Eta photoproduction in a combined analysis of pion- and photon-induced reactions
D. Rönchen, M. Döring, H. Haberzettl, J. Haidenbauer, U.-G. Meißner, K. Nakayama
aa r X i v : . [ nu c l - t h ] J un Eta photoproduction in a combined analysis of pion- and photon-induced reactions
D. R¨onchen, ∗ M. D¨oring, † H. Haberzettl, J. Haidenbauer,
3, 4
U.-G. Meißner,
1, 3, 4 and K. Nakayama
3, 5 Helmholtz-Institut f¨ur Strahlen- und Kernphysik (Theorie) and BetheCenter for Theoretical Physics, Universit¨at Bonn, 53115 Bonn, Germany Institute for Nuclear Studies and Department of Physics,The George Washington University, Washington, DC 20052, USA Institut f¨ur Kernphysik and J¨ulich Center for Hadron Physics,Forschungszentrum J¨ulich, 52425 J¨ulich, Germany Institute for Advanced Simulation, Forschungszentrum J¨ulich, 52425 J¨ulich, Germany Department of Physics and Astronomy, University of Georgia, Athens, Georgia 30602, USA
The ηN final state is isospin-selective and thus provides access to the spectrum of excited nucleonswithout being affected by excited ∆ states. To this end, the world database on eta photoproductionoff the proton up to a center-of-mass energy of E ∼ . πN → πN , ηN , K Λ and K Σ. For the analysis, the so-called J¨ulichcoupled-channel framework is used, incorporating unitarity, analyticity, and effective three-bodychannels. Parameters tied to photoproduction and hadronic interactions are varied simultaneously.The influence of recent MAMI T and F asymmetry data on the eta photoproduction amplitude isdiscussed in detail. PACS numbers: 11.80.Gw, 13.60.Le, 13.75.Gx.
I. INTRODUCTION
The determination of the spectrum of excited baryonsfrom experimental data is necessary to understand Quan-tum Chromodynamics at low and medium energies. Inthis non-perturbative regime, quark models [1, 2] andlattice calculations [3–5] predict more exited states thanfound so far in partial-wave analyses of experimentaldata. This dilemma is known as “missing resonance prob-lem” [6]. In the past, the dominant source of informa-tion on resonance properties was provided by elastic πN scattering [7–9]. The analysis of inelastic reactions is,however, essential [10, 11] when aiming at a reliable ex-traction of the entire spectrum and, consequently, at anidentification of missing states that might couple predom-inantly to channels other than πN .Among the inelastic channels accessible in πN scatter-ing, the ηN channel plays a crucial role. It couples ex-clusively to states with isospin I = 1 / N ∗ states unaffected by contributions from∆ ∗ states. Moreover, the ηN channel opens at relativelylow energies, in a region which is populated by numer-ous nucleon resonances. For example, some states likethe four-star N (1535)1 / − resonance are known to havea large ηN branching ratio. Less well-established reso-nances like the N (1710)1 / + , whose parameters are onlyweakly constrained from elastic πN scattering [9, 11–13],may show a noticeable signal in their ηN decay [10, 11].A narrow structure discovered in eta photoproduction on ∗ Electronic address: [email protected] † Electronic address: [email protected] the neutron [14–16] at around E = 1 .
68 GeV could alsoappear in eta production on the proton target.In this respect it is unfortunate that the database forthe reaction πN → ηN is problematic over the whole en-ergy range. Its coverage in scattering angles and energiesis too limited to perform a well-founded resonance anal-ysis. Furthermore, for energies around 100 MeV fromthe threshold and beyond, only some experiments wereperformed and many of those are known to suffer fromsystematic uncertainties [9, 10, 17].An alternative experimental window has opened in re-cent years with high-precision measurements of cross sec-tions and polarization observables in eta photoproduc-tion at photon-beam facilities like ELSA, JLab or MAMI,see Refs. [18–20] for reviews. For example, the first mea-surements from the JLab FROST target (polarization E for γp → π + n ) have appeared only recently [21], andmany more polarization data are expected. The databasefor eta photoproduction is not yet as large as the one forpion photoproduction, but it is rapidly growing. In addi-tion, the data already available are of much higher qual-ity than those for the pion-induced reaction πN → ηN .Recently, the first data for the beam-target asymmetry F in γp → ηp was presented by the A2 collaboration atMAMI, together with a measurement of the target asym-metry T [22].A key question is whether eta photoproduction datacan be used to access the poorly known N ∗ branching ra-tios to the ηN channel. The combined analysis of elastic πN scattering and pion photoproduction determines, atleast in principle, the helicity couplings and πN branch-ing ratios. With known helicity couplings, then indeedthe data on eta photoproduction allow one to pin downthe resonance ηN branching ratios, without having to re-sort to the problematic πN → ηN data [9, 10, 17]. Thisargumentation is, however, limited by the fact that