and (25)). First, in Fig. (22) we show the heicity-dependent tota photoabsorption cross section difference (σ P σ A ) for the semi-excusive channes d ± using the eementary operators from MAID (dashed curve) and our d(σ P -σ A )/dω [µb/sr] d(σ P -σ A )/dω [µb/sr] d(σ P -σ A )/dω [µb/sr] d(, )np d(, - )pp d(, + )nn E =55 MeV E =55 MeV E =55 MeV E =23 MeV E =23 MeV E =23 MeV E =35 MeV E =35 MeV E =35 MeV Fig. 23: Circuar photon poarization asymmetry for the separate channes of semi-excusive pion photoproduction on the deuteron as a function of the pion ange in the aboratory frame at various photon ab-energies using different eementary pion photoproduction operators and incuding FSI effects. Curve conventions: dashed, IA + + c using MAID [26]; dotted, IA ++ c using the dressed mutipoes of ELA [3]; soid, IA+ + using the bare eectromagnetic mutipoes of ELA. ELA mode (soid curve) incuding FSI effects. We compare to the atest experimenta data from the A2 and GDH@MAMI Coaborations [3] and we find an exceent agreement with data. Our ELA computation provides a sighty better agreement than the computation using MAID mode. In Fig. (23) we show the heicity differentia cross section difference d(σ P σ A )/dω using the bare ELA mode (soid), using MAID mode (dashed) and the dressed ELA mode (dotted) in the neutra channe. Both resuts are quite different, speciay at ow energies. At the peak position, we obtain arger vaues using MAID than using ELA. At extreme forward pion anges, arge differences between both resuts is obtained. This discrepancy is more noticeabe for energies cose to -threshod and shows up the differences among eementary operators. The difference between the dotted (dressed ELA) and soid (bare ELA) curves shows the effect of -rescattering, which is found to be important near -threshod. If we focus our attention on the E-asymmetry (Fig. (24)) and GDH integra (Fig. (25)), a sizeabe difference is obtained in the energy region near -threshod. Tabe c 25 SP atura Sciences Pubishing Cor.

19 App. Math. Inf. Sci. 9, o. 2, (25) / E-asymmetry E-asymmetry E-asymmetry.8.6 d(, )np d(, - )pp d(, + )nn E =55 MeV E =55 MeV E =55 MeV E =23 MeV E =23 MeV E =23 MeV E =35 MeV E =35 MeV E =35 MeV Fig. 24: The doube poarization E-asymmetry as a function of the pion ange in the aboratory frame at various photon abenergies for the separate channes of d(, ) using different eementary pion photoproduction operators and incuding FSI effects. Curve conventions as in Fig. (23). I GDH [µb] E [MeV] E [MeV] E [MeV] d(, )np d(, - )pp d(, + )nn Fig. 25: The Gerasimov-Dre-Hearn integra as a function of the upper integration imit for separate channes of the d(, ) reaction using different eementary pion photoproduction operators and incuding FSI effects. Curve conventions as in Fig. (23). (2) dispays the extracted vaues of the GDH integra up to 35 MeV for a three pion photoproduction channes and their sum for our resuts using MAID as eementary reaction operator. Compared the vaues in Tabe (2) using MAID to the vaues presented in Tabe () using ELA, we find that ELA: IIA GDH (35 MeV) = 6.78 µb whereas MAID: IIA GDH (35 MeV) = µb. After incuding FSI effects we obtained ELA: -2-4 Tabe 2: Same as Tabe () but using the MAID mode [26] as eementary reaction operator. Contribution pp + nn np IA IA IA + + c IIA++ GDH (35 MeV) = 4.29 µb whereas MAID: IIA GDH ++ c (35 MeV)=84.6 µb (see aso Ref. [29]). In the charged channes the resuts using MAID or ELA as eementary reaction operators are different in the near -threshod region, where the predicted E-asymmetry within the ELA is - in negative sign - smaer than those within MAID, but provide simiar resuts in the (232)-region. Athough both computations provide resuts for the GDH integra that differ in, approximatey, a factor two, the curves computed using MAID and ELA modes dispayed in Fig. (22) are not much different, and one must ook more in deep for the reasons in the different GDH vaues. This arge difference in the GDH - in spite of the reativey sma dissimiarity shown in Fig. (22) - shows the arge sensitivity of the GDH integra to the choice of the eementary operator empoyed. The GDH resuts from two arge contributions of negative and positive signs, that cance to a arge extent in the fina resuts. The curve computed with the MAID mode seems more symmetric and hence the canceation of the positive and the negative contributions to the integra is arger than for the ELA mode computation which presents a arger in absoute vaue negative part. This partia canceation aso means that the contribution of the processes considered in this work to the tota GDH integra up to 35 MeV is reativey sma compared to other contributions. For instance in Ref. [24] a vaue of about 24 µb for the tota GDH integra up to 35 MeV is found. Our pionic contributions gives 42 µb. The rest shoud be either coherent pion production or two-nuceon break-up. Summarizing, we can say that the MAID mode provides different predictions for poarization observabes than the ELA mode, particuary at ow photon energies, and that the GDH integra provides an exceent observabe to test different pion production operators. From the preceding discussion it is apparent that -rescattering and the choice of the eementary operator have a visibe effect on spin observabes. Therefore, we have to be carefu when incuding additiona -rescattering in cacuations that use MAID or SAID as eementary reaction operator. 5 Concusions and outook The main topic of this work was the investigation of poarization observabes in incoherent pion c 25 SP atura Sciences Pubishing Cor.

