aa r X i v : . [ nu c l - t h ] O c t EPJ manuscript No. (will be inserted by the editor)
Unitary Isobar Model - MAID2007
D. Drechsel , S. S. Kamalov , L. Tiator Institut f¨ur Kernphysik, Universit¨at Mainz, D-55099 Mainz JINR Dubna, 141980 Moscow Region, RussiaOctober 1, 2007
Abstract.
The unitary isobar model MAID2007 has been developed to analyze the world data of pionphoto- and electroproduction. The model contains both a common background and several resonance terms.The background is unitarized according to the K-matrix prescription, and the 13 four-star resonances withmasses below 2 GeV are described by appropriately unitarized Breit-Wigner forms. The data have beenanalyzed by both single-energy and global fits, and the transverse and longitudinal helicity amplitudes havebeen extracted for the four-star resonances below 2 GeV. Because of its inherent simplicity, MAID2007 iswell adopted for predictions and analysis of the observables in pion photo- and electroproduction.
PACS.
Our knowledge about the excitation spectrum of thenucleon was originally provided by elastic pion-nucleonscattering [1]. All the resonances listed in the ParticleData Tables [2] have been identified by partial-waveanalyses of this process with both Breit-Wigner and poleextraction techniques. From such analyses we know theresonance masses, widths, and branching ratios into the πN and ππN channels. These are reliable parametersfor the resonances in the 3- and 4-star tiers, with onlyfew exceptions. In particular, there remains some doubtabout the structure of two prominent resonances, theRoper P (1440), which appears unusually broad, andthe S (1535), where the pole can not be uniquelydetermined, because it lies close to the ηN threshold.On the basis of these relatively firm grounds, addi-tional information can be obtained for the electromagnetic(e.m.) γN N ∗ couplings through pion photo- and elec-troproduction. These couplings are described by electric,magnetic, and charge transition form factors, G ∗ E ( Q ), G ∗ M ( Q ), and G ∗ C ( Q ), or by linear combinations thereofas helicity amplitudes A / ( Q ), A / ( Q ), and S / ( Q ).So far we have some reasonable knowledge of the trans-verse amplitudes A / and A / at the real photon point,which are tabulated in the Particle Data Tables. For finite Q the information found in the literature is scarce anduntil recently practically nonexistent for the longitudinalamplitudes S / . But even for the transverse amplitudesonly few results have remained firm over the recentyears, such as the G ∗ M form factor of the P (1232) or ∆ (1232) resonance up to Q ≈
10 GeV , the A / ( Q )for the S (1535) up to Q ≈ , and the helicity asymmetry A ( Q ) for the resonances D (1520) and F (1680) up to Q ≈ [3]. Frequently also datapoints for other resonances, e.g., the Roper resonance, areshown together with quark model calculations. However,the statistical errors are often quite large and the modeldependence of the analysis may be even larger. In thiscontext it is worth mentioning that also the notion of a‘data point’ is somewhat misleading because the photoncouplings and amplitudes can only be derived indirectlyby a partial-wave analysis. It is in fact prerequisite toanalyze a particular experiment within a frameworkbased on the “world data”. The only exception from thiscaveat is the ∆ (1232) resonance. For this lowest-lyingand strongest resonance of the nucleon, the analysis isfacilitated by two important constraints: the validity of(I) the Watson theorem at the 1 % level and (II) thetruncation of the multipole series to S and P waves as agood first-order approximation. With these assumptionsthe e.m. couplings have been directly determined inthe real photon limit by a complete experiment withpolarized photons and detecting both neutral and chargedpions in the final state, thus allowing also for an isospinseparation [4]. Moreover, a nearly complete separationof the possible polarization observables has recentlyprovided the basis to extend such a “model-independent”analysis also to electroproduction [5]. However, we arestill far from such a situation for all the higher reso-nances. Neither are the mentioned constraints valid norare we close to a complete experiment. Until recentlythe data base was rather limited, the error bars werelarge, and no data were available from target or recoilpolarization experiments. Even now there exist only veryfew data points from double-polarization experiments atenergies above the ∆ (1232). However, the situation for D. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 unpolarized e + p → e ′ + p + π reaction has considerablyimproved, mainly by new JLab experiments in all threehalls A, B, and C. These data cover a large energy rangefrom the ∆ (1232) up to the third resonance region witha wide angular range in θ π . Furthermore, electron beampolarization has been used in several experiments atJLab, MIT/Bates, and MAMI/Mainz. Because of thelarge coverage in the azimuthal angle by the modernlarge-acceptance detectors, a separation of all 4 partialcross sections in the unpolarized experiment becomespossible. But even without a Rosenbluth separation ofthe transverse ( σ T ) and longitudinal ( σ L ) cross sections,there is an enhanced sensitivity to the longitudinalamplitudes due to the interference terms σ LT and σ LT ′ .Such data are the basis of our new partial-wave analysiswith an improved version of the Mainz unitary isobarmodel MAID.We proceed by presenting a brief history of the uni-tary isobar model in Sect. 2. The formalism of pion photo-and electroproduction is summarized in Sect. 3. In the fol-lowing Sect. 4 we present our results for photoproductionas obtained from the latest version MAID2007, and inSect. 5 this analysis is extended to electroproduction. Weconclude with a short summary in Sect. 6. – MAID98 In 1998 the first version of the Unitary IsobarModel was developed and implemented on the webto give an easy access for the community. MAID98was constructed with a limited set of nucleon reso-nances described by Breit-Wigner forms and a non-resonant background constructed from Born termsand t-channel vector-meson contributions [6]. In or-der to have the right threshold behavior and a rea-sonable description at the higher energies, the Bornterms were introduced with an energy-dependent mix-ing of pseudovector and pseudoscalar πN N coupling.Each partial wave was unitarized up to the two-pion threshold by use of Watson’s theorem. Specif-ically, the unitarization was achieved by introduc-ing additional phases φ R in the resonance ampli-tudes in order to adjust the phase of the total am-plitude. Only the following 4-star resonances were in-cluded: P (1232), P (1440), D (1520), S (1535), S (1650), F (1680), and D (1700). The e.m. ver-tices of these resonances were extracted from a bestfit to the VPI/GWU partial-wave analysis [7]. For the P (1232) resonance, we determined the following ra-tios of transition amplitudes: (I) electric quadrupole tomagnetic dipole transition, R EM = E /M − . R CM = C /M − . Q . The Q -dependence of the res-onance amplitudes in the second and third resonanceregions was expressed in terms of the quark electric and magnetic multipoles [8]. The non-unitarized back-ground contributions were determined using standardBorn terms and vector-meson exchange. In order topreserve gauge invariance, the Born terms were ex-pressed by the usual dipole form for the Sachs formfactors, and both the pion and the axial form fac-tor were set equal to the isovector Dirac form factor, F π ( Q ) = G A ( Q ) = F p ( Q ) − F n ( Q ). – MAID2000 In this version of MAID, the background contributionwas unitarized for the multipoles up to F waves ac-cording to the prescription of K-matrix theory. The S -wave multipoles E and L were modified in or-der to improve their energy dependence in the thresh-old region. With the new unitarization procedure, thepion photoproduction multipoles of SAID and someselected data for pion photo- and electroproduction inthe energy range up to W = 1 . ∆ (1232) multipoles were found to be R EM = − .
2% and R CM = − . Q . – MAID2003 In accordance with results of Ref. [10], the Q de-pendence of the electric and Coulomb excitationsof the ∆ (1232) resonance was modified. The ratio R EM was found to change sign at Q ≈ . from negative to positive values, and R CM decreasedfrom − .
5% at Q = 0 to − .