evenfor pion photoproduction the database is not yet fullycomplete. The extension of the analysis to eta photo-production will then improve the knowledge of the ηN branching ratios, but, due to incomplete databases, willalso lead to changes both in the helicity couplings andbranching ratios, as will be seen.Given the discussed mismatch in data quality, it isnot recommendable to only fit model parameters tiedto photoproduction and leave the hadronic interactionunchanged: The hadronic amplitude, poorly fixed fromthe πN → ηN data, appears as a sub-process in photo-production. Thus, to avoid any bias, an unconstrainedfit is needed. In this way, the higher statistical weightof the eta photoproduction data even provides a betterconstraint on the hadronic πN → ηN amplitude.The photoproduction of η mesons is also a prime can-didate for a “complete experiment”. From a mathemati-cal point of view, a complete experiment [23] consists of aset of eight carefully chosen observables, which resolve alldiscrete ambiguities up to an overall phase [24, 25]. Forexample, a complete set including F and T is given by { σ, Σ , T, P, E, F, C x , O x } [24]. For experiments with real-istic uncertainties, however, eight observables are not suf-ficient [26–28]. Less than eight observables are requiredin a truncated partial-wave analysis [29, 30].By contrast, for a complete experiment on the reac-tion π − p → ηn only four observables are needed [31,32]. Given the data situation for that reaction, a re-measurement would greatly advance our understandingof the ηN final state. Physics opportunities with a pionbeam are discussed in Ref. [33]. In any case, the currentdatabase does not contain a complete set of observables,neither for photon- nor pion-induced eta production. Onethen has to resort to other approaches that often combinedata from different initial and final states.Over the years, a variety of theoretical approaches hasbeen applied to analyze the pion- and photon-inducedproduction of η mesons. For example, photoproductionof η mesons in the resonance region not too far fromthreshold was studied in the framework of unitarizedchiral perturbation theory in Refs. [34–38]. Consider-ing a broader energy range up to and beyond 2 GeV, K -matrix [7, 39–44] and unitary isobar [45] models arepractical tools to perform an analysis of large amountsof data. Sometimes, the real part of the self-energiesis neglected and only on-shell intermediate states aremaintained, which reduces the complexity of the calcu-lations. For the purpose of a combined analysis of dif-ferent reactions over a wide energy range, so-called dy-namical coupled-channel (DCC) models provide a partic-ularly suited framework. Theoretical constraints of the S -matrix, like two- and three-body unitarity, analyticity,left-hand cuts and complex branch points, are manifestlyimplemented or at least approximated. This enables thereliable determination of the resonance spectrum in termsof pole positions, residues, and helicity couplings in the complex energy plane. The production of η mesons inDCC approaches was studied, e.g., in Refs. [46, 47].Here, we extend the J¨ulich model, a DCC approachpursued over many years [10, 48–51] starting withRef. [52], to perform a simultaneous analysis of the pion-induced reactions πN → πN , π − p → ηn , K Λ, K + Σ − , K Σ , π + p → K + Σ + , and the photon-induced reactions γp → π p , π + n , and ηp . We allow the hadronic ampli-tudes themselves to vary, in addition to the parameterstied to photoproduction. As discussed, this is necessarybecause the quality of the data in π − p → ηn is much in-ferior to the data in eta photoproduction. In a simultane-ous fit to all pion- and photon-induced data, we observethat, indeed, the eta photoproduction data have a stronginfluence. This influence reaches beyond the electromag-netic resonance properties and affects also resonance polepositions and hadronic branching ratios.The paper is organized as follows: In Sec. II, a shortoverview of the applied formalism is given. For a moredetailed introduction of the semi-phenomenological ap-proach to meson photoproduction we refer the reader toRef. [51]. In Ref. [10], an extensive description of thehadronic J¨ulich DCC framework is provided. In Sec. III Awe describe the data analysis and in Sec. III B the fit re-sults are shown. The extracted resonance parametersare presented and discussed in Sec. IV. Technical detailsabout the renormalization of the nucleon mass are sum-marized in an appendix. II. FORMALISM