20 546 E. M. Darwish: Review of Poarization Observabes in Incoherent Pion... photoproduction from the deuteron in the energy region from -threshod up to the (232)-resonance with incusion of a eading effects. Forma expressions for poarization observabes are derived and described by various beam, target and beam-target asymmetries for poarized photons and/or poarized deuterons. For the eementary pion photoproduction operator on the free nuceon, a reaistic effective Lagrangian approach is used which incudes seven nuceon resonances, in addition to Born and vector-meson exchange terms. The interactions in the fina two-body subsystems are taken from separabe representations of reaistic potentias. Resuts are given for the unpoarized cross sections, the douby poarized cross sections for parae and antiparae heicity states, the inear photon asymmetry, the doube poarization E-asymmetry, the vector and tensor deuteron asymmetries for the d pp, d + nn, and d np channes. Contributions from various channes to the spin asymmetry and the GDH integra are evauated by expicit integration up to a photon ab-energy of 35 MeV. The sensitivity to the eementary operator of the resuts has aso been investigated. Within our mode, we have found that the incusion of FSI effects is important for unpoarized cross sections. It eads to a reduction (an overestimation) of the tota cross section for ( ± ) production. -rescattering appears to be ess important compared to the -rescattering in both neutra and charged channes. Compared to experimenta data, we have found that the sizeabe discrepancies without rescattering are argey reduced and that a quite good agreement with the data is achieved. With respect to the ChPT prediction from Ref. [49] for the tota cross section of + production near -threshod, we have obtained good agreement when FSI is incuded. For the poarized tota cross sections σ P and σ A as we as spin asymmetry σ P σ A, we have found that the infuence of FSI stems predominanty from -rescattering, whereas -rescattering is much smaer and appears ony in σ A cose to -threshod. In the neutra channe, we have observed that FSI effects ead to a strong reduction of the spin asymmetry and that σ P is arger than σ A. In charged channes, we have found that FSI effects appear mainy in σ A near -threshod. We have aso found that σ P σ A starts out negative due to the E + mutipoe, and has a strong positive contribution at higher energies due to the M + mutipoe. The resuts of spin asymmetries are found to be very sensitive to the FSI effects, in particuar in the case of the production channe. We have compared our resuts to the presenty avaiabe experimenta data. A quaitative and quantitative agreement with the data of Σ-asymmetry from LEGS [4] are obtained after incuding the FSI effects. With respect to the prediction without FSI of Lee-Sato [5] for the Σ-asymmetry of -production channe, good agreement is found too. Regarding the resuts of the E-asymmetry, quaitativey a simiar behavior for a channes is found. We have aso found that the infuence of the FSI effects is strong, especiay in the neutra channe at forward anges and ow energies. The contributions of separate channes to the GDH integra are evauated by expicit integration up to 35 MeV. A tota vaue of the GDH integra of IIA++ GDH (35 MeV) = 4.29 µb has been computed after incuding FSI effects. A arge positive contribution to the GDH integra came from the -production channe whereas the charged pion channes gave a negative but in absoute vaue smaer contribution to the GDH integra. We have aso studied the infuence of the eementary operator on the unpoarized cross sections, poarized cross sections, spin asymmetries, and GDH integra. A observabes, speciay the spin asymmetries for both the neutra and charged channes, are found to be very sensitive to the eementary operator. In many cases the deviation among resuts obtained using different operators is very arge. It is shown aso that the process d(,) can serve as a fiter for different eementary operators since its cross section and asymmetry predictions show very different vaues when one varies the eementary pion production operators empoyed. The GDH integra is found to be an exceent observabe to discriminate among different eementary pion production operators. We obtain quite different vaues for the GDH integra for the MAID and the ELA modes, in spite of the fact that their predictions for the σ P σ A observabe seem simiar (see Fig. (22)). Due to an important canceation of the contributions beow and above a photon energy of 26 MeV the pionic contribution to the GDH integra up to 35 MeV computed in this work gives 42 µb, what is reativey sma compared to other contributions. The rest shoud be either coherent pion production or two-nuceon break-up [24]. Finay, we woud ike to point out that future improvements of the present mode (in particuar, its extension to higher energies) can be achieved by incuding the next eading correction from the intermediate,, and interactions. Poarization observabes in genera constitute more stringent tests for theoretica modes due to their sensitivity to sma ampitudes. At this point, a much needed measurement on the deuteron spin asymmetries wi certainy provide us with an important observabe to test our knowedge of the pion photoproduction on the neutron process and, hence, to provide us with vauabe information on the neutron spin asymmetry in an indirect way. An independent test within the framework of effective fied theory wi be aso of great interest. Moreover, the formaism can aso be extended to investigate poarization observabes of pion eectroproduction on the deuteron where the virtua photon has more degrees of freedom than the rea one. Therefore, it can be used to expore the reaction more deepy. c 25 SP atura Sciences Pubishing Cor.