5% at Q =4 GeV . Moreover, the following 4-star resonanceswere included in MAID2003: S (1620), D (1675), P (1720), F (1905), P (1910), and F (1950). Incontrast to previous versions, the helicity amplitudes ofall 13 resonances were input parameters and their Q dependence was parameterized by polynomials. Withthis new version of MAID we directly analyzed all thepion photo- and electroproduction data available since1960, and for the first time we made local (single en-ergy) and global (energy dependent) fits, independentof the GWU/SAID group. – MAID2005 The Q dependence of the Sachs form factors in theBorn terms was replaced by the more recent param-eterization of Ref. [11], and at the e.m. vertices ofthe pion-pole and seagull terms, realistic pion and ax-ial form factors were introduced. As a result the de-scription of charged pion electroproduction was muchimproved. On the basis of a large amount of newdata from MIT/Bates, ELSA/Bonn, Grenoble, Mainz,and Jefferson Lab, we performed new local and globalas well as single- Q fits and obtained a better de-scription of the data in the energy range 1.6 GeV
0) = W − m W , (5)describing the momentum of a real photon, and k R = k ( M R , , q R = q ( M R ) , (6)for the real photon and pion momenta at the resonanceposition, W = M R .The basic equations used for MAID2007 are taken fromthe dynamical Dubna-Mainz-Taipei (DMT) model [10,14,15]. In this approach the t-matrix for pion photo- andelectroproduction takes the form t γπ ( W ) = v γπ ( W ) + v γπ ( W ) g ( W ) t πN ( W ) , (7)with v γπ the transition potential for the reaction γ ∗ N → πN , t πN the πN scattering matrix, and g the free πN propagator. In a resonant channel the transition potential v γπ consists of two terms, v γπ ( W ) = v Bγπ ( W ) + v Rγπ ( W ) , (8) with v Bγπ the background transition potential and v Rγπ thecontribution of the “bare” resonance excitation. The re-sulting t-matrix can be decomposed into two terms [15] t γπ ( W ) = t Bγπ ( W ) + t Rγπ ( W ) , (9)where t Bγπ ( W ) = v Bγπ ( W ) + v Bγπ ( W ) g ( W ) t πN ( W ) , (10) t Rγπ ( W ) = v Rγπ ( W ) + v Rγπ ( W ) g ( W ) t πN ( W ) , (11)with t Bγπ including the contributions from both the non-resonant background and the γ ∗ N R vertex renormaliza-tion. The decomposition in resonance and backgroundcontributions is not unique, however, our definition hasthe advantage that all the processes starting with the e.m.excitation of a bare resonance are summed up in t Rγπ . The multipole decomposition of Eq. (10) yields the back-ground contribution to the physical amplitudes in thechannels α = ( ξ, ℓ, j, I ) [14], where ℓ , j and I denote theorbital momentum, the total angular momentum, and theisospin of the pion-nucleon final state, and ξ stands forthe magnetic ( ξ = M ), electric ( ξ = E ), and Coulomb or“scalar” ( ξ = S ) transitions, t B,αγπ ( q, k, Q ) = v B,αγπ ( q, k, Q ) (12)+ Z ∞ dq ′ q ′ t απN ( q, q ′ ; W ) v B,αγπ ( q ′ , k, Q ) W − W πN ( q ′ ) + iǫ , where W πN ( q ′ ) is the hadronic c.m. energy in the interme-diate state. The pion electroproduction potential v B,αγπ isconstructed as in Ref. [6] and contains contributions fromthe Born terms described by an energy-dependent mix-ing of pseudovector (PV) and pseudoscalar (PS) πN N coupling as well as t-channel vector meson exchange. Thequasi-potential v B,αγπ depends on 5 parameters: the PV-PSmixing parameter Λ m as defined in Eq. (12) of Ref. [6] and4 coupling constants for the vector-meson exchange. Theon-shell parts of v B,αγπ and t B,αγπ depend on two variablesonly, i.e., v B,αγπ ( q, k, Q ) = v B,αγπ ( W, Q ) (13) t B,αγπ ( q, k, Q ) = t B,αγπ ( W, Q ) . (14)The Q evolution of the s- and u-channel nucleon poleterms of the background is described by the form factorsof Ref. [5]. At the e.m. vertices of the pion-pole andseagull terms we apply a monopole form for the pionform factor and a dipole form for the axial form factor,while the standard dipole form factor is used for thevector-meson exchange.We note that the background contribution of MAID98was defined by t B,αγπ (MAID98) = v B,αγπ ( W, Q ) and as-sumed to be a real and smooth function. The unitariza-tion of the total amplitude was then provided by an ad-ditional phase φ α in the resonance contribution such that D. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 the phase of the total amplitude had the phase δ α of therespective πN scattering state. In MAID2007, however,the background contributions are complex functions de-fined according to K-matrix theory, t B,αγπ ( W, Q ) = v B,αγπ ( W, Q ) [1 + it απN ( W )] , (15)where the pion-nucleon elastic scattering amplitudes, t απN = [ η α exp(2 iδ α ) − / i , are described by the phaseshifts δ α and the inelasticity parameters η α taken fromthe GWU/SAID analysis [16]. The assumed structure ofthe background corresponds to neglecting the principalvalue integral in the pion-rescattering term of Eq. (12).Our previous studies of the P -wave multipoles in the(3,3) channel [10,15] showed that the “pion cloud”contributions of the principal value integral are effectivelyincluded by the dressing of the γN N ∗ vertex.Furthermore, the threshold behavior of the S waveswas improved. The results of the dynamical ap-proaches [17] show that the pion cloud contributionsare very important to obtain a good description of the E multipole in the π p channel. For this purpose wehave introduced the following phenomenological term: E corr0+ ( W, Q ) = A (1 + B q ) G D ( Q ) , (16)with A and B free parameters fixed by fitting the low-energy π photoproduction data, and G D the standardnucleon dipole form factor. The threshold correction forthe L multipole we will consider later in Sect. 5.3. As aresult the background contribution of MAID now dependson 8 parameters. We furthermore account for the cuspeffect in the π p channel appearing at the π + n thresholdby the term [18,19] E cusp0+ = − a πN ω c ReE γπ + s − ω π ω c , (17)where ω c = 140 MeV is the π c.m. energy at the cusp and a πN = 0 . /m π + the pion charge-exchange amplitude. For the resonance contributions we follow Ref. [6] and as-sume Breit-Wigner forms for the resonance shape, t R,αγπ ( W, Q ) = ¯ A Rα ( W, Q ) f γN ( W ) Γ tot M R f πN ( W ) M R − W − iM R Γ tot e iφ R , (18)where f πN ( W ) is the usual Breit-Wigner factor describingthe decay of a resonance with total width Γ tot ( W ), partial πN width Γ πN , and spin j , f πN ( W ) = C πN (cid:20) j + 1) π k W q mM R Γ πN Γ tot (cid:21) / , (19) with C πN = p / − / √ and , re-spectively. The energy dependence of the partial width isgiven by Γ πN ( W ) = β π Γ R (cid:18) qq R (cid:19) l +1 (cid:18) X R + q R X R + q (cid:19) ℓ M R W , (20)with Γ R = Γ tot ( M R ) and X R a damping parameter and β π the single-pion branching ratio. The expression for thetotal width Γ tot is given in Ref. [6]. The γN N ∗ vertex isassumed to have the following dependence on W : f γN ( W ) = (cid:18) k W k R (cid:19) n (cid:18) X R + k R X R + k W (cid:19) , (21)where n is obtained from a best fit to the real photondata, and with the normalization condition f γN ( M R ) = 1.The phase φ R ( W ) in Eq. (18) is introduced to adjustthe total phase such that the Fermi-Watson theorem isfulfilled below two-pion threshold. For the S - and P -wavemultipoles we extend this unitarization procedure up to W = 1400 MeV. Because of a lack of further information,we assume that the phases φ R are constant at the higherenergies. In particular we note that the phase φ R for the P (1232) excitation vanishes at W = M R = 1232 MeVfor all values of Q . For this multipole we may even applythe Fermi-Watson theorem up to W ≈ η α remains close to 1. For the D - and F -wave resonances, the phases φ R are assumedto be constant and determined from the best fit.Whereas MAID98 [6] included only the 7 mostimportant nucleon resonances, essentially with onlytransverse e.m. couplings, our present version containsall 13 resonances of the 4-star tier below 2 GeV withtransverse electric ( ¯ A Rα = ¯ E l ± ), transverse magnetic( ¯ A Rα = ¯ M l ± ), and Coulomb ( ¯ A Rα = ¯ S l ± ) couplings: P (1232), P (1440), D (1520), S (1535), S (1620), S (1650), D (1675), F (1680), D (1700), P (1720), F (1905), P (1910), and F (1950). Because we deter-mine the isovector amplitudes from the proton channels,the number of the e.m. couplings is 34 for the protonand 18 for the neutron channels, that is 52 parametersaltogether. These are taken to be constant in a single-Q analysis, e.g., in photoproduction but also at any fixed Q if sufficient data are available in the chosen energy andangular range. Alternatively, the couplings have also beenparameterized as functions of Q , as is discussed in Sec. 5.The more commonly used helicity amplitudes A / , A / , and S / are given by linear combinations of thee.m. couplings ¯ A Rα . These relations take the form A ℓ +1 / = −
12 [( ℓ + 2) ¯ E ℓ + + ℓ ¯ M ℓ + ] ,A ℓ +3 / = 12 p ℓ ( ℓ + 2)( ¯ E ℓ + − ¯ M ℓ + ) , (22) S ℓ +1 / = − ℓ + 1 √ S ℓ + . Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 5 Table 1.