In the approach (referred to as “J¨ulich model”), thehadronic scattering potential is iterated in a Lippmann-Schwinger equation formulated in time-ordered pertur-bation theory (TOPT) and two-body unitarity is, thus,automatically fulfilled. The three-body ππN states areparameterized through the channels ρN , σN and π ∆.These effective three-body channels are included dynam-ically, i.e., the ππ and πN subsystems match the cor-responding phase shifts [49]. The analytic structure ofthe amplitude is given through real and complex branchpoints [12] and the real, dispersive contributions of theintermediate states. Moreover, t - and u -channel ex-changes of known mesons and baryons constitute thenon-resonant part of the amplitude and serve as “back-ground”. While the u -channel diagrams approximate theleft-hand cuts, t -channel meson exchanges are essential toachieve three-body unitarity [53]. Note that the latter is,at the moment, only approximately satisfied in the J¨ulichmodel. By means of this explicit treatment of the back-ground, strong correlations between the different partialwaves and a non-trivial energy and angular dependence ofthe observables are generated. Although t - and u -channelprocesses are necessary for analytic structure and uni-tarity, they do not fully determine the amplitude. Bareresonance states are included as s -channel processes. Incontrast to previous versions of the approach, here wealso allow for additional contact interactions. Such inter-actions do not spoil the analytic properties ensured by s -, t - and u -channel interactions. They absorb physics be-yond the explicit processes and, thus, increase the model-independence of the approach at the cost of a few moreparameters. Practically, the changes in the amplitudesinduced by the contact terms are comparatively smalland the so-called background is still dominated by the t -and u -channel exchanges. Details are given in the follow-ing section. Note that contact terms are also included inRef. [32].In Ref. [50] the approach was extended to thestrangeness sector incorporating the K + Σ + final statein the analysis. In the J¨ulich2012 model of Ref. [10],the spectrum of nucleon and ∆ resonances was extractedfrom a simultaneous analysis of the reactions πN → πN , ηN , K Λ and K Σ. The extension of the J¨ulich ap-proach to pion photoproduction in a field-theoretical for-mulation, that respects the generalized off-shell Ward-Takahashi identity, was achieved in Ref. [54]. In Ref. [51],by contrast, the photon interaction is approximated in aphenomenological framework and the J¨ulich2012 analysisserves as final-state interaction. The flexible formulationof Ref. [51], used to study the world data on pion photo-production on the proton, proved to be capable of analyz-ing large amounts of data while at the same time main-taining the analytic properties of the J¨ulich approach.This framework, with the addition of contact terms, willbe applied in the present study.
A. Pion-induced reactions
The pion-induced reactions are treated within theJ¨ulich dynamical coupled-channel formalism [10]. The T -matrix which describes the scattering process of a baryonand a meson can be formulated in the partial-wave basisand reads T µν ( q, p ′ , E ) = V µν ( q, p ′ , E )+ X κ ∞ Z dp p V µκ ( q, p, E ) G κ ( p, E ) T κν ( p, p ′ , E ) . (1)In Eq. (1) and in the following, E always means thescattering energy in the center-of-mass (c.m.) frame and q ≡ | ~q | ( p ′ ≡ | ~p ′ | ) is the modulus of the outgoing (incom-ing) three-momentum that can be on- or off-shell. Thechannel indices ν , µ and κ represent the incoming, out-going and intermediate meson-baryon pairs, respectively.The propagators G κ for channels with stable particles,i.e. κ = πN , ηN , K Λ, or K Σ, are given by G κ ( p, E ) = 1 E − E a ( p ) − E b ( p ) + iǫ , (2)with E a = p m a + p and E b = p m b + p being the on-mass-shell energies of the intermediate particles a and b in channel κ with respective masses m a and m b . In caseof the channels with unstable particles ρN , σN and π ∆that parameterize the ππN channels, the propagators aremore complex; for details see Ref. [49].The scattering potentials V µν can be decomposed intoa pole and a non-pole part V µν = V NP µν + V P µν ≡ V NP µν + n X i =0 γ aµ ; i γ cν ; i E − m bi . (3)The quantity V NP denotes the sum of all t - and u -channelexchange diagrams, while V P comprises the s -channelresonance graphs. The functions γ cµ ; i ( γ aν ; i ) correspond tothe bare creation (annihilation) vertices of a resonance i with bare mass m bi and are constructed from an effectiveLagrangian which can be found in Table 8 of Ref. [50].Explicit expressions of γ cµ ; i and γ aν ; i are also given inRef. [50], cf. also Appendix A of Ref. [10]. The ex-change potentials that constitute V NP , also derived fromeffective Lagrangians, are compiled in Appendix B ofRef. [10]. The decomposition of Eq. (3) is slightly modi-fied in the present approach. We implement here contactterms that do not introduce any singularities and thatare used to absorb physics not explicitly contained in theparameterization through s -, t - and u -channel processes.The contact terms are introduced in a separable form andseparately for every partial wave, V CT µν = 1 m N γ CT; aµ γ CT; cν , (4)where the γ CT; cµ ( γ CT; aµ ) have the same functional formas the resonance vertices γ cµ ; i ( γ aµ ; i ) in Eq. (3). The asso-ciated couplings in the γ CT are now new free parametersto be adapted in the fit to the data.Formally, the