21 App. Math. Inf. Sci. 9, o. 2, (25) / Acknowedgements I am gratefuy acknowedge very usefu discussions with T.-S.H. Lee, T. Sato, H. Arenhöve, C. Fernández-Ramírez, E. Moya de Guerra, and J.M. Udías. I am deepy indebted to the members of the GDH and A2 Coaborations, especiay J. Ahrens and P. Pedroni, the TAPS Coaboration, speciay B. Krusche, and the LEGS Coaboration, speciay A.M. Sandorfi and M. Lucas, for providing me with their experimenta data. References [] G. F. Chew and H. W. Lewis, Phys. Rev. 84, 779 (95). [2] M. Lax and H. Feshbach, Phys. Rev. 88, 59 (952). [3] I. Bomqvist and J. M. Laget, uc. Phys. A 28, 45 (977). [4] J. M. Laget, uc. Phys. A 296, 388 (978). [5] J. M. Laget, Phys. Rep. 69, (98). [6] R. Schmidt, H. Arenhöve, and P. Wihem, Z. Phys. A355, 42 (996). [7] M.I. Levchuk, M. Schumacher, and F. Wissmann, arxiv:nucth/4. [8] E.M. Darwish, H. Arenhöve, and M. Schwamb, Eur. Phys. J. A 6, (23). [9] E.M. Darwish, H. Arenhöve, and M. Schwamb, Eur. Phys. J. A 7, 53 (23). 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22 548 E. M. Darwish: Review of Poarization Observabes in Incoherent Pion... [44] H. Garciazo and T. Mizutani, Systems, (Word Scientific, Singapore, 99). [45] P. Benz et a., uc. Phys. B 65, 58 (973). [46] G. Chiefari, E. Drago, M. apoitano, and C. Sciacca, Lett. uovo Cim. 3, 29 (975). [47] M.A. Quraan et a., Phys. Rev. C 57, 28 (998). [48] E.C. Booth, B. Chasan, J. Comuzzi, and P. Bosted, Phys. Rev. C 2, 27 (979). [49] V. Lensky, V. Baru, J. Haidenbauer, C. Hanhart, A. Kudryavtsev, and U.-G. Meißner, Eur. Phys. J. A 26, 7 (25). [5] I. Duck, Phys. Lett. B 59, 9 (975); J.V. obe, Phys. Lett. B 67, 39 (977). [5] T.-S.H. Lee and T. Sato, private communication. [52] T. Sato and T.-S.H. Lee, Phys. Rev. C 54, 266 (996); T. Sato and T.-S.H. Lee, Phys. Rev. C 63, 552 (2). Eed M. Darwish is an Associate Professor of Physics at Sohag University, Egypt. He received his PhD degree in Theoretica ucear Physics from the Johannes-Gutenberg University, Mainz, Germany as a DAAD schoarship. Then he became a Facuty Member at Sohag University for six years. He became Assistant Professor of Physics at Sohag University in 22 and Associate Professor in 23. He joined Taibah University, Saudi Arabia as an Associate Professor (on eave) in 28. He has served as Head of Appied Physics Department at Taibah University in His research incudes work on the study of sub-nucear degrees of freedom in eectromagnetic reactions in few-body systems, mainy the deuteron, e.g. photo- and eectro-production of pseudo-scaar mesons on the deuteron incuding poarization observabes. He has pubished more than 6 research papers in reputed internationa journas of physics. He is referee of severa internationa physics journas. c 25 SP atura Sciences Pubishing Cor.