The reduced e.m. amplitudes ¯ A α defined by Eq. (18) in terms of the helicity amplitudes. N ∗ ¯ E ¯ M ¯ SS / S − A / — −√ S / P / P
33 12 ( √ A / − A / ) − ( √ A / + A / ) − √ S / P / P — A / −√ S / D / D − ( √ A / + A / ) − ( √ A / − A / ) − √ S / D / D
35 13 ( √ A / − A / ) − ( √ A / + A / ) − √ S / F / F − ( √ A / + A / ) − ( √ A / − A / ) − √ S / F / F
37 14 ( p A / − A / ) − ( p A / + A / ) − √ S / for resonances with total spin j = ℓ + , and A ℓ − / = 12 [( ℓ + 1) ¯ M ℓ − − ( ℓ −
1) ¯ E ℓ − ] ,A ℓ − / = − p ( ℓ − ℓ + 1)( ¯ E ℓ − + ¯ M ℓ − ) , (23) S ℓ − / = − ℓ √ S ℓ − for total spin j = ℓ − . The inverse relations for the partialwaves are listed in Table 1. The helicity amplitudes arerelated to matrix elements of the e.m. current J µ betweenthe nucleon and the resonance states, e.g., as obtained inthe framework of quark models, A / = − r πα em k W < R, | J + | N, − > ζ ,A / = − r πα em k W < R, | J + | N, > ζ , (24) S / = − r πα em k W < R, | ρ | N, > ζ , where J + = − √ ( J x + iJ y ) and α em = 1 / ζ , which in principle can be obtained from the pionic de-cay of the resonance calculated within the same model.Because this phase is ignored in most of the literature,the comparison of the sign is not always meaningful, espe-cially in critical cases such as the Roper resonance whosecorrect sign is not obvious from the data. In contrast withMAID98 and MAID2000, our present version uses the he-licity amplitudes A / , A / , and S / for photoproductionas input parameters, except for the P (1232) resonancewhich is directly described by the 3 e.m. amplitudes ¯ A α . The unitary isobar model MAID2007 has been developedto analyze the world data of pion photo- and electropro-duction. In this section we fix (I) the background parame-ters and the helicity amplitudes for pion photoproduction( Q = 0) and (II) the dependence of the resonance con-tributions on the c.m. energy W . These results are thengeneralized to pion electroproduction in the next section. The main part of the photoproduction data was takenfrom the GWU/SAID compilation of SAID2000, whichincludes the data published between 1960 and 2000, a to-tal of 14700 data points. A separation of these data indifferent physical channels and observables is given in Ta-ble 2. In the following years the data base was extendedby including recent results from MAMI (Mainz) [20,21,22,23], GRAAL (Grenoble) [24,25], LEGS (Brookhaven) [26],and ELSA (Bonn) [27] as listed in Table 3. Altogether4976 more data points were added. As a result our fulldata base contains 19676 points within the energy range140 MeV < E γ < Table 2.
Number of data points from the SAID2000 data basefor differential cross sections ( dσ ), photon asymmetries ( Σ ),target asymmetries ( T ), and recoil asymmetries ( P ).channel dσ Σ T P total nπ + pπ pπ − Table 3.
Number of data points collected after 2000 for differ-ential cross sections ( dσ ), photon asymmetries ( Σ ), and helicityasymmetries ∆σ = dσ / − dσ / .channel range E γ (MeV) data points (observable) Ref. pπ dσ ) + 357 ( Σ ) [20] pπ dσ ) + 138 ( ∆σ ) [21] pπ + dσ ) + 129 ( ∆σ ) [22] pπ + dσ ) + 102 ( ∆σ ) [23] pπ + Σ ) [24] pπ dσ ) + 469 ( Σ ) [25] nπ − dσ ) [26] pπ dσ ) [27]total 4976 D. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 Our strategy for the data analysis is as follows. First,we try to find a global (energy dependent) solution by fit-ting all the data in the range 140 MeV ≤ E γ ≤ t αγπ / Re t αγπ above the two-pion thresh-old. At the lower energies this phase is constrained by the πN scattering phase. In a second step we perform local(single energy) fits to the data, in energy bins of 10 MeVin the range 140 MeV ≤ E γ ≤
460 MeV and of 20 MeVfor the higher energies, by varying the absolute values ofthe multipoles but keeping the phase as previously deter-mined. Similar to the prescription of the SAID group weminimize the modified χ function χ = N data X i (cid:18) Θ i − Θ exp i δΘ i (cid:19) + N mult X j (cid:18) X j − ∆ (cid:19) . (25)The first term on the r.h.s. of this equation is the stan-dard χ function with Θ i the calculated and Θ exp i the mea-sured observables, δΘ i the statistical errors, and N data thenumber of data points. In the second term, N mult is thenumber of the varied multipoles and X j describes the de-viation from the global fit. The fitting procedure startswith the initial value X j = 1 corresponding to the globalsolution, and the quantity ∆ enforces a smooth energydependence of the single-energy solution. In the limit of ∆ → ∞ we obtain the standard χ , and for ∆ → ∆ is chosen from the condition1 < χ /χ < .
05. The described two-step fitting proce-dure can be repeated several times by adjusting the energydependence of the global solution, for example by chang-ing the parameters X R and n in Eqs. (20-21) in order toimprove the agreement between the global and local solu-tions. Our results for χ are summarized in Table 4 by com-paring the local and global solutions for different energyranges and channels. We recall that the number of variedmultipoles N mult in the proton and neutron channels isdifferent. Since the number of data points in the protonchannels ( γπ and γπ + ) is about one order of magnitudelarger than for the neutron channel ( γπ − ), we proceed asfollows. First, we analyze the proton channel and extractthe multipoles p E / l ± , p M / l ± , E / l ± , and M / l ± as definedin Ref. [6]. Second, with the thus obtained values for theisospin 3/2 multipoles, we extract the multipoles n E / l ± and n M / l ± from the neutron channel. In this way weminimize the pressure from the large number of protondata on the results in the neutron channel. The numberof varied multipoles also depends on the energy. For E γ <
450 MeV we vary all the S - and P -wave multipolesplus p,n E / − and E / − . At the higher energies we includeall the multipoles up to the F waves. Table 4.
Results for χ from single-energy (se) and global (gl)solutions. proton E γ [MeV] N mult N data χ χ E γ [MeV] N mult N data χ χ The best fit for the background and the resonanceparameters yields the results listed in Tables 5 and6, respectively. The PS-PV mixing parameter and thevector-meson coupling constants are defined as in Ref. [6].However we note that in the present version we donot use form factors at the hadronic vertices involvingvector-meson exchange. In Tables 7 and 8 we com-
Table 5.