numerator structure of Eq. (4) is thesame as the one of s -channel pole terms, to ensure thecorrect threshold behavior. Also, the contact terms carrychannel indices. Formally, we can treat the contact termsas bare s -channel processes and absorb the contributionsin the definition of V P : V P µν → n X i =0 γ aµ ; i γ cν ; i E − m bi + V CT µν . (5)For a compact notation, we will no longer distinguishbetween bare resonance vertices γ and structures of thecontact term γ CT in the following. Then, the index i canrefer to a bare s -channel resonance vertex γ or one of theterms in the numerator of Eq. (4).Similar to the potential V µν , the scattering matrix T µν can also be written as the sum of a pole and a non-polepart, T µν = T P µν + T NP µν (6)with the unitary T NP µν defined through T NP µν = V NP µν + X κ V NP µκ G κ T NP κν . (7)Here and in the following, we do not display functionarguments and integration symbols in favor of a morecompact notation. Besides the resonance poles arisingfrom s -channel diagrams in T P , a dynamical generationof poles in T NP is also possible, as explained in Refs. [10,48].Note that the pole part T P can be evaluated fromthe non-pole part T NP . To this purpose one defines thedressed creation (annihilation) vertex Γ cµ ; i (Γ aµ ; i ) viaΓ cµ ; i = γ cµ ; i + X ν γ cν ; i G ν T NP νµ , Γ aµ ; i = γ aµ ; i + X ν T NP µν G ν γ aν ; i , Σ ij = X µ γ cµ ; i G µ Γ aj ; µ , (8)where Σ is the self-energy. The indices i and j label the s -channel states or a contact diagram in a given partialwave.The pole part is then given by T P µν = Γ aµ ; i D ij Γ cν ; j (9)with the resonance propagator D ij . For example, if thereare two s -channel resonances with masses m and m (indices i, j ∈ { , } ) plus one contact term (indices i, j =3), the expression readsΓ aµ = (Γ aµ ;1 , Γ aµ ;2 , Γ aµ ;3 ) , Γ cµ = Γ cµ ;1 Γ cµ ;2 Γ cµ ;3 ,D − = E − m b − Σ − Σ − Σ − Σ E − m b − Σ − Σ − Σ − Σ m N − Σ . (10)The decomposition into pole and non-pole part per-formed here has mostly technical reasons: The numer-ical evaluation of the non-pole part is much more time-consuming than the evaluation of the pole part. Thisleads to an effective, nested fitting workflow as discussedin detail in Refs. [10, 51]. Here, we have introduced thecontact terms technically on the same footing as reso-nances, which allows to fit those terms computationallymore effectively.In Sec. 2.2 of Ref. [10], the renormalization of the nu-cleon mass and coupling in the presence of two bare s -channel states in the P partial wave was derived. Withthe implementation of contact diagrams, this procedurehas to be extended to two resonances and one contactdiagram. This is addressed in Appendix A. B. Photon-induced reactions
A field-theoretical description of the photoproductionamplitude within a gauge-invariant framework that re-spects the generalized off-shell Ward-Takahashi iden-tity [55–57] was successfully applied in the analysis of pion photoproduction in Ref. [54]. In this study an ear-lier version of the J¨ulich model was utilized to providethe hadronic final-state interaction. This field-theoreticalmethod allows one to gain insight into the microscopicreaction dynamics of the photo-interaction. By contrast,the semi-phenomenological approach to pseudoscalar me-son photoproduction, developed in Ref. [51] and usedhere, is more flexible and facilitates the analysis of alarge amount of data. Here, the photo-interaction ker-nel is approximated by energy-dependent polynomials,while the hadronic final-state interaction is provided bythe J¨ulich DCC model described in Sec. II A. The for-malism is inspired by the GW-SAID CM12 parameter-ization [58] and will be applied in the present study.Nonetheless, we consider the present analysis as an inter-mediate step towards an expansion of the field-theoreticalframework of Ref. [54]. A detailed introduction to thesemi-phenomenological approach was given in Sec. 2.2 ofRef. [51]. In the following, we recapitulate the basic ele-ments.The multipole amplitude of the photoproduction pro-cess is given by M µγ ( q, E ) = V µγ ( q, E )+ X κ T µκ ( q, p, E ) G κ ( p, E ) V κγ ( p, E ) , (11)where the index γ denotes the initial γN state. T µκ is thehadronic half-off-shell T -matrix introduced in Sec. II Awith the intermediate (final) meson-baryon channel κ ( µ ) and the corresponding off-shell momentum p (on-shell momentum q ). Integration over the intermediateoff-shell momentum p similar to Eq. (1) is suppressedhere in the notation, following the convention of Eq. (7).In the present analysis the photon is allowed to coupleto the intermediate channels κ = πN , ηN and π ∆, whilewe have πN and ηN as final states µ .The photoproduction kernel V µγ is written as V µγ ( p, E ) = α NP µγ ( p, E ) + X i γ aµ ; i ( p ) γ cγ ; i ( E ) E − m bi . (12)Here, α NP µγ stands for the photon coupling to the non-polepart of the photoproduction kernel. The tree-level cou-pling of the γN channel to the nucleon and ∆ resonancesis represented by the vertex function γ cγ ; i , where i de-notes the resonance number in a given partial wave. Thehadronic resonance annihilation vertex γ aµ ; i is exactly thesame as in Eq. (3), which results in the cancellation of theexplicit singularity at E = m bi . Both quantities, α NP µγ and γ cγ ; i , are approximated by energy-dependent polynomials P , α NP µγ ( p, E ) = ˜ γ aµ ( p ) √ m N P NP µ ( E ) γ cγ ; i ( E ) = √ m N P P i ( E ) . (13)In Eq. (13), the vertex function ˜ γ aµ has the same formas γ aµ,i in Eq. (12) but without any dependence on theresonance number i . The polynomials P , for a givenmultipole, are parameterized as P P i ( E ) = ℓ i X j =1 g P i,j (cid:18) E − E s m N (cid:19) j e − λ P i ( E − E s ) P NP µ ( E ) = ℓ µ X j =0 g NP µ,j (cid:18) E − E s m N (cid:19) j e − λ NP µ ( E − E s ) , (14)with g and λ > j , ℓ i and ℓ µ , are chosen so as to permit a good data descrip-tion, but are restricted to be less than 4. In order to ful-fill the decoupling theorem, which states that resonancecontributions are parametrically suppressed at threshold,the summation for P P starts with j = 1. The expan-sion point E s is chosen to be close to the πN threshold, E s = 1077 MeV. In this way, the factor e − λ ( E − E s ) ab-sorbs the potentially strong energy dependence at the γN threshold, which is not too far from the πN threshold.Moreover, this factor guarantees a well-behaved multi-pole amplitude in the high-energy limit, although a morequantitative matching to Regge amplitudes remains tobe done [59].In order to achieve a good description of the high-precision data for pion photoproduction close to thresh-old, we take into account some isospin breaking effects,i.e. we apply different threshold energies for the π p andthe π + n channels, as explained in Sec. 2.3 in Ref. [51].Similarly, we take the physical threshold of the ηp finalstate when calculating observables. Note that, in general,isospin-averaged masses are used in the J¨ulich model.A multipole decomposition of the photoproduction am-plitude of pseudoscalar mesons can be found in Ap-pendix A of Ref. [51]. III. RESULTSA. Database and free parameters
The database of the present study comprises thehadronic data used in Ref. [10]. This represents the worlddatabase on the reactions πN → ηN, K Λ, and K Σ upto E ∼ . πN → πN WI08 energy-dependent solution of the GWU/INS SAID group [60].In addition, we include almost all published data on pionphotoproduction off the proton up to E ∼ . γp → ηp , again up to E ∼ . T and thebeam-target asymmetry F by the A2 collaboration at Fit A Fit B πN → πN PWA GW-SAID WI08 [60] π − p → ηn dσ/d Ω, Pπ − p → K Λ dσ/d Ω, P , βπ − p → K Σ dσ/d Ω, Pπ − p → K + Σ − dσ/d Ω π + p → K + Σ + dσ/d Ω, P , β ∼ γp → π p dσ/d Ω, Σ, P , T , ∆ σ , G , Hγp → π + n dσ/d Ω, Σ, P , T , ∆ σ , G , Hγp → ηp dσ/d Ω, P , Σ dσ/d Ω, P , Σ, T , F T and F [22]. MAMI [22]. These data are added in a second fit, fitB. Performing these two fits allows us to estimate theinfluence of the new polarization data on the extractedresonance spectrum. An overview of the data includedin the fits can be found in Table I.Compared to the high-precision data nowadays avail-able in case of pseudoscalar meson photoproduction, thedata situation for the pion-induced reactions is difficult inlarge parts, the lack of polarization measurements beingone of the major issues. In Sec. 3 of Ref. [10] the situ-ation for the individual hadronic channels was discussedin detail. In the present study, we adopt the system-atic errors and mainly also the special weights applied tocertain data sets.Moreover, we continue along the lines of Ref. [51] andapply an additional systematic error of 5 % to all photon-induced data in order to account for discrepancies in thedata. The amount of 5% is an estimate at this point.More advanced techniques have been applied, e.g., bythe GWU/SAID group allowing for normalization cor-rections [9, 58]. We plan to improve our analysis alongthese lines in the future. To compensate for the smallernumber of data points for T and F in eta photoproduc-tion, those data are weighted in fit B with generic factorsaround 10 as found necessary for obtaining satisfactoryfit results. To achieve a good description of the dataat higher energies, additional weights have to be applied.The situation is similar in case of the beam asymmetry Σin γp → ηp . Compared to the number of data points forthe differential cross section, 5680, only a few are avail-able for Σ, namely 189. Furthermore, we did not attemptto achieve a good description of the recoil polarization P for γp → ηp as only seven data points are available andtheir influence on the fit is very limited. The data are,however, included in the fit but no special weights wereapplied.In the J¨ulich approach the free parameters tied tothe hadronic interaction are the bare coupling constantsand masses of the s -channel resonances, the strengths ofthe contact terms, and the cut-off parameters in the t -and u -channel diagrams constituting T NP . In addition,there are