Masses and coupling constants for vector mesons,PS-PV mixing parameter Λ m , and parameter A for the low-energy correction of Eq. (16). m V [MeV] λ V ˜ g V ˜ g V / ˜ g V ω
783 0.314 16.3 -0.94 ρ
770 0.103 1.8 12.7 Λ m = 423 MeV A = 1 . × − /m + π B = 0 . fm pare the helicity amplitudes obtained from MAID2003and MAID2007 with the results of the PDG [2] andGWU/SAID [28,29] analysis. As is very typical for aglobal analysis with about 20,000 data points fitted toa small set of 20-30 parameters, the fit errors appearunrealistically small. However, one should realize thatthese errors only reflect the statistical uncertainty of theexperimental error, whereas the model uncertainty canbe larger by an order of magnitude. We therefore do notlist our fit errors, which in fact are very similar in theGW02 or GW06 fits of the SAID group [28,29]. The onlyrealistic error estimate is obtained by comparing differentanalysis, such as SAID, MAID, and coupled-channelsapproaches.Next we present our results for the multipoles startingwith the threshold region. In Fig. 1 we demonstratethe effects of the low-energy correction and the cusp . Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 7 Table 6.
Resonance masses M R , widths Γ R , single-pion branching ratios β π , and angles φ R as well as the parameters X R , n E ,and n M of the vertex function Eq. (21). proton neutron N ∗ M R [MeV] Γ R [MeV] β π φ R [deg] X R [MeV] n E n M n E n M P (1232) 1232 130 1.0 0.0 570 -1 2 -1 2 P (1440) 1440 350 0.70 -15 470 — 0 — -1 D (1520) 1530 130 0.60 32 500 3 4 7 2 S (1535) 1535 100 0.40 8.2 500 2 — 2 — S (1620) 1620 150 0.25 23 470 5 — 5 — S (1650) 1690 100 0.85 7.0 500 4 — 4 — D (1675) 1675 150 0.45 20 500 3 5 3 4 F (1680) 1680 135 0.70 10 500 3 3 2 2 D (1700) 1740 450 0.15 61 700 4 5 4 5 P (1720) 1740 250 0.20 0.0 500 3 3 3 3 F (1905) 1905 350 0.10 40 500 4 5 4 5 P (1910) 1910 250 0.25 35 500 — 1 — 1 F (1950) 1945 280 0.40 30 500 6 6 6 6 Table 7.
Proton helicity amplitudes at Q = 0 for the majornucleon resonances, in units 10 − GeV − / . The results withMAID2003 and MAID2007 are compared to the PDG [2] andGWU/SAID [29] analysis.PDG GW06 2003 2007 P (1232) A / -135 ± ± A / -250 ± ± E /M ± P (1440) A / -65 ± ± D (1520) A / -24 ± ± A / ± ± S (1535) A / ±
30 91.0 ± S (1620) A / ±
11 49.6 ± S (1650) A / ±
16 22.2 ± D (1675) A / ± ± A / ± ± F (1680) A / -15 ± ± A / ±
12 133.6 ± D (1700) A / ±
15 125.4 ± A / ±
22 105.0 ± P (1720) A / ±
30 96.6 ± A / -19 ±
20 -39.0 ± F (1905) A / ±
11 21.3 ± A / -45 ±
20 -45.6 ± F (1950) A / -76 ±
12 -78 -94 A / -97 ±
10 -101 -121 effect for π photoproduction, as described by Eqs. (16)and (17), respectively. The prediction of MAID98 for π photoproduction at threshold (dotted lines) lies sub-stantially below the data. In accordance with Ref. [17],the phenomenological term E corr simulates the pionoff-shell rescattering or pion-loop contributions of ChPT.The cusp term of Eq. (17) describes the strong energydependence near π + threshold, which has its origin inthe pion mass difference and the strong coupling with Table 8.
Neutron helicity amplitudes at Q = 0 for the majornucleon resonances. GW02 are the results GWU/SAID analy-sis [28]. Further notation as in Tab. 7.PDG GW02 2003 2007 P (1440) A / ±
10 47 ± D (1520) A / -59 ± ± A / -139 ±
11 -112 ± S (1535) A / -46 ±
27 -16 ± S (1650) A / -15 ±
21 -28 ± D (1675) A / -43 ±
12 -50 ± A / -58 ±
13 -71 ± F (1680) A / ±
10 29 ± A / -33 ± ± P (1720) A / ±
15 17 -3 A / -29 ±
61 -75 -31 the π + n channel. The figure shows that the off-shellpion rescattering substantially improves the agreementwith the data. However, one problem still remains in thethreshold region. The experimental photon asymmetry Σ in π photoproduction at E γ ≈
160 MeV takes positivevalues, whereas the MAID results are negative in thisregion. As has been demonstrated in Refs. [17,32],this observable is very sensitive to the M − multipolewhich strongly depends on the details of the low-energybehavior of Roper resonance, vector meson and off-shellpion rescattering contributions. Therefore, a slight modi-fication of one or all of these mechanisms can drasticallychange the photon asymmetry.Figures 2-4 display the results for the most important S and P waves in the ∆ (1232) region. However, a lookat Fig. 5 shows that also the D -wave amplitudes p E / − , E / − , and n E / − , give sizable contributions in this region,in particular through their real parts. In these figures, wepresent the MAID and SAID global (energy dependent) D. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 solutions, together with our local (single energy) fitobtained for energy bins of 10 MeV. In general the MAIDand SAID results are close, which is not too surprisingbecause the phases are constrained by the Fermi-Watsontheorem. However, there are much larger differences inthe p E / and p E / − amplitudes, which indicates that thepresent data base is still too limited to determine thesebackground amplitudes in a reliable way.More substantial discrepancies between the MAID andSAID analyses are found in the second and third resonanceregions. A detailed comparison of the two models is shownin Figs. 6-13. As pointed out in Sect. 3.1, it is prerequisiteto know the phases of the multipoles in order to get correctsingle-energy solutions above the two-pion threshold. InMAID2007 these phases are determined by Eqs. (9), (15),and (18). The SAID analysis is based on the followingparametrization of the partial wave amplitudes: t γπ = (Born + A ) (1 + it πN ) + B t πN +( C + iD ) (Im t πN − | t πN | ) , (26)where A , B C and D are polynomials in the energywith real coefficients, and t πN is the pion-nucleon elasticscattering amplitude of Eq. (15). As seen in the firstresonance region, the most serious differences betweenMAID and SAID are again found for the real parts of themultipoles p E / and p E / − . We have checked the phasesof these multipoles by independent calculations on thebasis of dispersion relations [33,34]. The result confirmedour phase relations. Concerning the small amplitudes, themost sizable differences between SAID and MAID are inthe M / − p M / and p E / multipoles. In the neutronchannel, the largest differences are in the multipoles n E / , n E / − , n E / , and n M / . In the last two casesthis is due to the large contribution from the P (1720)resonance which is not found in the SAID analysis (seeTable 7).Let us finally discuss the possible contributions of theweaker resonances. As discussed in Ref. [35], the two addi-tional S resonances found with masses of about 1800 and2000 MeV might also show up in pion photoproduction.This conclusion was mainly based on the single-energy so-lution of the SAID group. As illustrated by Fig. 14, ourpresent analysis requires only one additional S resonancewith mass M R ≈ M R ≈ ω and ρ exchange) for all the partial waves. Another in-teresting topic deserving further experimental and the-oretical studies, concerns the Roper or P channel. Asclearly seen in Fig. 15, both our and the SAID analy-sis yield a second resonance structure of the p M / − mul- tipole at E γ ≈ W ≈ Γ tot ≈ Γ tot ≈ A / ≈ − .