certain couplings in some of the latter diagramsthat cannot be connected to other coupling constants viaSU(3) flavor symmetry and have to be fitted to data aswell. In the present study we do not alter the T NP pa-rameters but employ the values found in the J¨ulich2012analysis of pion-induced reactions [10].We do, however, vary the parameters tied to hadronicresonances and contact terms. This is a necessity in thecurrent situation, given that the data of pion-inducedeta production is of less quality than the eta photopro-duction data. The higher-quality photoproduction datacan, thus, help to constrain the hadronic amplitude. Inboth fits A and B the number of s -channel resonance pa-rameters amounts to 128. For each of the 11 genuine I = 1 / I = 3 / πN , ρN , ηN , π ∆, K Λ and K Σ as allowed by isospin. Contact termsintroduced in Eq. (4) are switched on in both fits in the S and P partial waves for the πN and ηN channel,and in the P partial wave for the πN , ηN , π ∆, K Λand K Σ channel, giving rise to 9 additional fit param-eters. The free parameters that are used to tune theinteraction of the photon with hadrons are the resonanceparameters g P i,j and λ P i and the non-pole couplings g NP µ,j and λ NP µ,j with µ = πN , π ∆ or ηN , cf. Eq. (14). For-mally, all fit parameters up to j = 3 are implementedin the computer code. However, the actual number of fitparameters is chosen as required by data and the remain-ing parameters are set to zero. In order to achieve a gooddescription of T and F in fit B, additional fit parametersthat were set to zero in fit A had to be released. Thus, wehave 443 parameters tied to the photon interaction in fitA and 456 in fit B. In total, the number of fit parametersadds up to 580 in fit A and 593 in fit B.The free parameters are adjusted to the data in simul-taneous fits of all pion- and photon-induced reactionsusing MINUIT on the JUROPA supercomputer at theForschungszentrum J¨ulich. B. Fit results
In the following, only selected fit results for the reac-tions γp → ηp and π − p → ηn are shown. Data setswith energies that differ by less than 5.5 MeV are some-times displayed in the same graph. The full fit results forall pion- and photon-induced reactions included in this analysis can be found online [61].The definition of the various photoproduction observ-ables in terms of the multipole amplitudes M µγ is givenin Appendix B of Ref. [51]. The convention agrees withthe one of the SAID group [62, 63]. γp → ηp In Figs. 1 and 2 we show selected results for the dif-ferential cross section. In the threshold region only verysmall differences can be observed between fits A and B.Starting at E ∼ E = 1629 MeV in Fig. 1. At higher energies, the differ-ences are most apparent at very forward angles. However,the data situation does not allow for an assessment re-garding which of the fits is best. This can be seen, e.g., at E = 2006 MeV in Fig. 2 where both fits seem to describethe data equally well.The situation is similar in case of the beam asymmetryΣ in Fig. 3. Disagreements of the fits A and B are visiblepredominantly at higher energies and at forward angleswhere there are no data.Only seven data points are available for the recoil po-larization P , cf. Fig. 4. Although the results of our twofits are very different, especially at higher energies, bothdescribe the data more or less well but with some defi-ciencies. A larger database of this observable may helpconstrain the partial wave content.The fit results for the transverse target asymmetry T and the beam-target asymmetry F can be found in Fig. 5.These data were only included in fit B, meaning that theblue dashed line in Fig. 5 (fit A) represents a predictionfor these observables. Note that older data for T fromRef. [77], that are partially in conflict with the MAMIdata, were not fitted. The prediction of fit A at lower en-ergies E < T and good for F ,while the shortcomings of the prediction are considerableat higher energies. However, once the data are included(fit B) a good description over the whole energy range isachieved. π − p → ηn In Fig. 6 we show selected fit results for the differentialcross section of the reaction π − p → ηn . In addition tofits A and B, we display results from the old fit A of theJ¨ulich2012 analysis [10], in which only pion-induced re-actions were analyzed. We call the latter fit A had in thefollowing, where the subscript serves as a reminder thatthe fit in Ref. [10] included only hadronic data. Sincethe resonance vertex function γ aηN appears in the con-struction of the hadronic amplitude in Eq. (3) and alsoin the photoproduction kernel in Eq. (12), the bare pa-rameters of γ aηN have influence on the pion- as well as on BA07KR95 NA06
DY950.60.811.21.4 CR05DU020.60.811.21.40.40.81.2 CR090.40.80.20.4
WI09 d σ / d Ω [ µ b / s r ] θ [deg]FIG. 1: Differential cross section of the reaction γp → ηp . Dashed (blue) line: fit A; solid (red) line: fit B; data: MC10 [64],BA07 [65], KR95 [66], NA06 [67], DY95 [68], CR05 [69], DU02 [70], CR09 [71], WI09 [72]. In this and all following figures, thenumbers in the plots specify the pertinent center-of-mass energy E [MeV]. the photon-induced production of the ηN final state. Asmuch less data are available for π − p → ηn compared to γp → ηp , the photon-induced eta production does imposeconstraints on the fit results of the hadronic ηN channel.As can be seen in Fig. 6, starting already at energiesof E ∼ . E > π − p → ηn stem from a measurement [78] deemed prob-lematic due to a miscalibration of the beam momentum,see Ref. [17] for details. Those data enter the fit with amuch reduced weight and all three fits yield different re-sults. This is most apparent at higher energies, where fitsA and B are dominated by photon-induced data. Here,fits A and B mostly differ at extreme forward angles whilefor other angles they coincide reasonably well. This dis-crepancy has been also noted for the differential crosssection and beam asymmetry in fits A and B in eta pho- toproduction, mostly due to less precise data at forwardangles.In Fig. 7 we present results of the recoil polarizationin π − p → ηn . In principle, the only published data forthis observable [79] exhibit the same problems as the dif-ferential cross sections of Ref. [78] because the same ex-perimental set up was used. Those data are fitted witha very low weight, too. As in the case of the differentialcross section, the differences in the three fits are obvious.However, in view of the discussed quality issues with theavailable data for π − p → ηn , we see no reason to enforcea better data description.The comparison made here demonstrates the need fora pion beam to re-measure the reaction π − p → ηn . Notethat only four observables are needed for a complete ex-periment, as discussed in the Introduction. SU09 d σ / d Ω [ µ b / s r ] θ [deg]FIG. 2: Differential cross section of the reaction γp → ηp . Dashed (blue) line: fit A; solid (red) line: fit B; data: cf. Fig. 1 andSU09 [73]. -1-0.500.51 BA07 ER07-1-0.500.510 30 60 90 120150-1-0.500.51 0 30 60 90 120150 0 30 60 90 120150 0 30 60 90 120150 0 30 60 90 120150 0 30 60 90 120150 0 30 60 90 120150 Σ θ [deg]FIG. 3: Beam asymmetry of the reaction γp → ηp . Dashed (blue) line: fit A; solid (red) line: fit B; data: BA07 [65], ER07 [74]. -1-0.500.51 HE70 HO71 -1-0.500.51 θ [deg] P FIG. 4: Recoil polarization of the reaction γp → ηp . Dashed(blue) line: fit A; solid (red) line: fit B; data: HE70 [75],HO71 [76]. C. Multipoles
In Fig. 8 the electric and magnetic multipoles from fitsA and B for the reaction γp → ηp are shown. Addi-tionally, we display the multipoles of the Bonn-GatchinaBG2014-02 solution [85]. Note that those amplitudeswere not included in the fit.Generally speaking, the multipoles extracted in ouranalysis and the ones of the Bonn-Gatchina group exhibitlarge differences. An exception is the E and, to a cer-tain degree, also the M − multipole. While for E dif-ferences between our fits A and B are hardly noticeable atall, the M − multipole features clearly visible deviationsin the two fits at energies higher than E ∼ N (1710)1 / + resonance is located.Among the lower multipoles, the M shows the moststriking discrepancies between fits A and B, see also thediscussion on the influence of variations in the P par-tial wave on the description of T and F in Sec. IV A.For higher multipoles fit B sometimes shows a strongerenergy dependence than fit A, cf. E − , M − and E , M . In summary, the new MAMI data for T and F have a large impact on the multipole amplitudes.We observe that for lower partial waves, the eta pho-toproduction multipoles of the present study exhibitless agreement with the Bonn-Gatchina multipoles thanour pion photoproduction multipoles with the Bonn-Gatchina BG2014-02 [85] or the GW-SAID CM12 solu-tion [58]. This suggests that the multipole content of thereaction γp → ηp is much less established than in thecase of pion photoproduction where the various analysesagree better. Figures showing the pion photoproductionmultipoles can be found online [61]. IV. RESONANCE SPECTRUM
A resonance state is uniquely defined by its pole posi-tion in the complex energy plane, the residues associatedwith the channel transitions, and the Riemann sheet thepole is located on. With the exception of the physicalsheet of the lowest lying channel, the poles can appearon various Riemann sheets, but not all of them are ofphysical interest. Usually, only the poles on the sheetwhich is closest to the physical axis are considered. Weselect this sheet by rotating the right-hand cuts of allchannels in the direction of the negative imaginary E axis. In this way, we define the second sheet where allpoles extracted in the present study lie. See Ref. [49] fora detailed discussion.In order to determine the pole positions, the scatter-ing amplitude has to be continued to the second Rie-mann sheet. For this purpose we apply the method ofanalytic continuation following Ref. [49] where the am-plitude on the second sheet is accessed via a contour de-formation