024 GeV − / , thePDG lists 0 . ± .
022 GeV − / . Of course, these num-bers do strongly depend on the values for the single-pionbranching ratio. On the other hand, we do not anticipatelarge effects from different definitions of the backgroundin this channel, because the background contribution isvery small in the resonance region (see Fig. 7). In most of the pion electroproduction experiments the five-fold differential cross section was measured. However, dif-ferent conventions exist for the partial cross sections, andtherefore we recall the definitions used in MAID. For anunpolarized target the cross sections written as the prod-uct of the virtual-photon flux factor Γ v and the virtualphoton cross section dσ v /dΩ π [36], dσdΩ Lf dE Lf dΩ π = Γ v dσ v dΩ π , (27) dσ v dΩ π = dσ T dΩ π + ǫ dσ L dΩ π + p ǫ (1 + ǫ ) dσ LT dΩ π cos Φ π (28)+ ǫ dσ T T dΩ π cos 2 Φ π + h p ǫ (1 − ǫ ) dσ LT ′ dΩ π sin Φ π , where ǫ and h describe the polarizations of the virtualphoton and the electron, respectively. We further notethat the hadronic kinematics is expressed in the c.m. sys-tem, whereas the electron and virtual photon kinematicsis written in the lab frame, as indicated by L in the fol-lowing variables: the initial and final electron energies E Li and E Lf , respectively, the electron scattering angle θ L , thephoton energy ω L = E Li − E Lf , and the photon three-momentum k L . With these definitions the virtual photonflux and the transverse photon polarization take the form ǫ = 11 + 2 k L Q tan θ L , Γ v = α em π E Lf E Li KQ − ǫ . (29)As in our previous notation [36], the flux is denotedby the photon “equivalent energy” in the lab frame, K = K H = ( W − m ) / m as originally introduced byHand [37]. Another definition was given by Gilman [38]who used K = K G = | k L | .The first two terms on the r.h.s. of Eq. (28) are thetransverse ( T ) and longitudinal ( L ) cross sections. Theydo not depend on the pion azimuthal angle Φ π . The thirdand fifth terms describe longitudinal-transverse interfer-ences ( LT , LT ′ ). They contain an explicit factor sin θ π and . Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 9 therefore are vanishing along the axis of momentum trans-fer. The same is true for the fourth term, a transverse-transverse interference ( T T ) proportional to sin θ π . It isuseful to express these 5 cross sections in terms of hadronicresponse functions depending only on 3 independent vari-ables, i.e., R i = R i ( Q , W, θ π ). The corresponding rela-tions take the form dσ T dΩ π = qk W R T , dσ T T dΩ π = qk W R T T , dσ L dΩ π = qk W Q ω γ R L ,dσ LT dΩ π = qk W Qω γ R LT , dσ LT ′ dΩ π = qk W Qω γ R LT ′ . (30)As a result of this equation, the longitudinal ( L ) andlongitudinal-transverse ( LT and LT ′ ) response functionsmust be proportional to ω γ and ω γ , respectively, in orderto avoid non-physical singularities at the energy for whichthe c.m. virtual photon energy passes through zero. The5 response functions may be expressed in terms of 6independent CGLN amplitudes F , ..., F [39], or in termsof the helicity amplitudes H , ..., H , which are linearcombinations of the CGLN amplitudes. The relevantexpressions can be found in Refs. [36,40]. The main part of our data base for pion electroproductionincludes the compilation of the GWU/SAID group [41] in2000 and recent data from Bonn and JLab (see Table 9).Altogether this base contains about 70000 data pointswithin the energy range 1.074 GeV < W < ≤ Q ≤ . In addi-tion we have analyzed high precision data from Bates [47,48], Mainz [49,50], and JLab [5,51]. Our fitting proce-dure was as follows. In a first step we fitted the data setsat constant values of Q (single- Q fit). This procedureis similar to the partial-wave analysis for pion photopro-duction except for the additional longitudinal couplings ofthe resonances. Second, we introduced a smooth Q evolu-tion of the e.m. transition form factors and parameterizedthe 3 helicity amplitudes accordingly. In a combined fitwith the complete electroproduction data base and infor-mation from the single- Q fits we finally constructed the Q -dependent solution (super-global fit). This new solu-tion (MAID2007) was then compared with the previoussolution (MAID2003) in terms of χ as presented in Ta-ble 9. In most cases the new fit improves the descriptionof the data, in particular for the nπ + channel. ∆ (1232) form factors In the literature the e.m. properties of the
N ∆ (1232) tran-sition are described by either the magnetic ( G ∗ M ), electric( G ∗ E ), and Coulomb ( G ∗ C ) form factors or the helicity am-plitudes A / , A / , and S / , which can be derived from Table 9.
The number of data points, N data , and the χ valueper data point obtained with MAID2003 and MAID2007.Ref. W (MeV) N data χ /N data (2003)channel Q (GeV ) observables χ /N data (2007)SAID00 1074-1895 13152 3.238 pπ σ , ... 3.172SAID00 1125-1975 5464 3.297 nπ + σ , ... 4.188Bonn02 [42] 1153-1312 4914 1.378 pπ σ pπ σ pπ σ LT ′ nπ + σ LT ′ nπ + σ pπ σ pπ , nπ + σ , ... 2.437SAID00 1253-1976 799 2.100 pπ − σ the reduced e.m. amplitudes ¯ A α as defined by Eq. (18). Itis worthwhile pointing out that these amplitudes are re-lated to the multipoles over the full energy region, that is,they are the primary target of the fitting procedure. Theform factors and helicity amplitudes are then obtained byevaluating the reduced e.m. amplitudes at the resonanceposition W = M ∆ =1232 MeV. The respective relationstake the following form: G ∗ M ( Q ) = − c ∆ ( A / + √ A / ) = 2 c ∆ ¯ A ∆M ( M ∆ , Q ) ,G ∗ E ( Q ) = c ∆ ( A / − √ A / ) = − c ∆ ¯ A ∆E ( M ∆ , Q ) ,G ∗ C ( Q ) = √ c ∆ M ∆ k ∆ S / = − c ∆ M ∆ k ∆ ¯ A ∆S ( M ∆ , Q ) , with c ∆ = (cid:18) m k ∆W πα em M ∆ k ∆ (cid:19) / , (31)and where k ∆ = k ∆ ( Q ) = k ( M ∆ , Q ) and k ∆W = k ( M ∆ ,
0) are the virtual photon momentum and the pho-ton equivalent energy at resonance. Because the ∆ (1232)is very close to an ideal resonance, the real part of theamplitudes vanishes for W = M ∆ and the form factorscan be directly expressed by the imaginary parts of thecorresponding multipoles at the resonance position, G ∗ M ( Q ) = b ∆ Im { M (3 / ( M ∆ , Q ) } ,G ∗ E ( Q ) = − b ∆ Im { E (3 / ( M ∆ , Q ) } , (32) G ∗ C ( Q ) = − b ∆ M ∆ k ∆ Im { S (3 / ( M ∆ , Q ) } , where b ∆ = (cid:18) m q ∆ Γ ∆ α em k ∆ (cid:19) / , and with Γ ∆ = 115 MeV and q ∆ = q ( M ∆ ) the pion mo-mentum at resonance. The above definition of the formfactors is due to Ash [52]. The form factors of Jones andScadron [53] are obtained by multiplying our form fac-tors with p Q / ( M N + M ∆ ) . We note that the formfactor G ∗ C differs from our previous work [54] by the fac-tor 2 M ∆ /k ∆ in Eq. (32). With these definitions all 3transition form factors remain finite at pseudo-threshold(Siegert limit). In the literature, the following ratios ofmultipoles have been defined: R EM = − G ∗ E G ∗ M = A / − √ A / A / + √ A / , (33) R SM = − k ∆ M ∆ G ∗ C G ∗ M = √ S / A / + √ A / . (34)In MAID2003 the Q dependence of the e.m. ampli-tudes ¯ A ∆α was parameterized as follows:¯ A ∆α ( W, Q ) = A α (1 + β α Q n α ) kk W e − γ α Q G D ( Q ) , (35)where G D ( Q ) = 1 / (1 + Q / .
71 GeV ) is the dipoleform factor. MAID2007 follows this parametrization forthe magnetic and electric amplitudes, although with some-what different values of the parameters (see Table 10).In order to fulfill the Siegert theorem, we have howeverchanged the description of the Coulomb amplitude asspecified below. The results of MAID2003 and MAID2007 Table 10.
Parameters for the N∆ amplitudes given byEqs. (35) and (42). The amplitudes A α are in units 10 − GeV − / , the parameters β and γ in GeV − . For the Coulombamplitude in MAID2007 we use Eq. (42) with d =4.9.M E S model A α
300 -6.50 -19.50 2003300 -6.37 -12.40 2007 β α γ α n α for G ∗ M ( Q ) are compared in Fig. 16. Because our single- Q analysis follows the global fit closely, it is not shownin the figure. We find an excellent agreement with thedata, which also include the new high- Q data of theJLab/CLAS Collaboration [13]. At this point a word ofcaution is in order. Because the form factors are extractedfrom the multipoles by Eq. (32), they are proportionalto √ Γ ∆ . The MAID fit to the experimental data yields Γ ∆ =130 MeV, which is different from the usually assumedvalue of about 115 MeV. Therefore, in order to comparewith form factor values of other analyses, we scale our pre-dicted form factor with p / G ∗ M (0) / N → ∆ magnetic transition moment, µ N∆ = 3 . ± .
03, in unitsof the nuclear magneton. R EM and R SM Let us next discuss our results for the R EM and R SM ratios. In all previous solutions these ratios were nearlyconstant for Q < . However, calculations in ef-fective field theories [55,56] and dynamical models [10,15,57] indicated a rapid rise of R SM for Q → +0. Thisdependence is rather model-independent, because it re-flects the behavior of the multipoles at physical threshold(pion momentum q →
0) and pseudothreshold (Siegertlimit, photon momentum k →
0) [58]. The longitudinaland Coulomb multipoles are related by gauge invariance, k · J = ω γ ρ , which leads to | k | L Iℓ ± ( W, Q ) = ω γ S Iℓ ± ( W, Q ) . (36)Since the photon c.m. energy ω γ vanishes for Q = Q = W − m , the longitudinal multipole must have a zeroat that momentum transfer, L Iℓ ± ( W, Q ) = 0. Further-more, gauge invariance implies that the longitudinal andCoulomb multipoles take the same value in the real pho-ton limit, L Iℓ ± ( W, Q = 0) = S Iℓ ± ( W, Q = 0). Finally,the multipoles obey the following model-independent re-lations at physical threshold ( q →
0) and pseudothreshold( k → E Iℓ + , L Iℓ + ) → k ℓ q ℓ ( ℓ ≥ M Iℓ + , M Iℓ − ) → k ℓ q ℓ ( ℓ ≥
1) (37)( L Iℓ − ) → kq ( ℓ = 1)( E Iℓ − , L Iℓ − ) → k ℓ − q ℓ ( ℓ ≥ . According to Eq. (36) the Coulomb amplitudes acquire anadditional factor k at pseudothreshold, i.e., S Iℓ ± ∼ kL Iℓ ± .This limit is reached at Q = Q = − ( W − m ) (pseudo-threshold), and because no direction is defined for k =0, the electric and longitudinal multipoles are no longerindependent at this point, E Iℓ + /L Iℓ + → E Iℓ − /L Iℓ − → − ℓ/ ( ℓ −
1) if k → . (38)In the case of the N ∆ multipoles, Eq. (38) yields the fol-lowing relation in the limit k → L / → E / → O ( k )and consequently S / = kE / /ω γ → O ( k ). Althoughthe pseudo-threshold is reached at the unphysical point Q = − ( M ∆ − m ) ≈ − .
084 GeV , it still influencesthe multipoles near Q = 0 because of the relatively smallexcitation energy of the ∆ (1232). In particular we get the . Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 11 following relation for Q → Q : R SM = S (3 / M (3 / = kω γ E (3 / M (3 / → kM ∆ − m R EM . (39)With increasing value of Q , the Siegert relation fails todescribe the experimental data. Moreover, it contains asingularity at ω γ =0, which occurs in ∆ (1232) electropro-duction already at Q = 0 .
64 GeV . However, we obtaina good overall description by using the idea of Ref. [59]that the ratio R SM is related to the (elastic) form factorsof the neutron, R SM ( Q ) = m k ∆ ( Q ) G nE ( Q )2 Q G nM ( Q ) . (40)This relation gives the necessary proportionality to thephoton momentum at small Q , describes the experimen-tal value of the ratio over a wide range of Q , and yields anasymptotic behavior consistent with the prediction of per-turbative QCD that R SM should approach a constant for Q → ∞ . This leads to the following simple parametriza-tion: R SM ( Q ) = − k ∆ ( Q )8 m a dτ , (41)with τ = Q / (4 m ), and the parameters a and d to bedetermined by a fit to the data. On the basis of this ansatz,the Coulomb coupling has been modified as follows:¯ A ∆S ( W, Q ) = A S β S Q dτ k k W k ∆W e − γ S Q G D ( Q ) , (42)with parameters given in Table 10. This leads to the mul-tipole ratio R SM ( Q ) = A S A M
11 + dτ (cid:18) β S Q β M Q (cid:19) k ∆ k ∆W . (43)By construction this ratio vanishes in the Siegert limit, Q → Q , and approaches a (negative) constant for Q → ∞ in agreement with perturbative QCD. However,a word of caution has to be added at this point. Thepolynomials and gaussians used to fit the data in therange of low and intermediate virtualities, Q <
10 GeV ,should not be expected to yield realistic extrapolationsto the higher values of Q .The correct Siegert limit is even more important forpion S -wave production in the threshold region, in whichcase the pseudo-threshold comes as close as Q = − m π ≈− .
02 GeV . The term describing the pion cloud contri-bution has therefore been parameterized as follows: L corr0+ ( W, Q ) = ω γ ω pt e − β ( Q − Q pt ) E corr0+ ( W, Q ) , (44)where ω = − Q = ( W − m ) . From a fit to π electroproduction data near threshold [61], we obtain β = 10 GeV − . In the future we intend to study pion electroproduction near threshold in more detail.Figures 17 and 18 display the super-global solutions ofMAID2003 (dashed lines) and MAID2007 (solid lines) forthe ratios R EM and R SM in comparison with other analy-ses. Different from our previous solution, the ratio R EM ofMAID2007 stays always below the zero line, in agreementwith the original analysis of the data [13,62] and also withthe dynamical model of Sato and Lee [57] who concludedthat R EM remains negative and tends towards more nega-tive values with increasing Q . This indicates that the pre-dicted helicity conservation at the quark level is irrelevantfor the present experiments. We also analyzed the newdata of Ref. [13] in the range of 3 GeV ≤ Q ≤ and found slightly decreasing values of R EM from oursingle- Q analysis. In this analysis we varied both the ∆ and the Roper multipoles. For the ratio R SM both thesuper-global and the single- Q solutions yield ratios thatasymptotically tend to a negative constant. This result isin good agreement with the prediction of Ref. [59] (dash-dotted curve in Fig. 18) but disagrees with our previoussolution and with the analysis of Ref. [13]. As discussedbefore, the new solution has a large slope at small Q as aconsequence of the Siegert theorem. The following Fig. 19displays the Q dependence of the helicity amplitudes forthe N ∆ (1232) transition. Our single- Q fit is in excellentagreement with the super-global solution, except for thevalues of S / at Q = 0 . . Above the two-pion threshold we can no longer applythe two-channel unitarity and consequently the Watsontheorem does not hold. Therefore, the backgroundamplitude of the partial waves does not vanish at reso-nance as was the case for the ∆ (1232) resonance. As animmediate consequence the resonance-background sep-aration becomes more model-dependent. In MAID2007we choose to separate the background and resonancecontributions according to the K-matrix approximation.Furthermore, we recall that the absolute values of thehelicity amplitudes are correlated with the values usedfor the total resonance width Γ R and the single-pionbranching ratio β π . On the experimental side, the dataat the higher energies are no longer as abundant asin the ∆ region. However, the large data set recentlyobtained by the CLAS collaboration (see Table 9)enabled us to determine the transverse and longitudinalhelicity couplings as functions of Q for all the 4-starresonances below 1700 MeV. These data are available inthe kinematical region of 1100 MeV < W < . < Q < . .The helicity amplitudes for the Roper resonance P (1440) are shown in Fig. 20. Our latest super-globalsolution (solid lines) is in reasonable agreement with thesingle- Q analysis. The figure shows a zero crossing ofthe transverse helicity amplitude at Q ≈ . anda maximum at the relatively large momentum transfer Q ≈ . . The longitudinal Roper excitation risesto large values around Q ≈ . and in fact pro-duces the strongest longitudinal amplitude we can find inour analysis. This answers the question raised by Li andBurkert [63] whether the Roper resonance is a radiallyexcited 3-quark state or a quark-gluon hybrid, becausein the latter case the longitudinal coupling should van-ish completely. From the global fit we find the followingparametrization for the Q dependence of the Roper am-plitudes for the proton and neutron channels: A p / ( Q ) = A ,p / (1 − . Q − . Q ) e − . Q ,S p / ( Q ) = S ,p / (1 + 40 Q + 1 . Q ) e − . Q , (45) A n / ( Q ) = A ,n / (1 + 0 . Q ) e − . Q ,S n / ( Q ) = S ,n / (1 + 2 . Q ) e − . Q , (46)where Q should be inserted in units of GeV . Thenumerical values of the helicity amplitudes for realphotons are given in Table 11. At Q =0 the fit yields alarge neutron value for the Coulomb amplitude S / , butwith increasing Q the proton and neutron amplitudesbecome comparable. Table 11.
Helicity amplitudes for the P (1440) resonance at Q =0 in units 10 − GeV − / . A / S / proton neutron proton neutron P (1440) -61.4 54.1 4.2 -41.5 For all the higher resonances the transverse and lon-gitudinal helicity amplitudes are simply parameterized bythe form A λ ( Q ) = A λ (1 + α Q ) e − βQ . (47)The values of the fit parameters A λ , α and β are listedin Tables 12 and 13. In the following Fig. 21 we presentthe results for the S (1535). As is also known from η electroproduction, the transverse form factor falls off veryslowly. At a virtuality of Q ≈ this resonanceis much stronger than the ∆ (1232) or the D (1520)and only comparable to the Roper. However, due to itsmuch smaller width as compared to the Roper, the S dominates over the Roper at large Q . This result isin agreement with the inclusive electroproduction crosssection on the proton, which clearly shows the dominanceof the ∆ (1232) at small momentum transfer whereas atthe larger momentum transfers the second resonanceregion takes over.In Fig. 22 we compare our results to those of Az-nauryan et al. [64] who used a similar set of the CLASdata in the second resonance region. Our super-global so-lutions (solid lines) agree generally quite well with the JLab-Yerevan analysis, which was performed with bothan isobar model and dispersion analysis. The following Table 12.
The proton parameters for the higher resonances: α and β as defined by Eq. (47), in units GeV − , and S / , thelongitudinal amplitude at Q = 0, in units 10 − GeV − / . Thevalues for the transverse amplitudes A / , / are determinedby the real photon physics and listed in Table 7. A / A / S / S / proton α β α β α βD (1520) 7.77 1.09 0.69 2.10 4.19 3.40 -63.6 S (1535) 1.61 0.70 — — 23.9 0.81 -2.0 S (1620) 1.86 2.50 — — 2.83 2.00 16.2 S (1650) 1.45 0.62 — — 2.88 0.76 -3.5 D (1675) 0.10 2.00 0.10 2.00 0.00 0.00 0.00 F (1680) 3.98 1.20 1.00 2.22 3.14 1.68 -44.0 D (1700) 1.91 1.77 1.97 2.20 0.00 0.00 0.00 P (1720) 1.89 1.55 16.0 1.55 2.46 1.55 -53.0 Table 13.
The neutron parameters for the higher resonances.The values for the transverse amplitudes A / , / are given inTable 8. Further notation as in Table 12. A / A / S / S / neutron α β α β α βD (1520) -0.53 1.55 0.58 1.75 15.7 1.57 13.6 S (1535) 4.75 1.69 — — 0.36 1.55 28.5 S (1650) 0.13 1.55 — — -0.50 1.55 10.1 D (1675) 0.01 2.00 0.01 2.00 0.00 0.00 0.00 F (1680) 0.00 1.20 4.09 1.75 0.00 0.00 0.00 P (1720) 12.7 1.55 4.99 1.55 0.00 0.00 0.00 Fig. 23 displays our super-global and single- Q fits for the D (1520) and F (1680) resonances. The figure demon-strates that (I) the helicity non-conserving amplitude A / dominates for real photons and (II) with increasing val-ues of Q , A / drops faster than the helicity conservingamplitude A / . As a consequence the asymmetry A ( Q ) = | A / | − | A / | | A / | + | A / | (48)changes rapidly from values close to − Q range. As is seen in Fig. 24, the asymmetrycrosses the zero line at Q ≈ . for the D (1520)resonance and at Q ≈ . for the F (1680). As acomparison, the asymmetry A for the ∆ (1232) resonanceis practically constant over this Q range with a value ≈− .
5. This again shows the special role of the ∆ resonance,where the helicity conservation is not observed. . Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 13 Using the world data base of pion photo- and electropro-duction and recent data from Bates/MIT, ELSA/Bonn,MAMI/Mainz, and Jefferson Lab, we have extracted thelongitudinal and transverse helicity amplitudes of nucleonresonance excitation for all the 4-star resonances below W = 2 GeV. For this purpose we have extended ourunitary isobar model MAID and parameterized the Q dependence of the transition amplitudes. The comparisonbetween such super-global solutions with the correspond-ing single- Q fits gives us confidence in the obtainedhelicity couplings for the P (1232), P (1440), S (1535), D (1520), and F (1680) resonances, even though themodel uncertainty is still quite large, particularly for thelongitudinal amplitudes.For the higher 4-star and all 3-star resonances thesituation is less clear. This deplorable situation reflectsthe fact that a model-independent analysis requiresprecision data over a large kinematical range. In somecases double-polarization experiments will be helpful,as has already been shown for pion photoproduction.Furthermore, charged pion electroproduction data areneeded with the same quantity and quality as for neutralpions, in order to resolve the ambiguities in the isospinstructure, in particular for the S and S resonances.While we have mostly discussed the electroproductionfrom proton targets, also the existing neutron data havebeen analyzed. The latter are of course less abundant,and moreover no new neutron data have been reportedover the recent years. Because the isospin symmetry ismost likely on safe grounds in the resonance region, onlythe electromagnetic neutron couplings with isospin 1 / – Dedicated experiments to investigate the higher en-ergy region, which have to include an intense studyof the polarization degrees of freedom. Experience hasshown that even the physics of the ∆ (1232) requiresa full-fledged program to measure the spin observablesin order to understand the background of the non-resonating multipoles. – A fresh approach to also determine the excitation spec-trum of the neutron. As an example, the comparisonof the Roper or P (1440) helicity amplitudes for pro-ton and neutron will shed light on the structure of thisenigmatic resonance. – The open question of the excitation spectrum in thethird resonance region and above deserves furtherstudies in both theory and experiment. This includes“missing” and “exotic”, e.g., 5-quark resonances aswell as more mundane second and third resonances ina multipole, which show up in a particular analysis andnot in another one.In conclusion we hope that MAID2007, just as otherapproaches based on partial-wave analysis, dynamicmodels, coupled-channels calculations, and dispersiontheory, will contribute to settle the mentioned issues andthus to improve our still somewhat vestigial knowledge ofthe nucleon’s resonance structure.
Acknowledgment
This work was supported by the Deutsche Forschungs-gemeinschaft through the SFB 443, by the joint projectNSC/DFG 446 TAI113/10/0-3 and by the joint Russian-German Heisenberg-Landau program.
References
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The E multipole for γp → π p . Real part: the re-sults of MAID98 (dotted line) and MAID2007 without the cuspeffect (dashed-dotted line) as well as the full MAID2007 cal-culation (red solid line). The red dashed line is the imaginarypart of the full MAID2007 solution. The data points are fromRefs. [30]( • ) and [31]( ◦ ).6 D. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 Fig. 2.
The global solutions of MAID2007 (solid lines) and GWU/SAID [29] (red dashed-dotted lines, solution FA06K) for themultipoles p E / , E / , M / , and E / as function of the photon lab energy E γ in the first resonance region. The blue opencircles show our single-energy solution. The dashed lines represent our unitarized background contributions in the M / and E / multipoles.. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 17 Fig. 3.
The multipoles p M / − , M / − , p E / , and p M / as function of the photon lab energy E γ . Further notation as in Fig. 2.8 D. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 Fig. 4.
The multipoles n E / , n M / − , n E / , and n M / as function of the photon lab energy E γ . Further notation as in Fig. 2.. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 19 Fig. 5.
The multipoles p E / − , E / − , and n E / − as function of the photon lab energy E γ . Further notation as in Fig. 2.0 D. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 Fig. 6.
The global solutions of MAID2007 (solid lines) and GWU/SAID [29] (red dashed-dotted lines, solution FA06K) for themultipoles p E / , E / , M / , and E / as function of the photon lab energy E γ in the second and third resonance regions.The blue open circles show our single-energy solution. The dashed lines represent our unitarized background given by Eq. (15).Note that the background for Re M / is out of scale.. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 21 Fig. 7.
The global solutions of MAID2007 (black solid lines) and GWU/SAID [29] (red dashed-dotted lines) for the multipoles p M / − , M / − , p E / , and p M / as function of the photon lab energy E γ in the second and third resonance regions. Furthernotation as in Fig. 6.2 D. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 Fig. 8.
The global solutions of MAID2007 (black solid lines) and GWU/SAID [29] (red dashed-dotted lines) for the multipoles p E / − , p M / − , E / − , and M / − as function of the photon lab energy E γ in the second and third resonance regions. Furthernotation as in Fig. 6.. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 23 Fig. 9.
The global solutions of MAID2007 (solid lines) and GWU/SAID [29] (red dashed-dotted lines) for the multipoles p E / , p M / , p E / − , and p M / − as function of the photon lab energy E γ in the second and third resonance regions. Further notationas in Fig. 6. The resonance contribution to the p E / multipole is very small, and therefore the solid and dashed lines coincide.4 D. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 Fig. 10.
The global solutions of MAID2007 (solid lines) and GWU/SAID [29] (red dashed-dotted lines) for the multipoles E / − , M / − , E / , and M / as function of the photon lab energy E γ in the second and third resonance regions. Further notation asin Fig. 6. The resonance contribution to the E / multipole is very small, and therefore the solid and dashed lines coincide.. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 25 Fig. 11.
The global solutions of MAID2007 (solid lines) and GWU/SAID [29] (red dashed-dotted lines) for the multipoles n E / , n M / − , n E / , and n M / as function of the photon lab energy E γ in the second and third resonance regions. Furthernotation as in Fig. 6.6 D. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 Fig. 12.
The global solutions of MAID2007 (solid lines) and GWU/SAID [29] (red dashed-dotted lines) for the multipoles n E / − , n M / − , n E / , and n M / as function of the photon lab energy E γ in the second and third resonance regions. Furthernotation as in Fig. 6.. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 27 Fig. 13.
The global solutions of MAID2007 (solid lines) and GWU/SAID [29] (red dashed-dotted lines) for the multipoles n E / − and n M / − as function of the photon lab energy E γ in the second and third resonance regions. Further notation as inFig. 6. Fig. 14.
The contribution of a third S resonance in the p E / multipole with M R =1950 MeV, Γ R =200 MeV, single-pionbranching ratio β π =0.4, and helicity amplitude A / =0.028 GeV − / . The solid and dashed lines are our global solutionswith and without this resonance, respectively. The red dashed-dotted lines represent the global SAID solution. The blue opencircles and green crosses are the single-energy solutions of MAID2007 and the SAID, respectively. This resonance is included inMAID2007.8 D. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 Fig. 15.
The contribution of a second P resonance in the p M / − multipole with M R =1700 MeV, Γ tot =47 MeV, single-pionbranching ratio β π =0.1, and helicity amplitude A / = − .
024 GeV − / . The solid and dashed lines are our global solutionswithout and with this resonance, respectively. Further notation as in Fig. 14. The P (1700) is not included in MAID2007.. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 29 Fig. 16.
The Q dependence of the magnetic form factor G ∗ M for the N∆ (1232) transition divided by 3 G D ( Q ). Thesolid and dashed blue lines are the results of MAID2007 andMAID2003, respectively. The red open triangles represent thenew JLab data of Ungaro et al. [13]. See Ref. [10] for the no-tation of the other data points.0 D. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 Fig. 17.
The Q dependence of the ratio R EM at the ∆ (1232) resonance. The blue solid and dashed lines are the super-globalsolutions from MAID2007 and MAID2003, respectively. The data points are from Refs. [42] (open square), [43] (solid triangles),[47] (open diamond), [48] (cross), [5] (open circle), [62] (asterisks), and [13] (open triangles). The green solid circle at Q = 0 inthe left panel is from Ref. [4], and the black solid circles in the right panel are obtained by our single- Q analysis of the datafrom Ref. [13]. Fig. 18.
The Q dependence of the ratio R SM at the ∆ (1232) resonance position. The blue solid and dashed lines are theMAID2007 and MAID2003 super-global solutions, respectively, the dashed-dotted line is obtained using Eq. (41) with a = 0 . d = 1 .
75. The data point of Ref. [49] (diamond) at Q = 0 . is practically identical to the Bates result [47], the fullcircle at Q = 0 . is from Ref. [50]. See Fig. 17 for the notation of the further data points. Fig. 19.
The Q dependence of the 3 helicity amplitudes for the ∆ (1232) resonance, in units 10 − GeV − / . The solid anddashed lines are the MAID2007 and MAID2003 super-global solutions, respectively. The data points are from our single- Q fitsto the π and π + CLAS data (red full and blue open circles, see Table 9 for references), Ref. [62] (blue full circles), Ref. [47](black full circles at Q = 0 .
127 GeV ), and Ref. [2] (green full circles at Q = 0).. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 31 Fig. 20.
The Q dependence of the helicity amplitudes for the P (1440) resonance of the proton. Further notation as in Fig. 19. Fig. 21.
The Q dependence of the helicity amplitudes for the S (1535) resonance of the proton. Further notation as in Fig. 19.2 D. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 Fig. 22.
The Q dependence of the helicity amplitudes for the P (1440) and S (1535) resonances of the proton. The MAID2007super-global analysis (solid lines) and the single- Q fits (red full circles with error bars) are compared to the results of Aznau-ryan [64] obtained from a similar data set within an isobar model (full triangles) and dispersion theory (open triangles). Furthernotation as in Fig. 19. Fig. 23.
The Q dependence of the helicity amplitudes for the D (1520) and F (1680) resonances of the proton. Notation asin Fig. 19.. Drechsel, S. S. Kamalov, L. Tiator: Unitary Isobar Model - MAID2007 33 Fig. 24.
The helicity asymmetry A ( Q ) of Eq. (48) for the D (1520) and F (1680) resonances of the proton. The solid anddashed curves are the super-global MAID2007 and MAID2003 solutions, respectively. The data are the results of our single- Q2