of the momentum integration. The calculationof the residues proceeds via the formalism illustrated inAppendix C of Ref. [50]. Definitions of the normalizedresidue and the branching ratio into a specific channelare given in Sec. 4.1 of Ref. [10]. In case of the lattertwo quantities we use the same definitions as the ParticleData Group [86]. For a reliable extraction of the reso-nance parameters, the correct structure of branch points,including the complex branch points of the channels in-cluding unstable particles π ∆, σN and ρN , is crucial.In Ref. [12] it was shown that the absence of the lattermight lead to false resonance signals.The definition of the photocouplings at the pole ˜ A h pole ,˜ A h pole = A h pole e iϑ h , (15)can be found in Appendix C of Ref. [51] and is identicalto the definition given in Ref. [87]. The photocouplingat the pole characterizes the coupling of the γN channelto a resonance, independently of the final state in thereaction under consideration. Note that, in general, thecomplex ˜ A h pole cannot be compared to the real-valued he-licity amplitudes A h , see Sec. D of Ref. [51] for furtherremarks.In Tables II to IV we list the pole positions, residuesand the photocouplings at the pole of the present study.In addition to the values extracted in the current fits Aand B we list the pole positions and residues found in fitA of the J¨ulich2012 analysis [10], called fit A had in thepresent study, and the photocouplings of fit 2 from theJ¨ulich2013 analysis [51]. Note that in the latter study,the parameters of the hadronic T -matrix were not al-tered, i.e. the resonance pole positions and hadronicresidues are the same as in fit A had of Ref. [10]. Anoverview of the pole positions of fit A and B is also givenin Fig. 9.In Table III the π ∆ channel labeled (6) correspondsto the case where | J − L | = 1 / -1-0.500.51 -1-0.500.51 T θ [deg] -1-0.500.51 -1-0.500.51 F θ [deg] FIG. 5: Transverse target asymmetry T and beam-target asymmetry F in the reaction γp → ηp . Dashed (blue) line: predictionof fit A; solid (red) line: fit B; data: Ref. [22]. to | J − L | = 3 /
2. Also the ρN channel can couple to aresonance with a given J P in multiple ways, cf. Table 11in Ref. [10]. Here, we only quote normalized residuesfor π ∆, since at energies well above the π ∆ thresholdthis channel can be regarded as being composed of thetwo stable particles π and ∆. In general, the resonancecoupling at the pole to a channel like π ∆ is a functionof the center-of-mass momentum of the stable particle(that equals the summed momenta of the decay prod-ucts of the unstable particle). Here, we do not quotethis function of q c . m . but choose q c . m . as on-shell three-momentum of a stable ∆ of mass m = 1232 MeV and apion. Obviously, this prescription does not lead to mean-ingful results for the very broad σ in the σN channel, orthe ρN channel. In the latter channel, most resonancesare not far above the threshold that is situated around E = (1 . − i . ρ cannot be considereda stable particle. A. Discussion of specific resonances
Compared to the earlier analysis of Ref. [10], the exten-sion of the model to eta photoproduction did not requirethe inclusion of additional bare s -channel states, and wefind no new dynamically generated resonances either. Inparticular, there is no need to include a narrow state ataround E ≈ .
68 GeV. The narrow structure discoveredin eta photoproduction on the neutron [14–16] is absentin the present analysis of eta photoproduction on theproton.In the following, when discussing selected resonancestates, we always refer to the values quoted in Tables IIto IV. S : While the real part of the pole position of the N ( ) − is very stable throughout all three fitsA, B and A had , the width is by 30 MeV larger in thenew fits that include eta photoproduction data. The newvalue is close to the one found in a recent analysis [88] ofthe GW-SAID WI08 solution [60], where elastic πN and1 d σ / d Ω [ m b / s r ] -1 -0.5 0 0.5 1cos θ d σ / d Ω [ m b / s r ] -1 -0.5 0 0.5 1cos θ θ FIG. 6: Differential cross section of the reaction π − p → ηn .Dashed (blue) line: fit A; solid (red) line: fit B; dash-dotted(green) line: fit A had of Ref. [10]. Data: filled circles fromRef. [80]; filled squares from Ref. [81]; empty triangles up fromRef. [82]; stars from Ref. [83]; empty squares from Ref. [84];empty diamonds from Ref. [78]. Note that the data situationfor this reaction is problematic, see text. πN → ηN data were fitted. Also the mass in the latteranalysis is very similar to our fits. We obtain the samevalues for the normalized ηN residue in all three fits.While the elastic πN residue is larger in the new fits,the coupling to this channel is still considerably smallerthan the coupling to ηN . In both current fits A and B,the magnitude of the photocoupling A h pole is more thantwice as large as in the previous fit 2 of Ref. [51] whereonly pion photoproduction data were considered. Thischange is related to the increase of the width for the N (1535) 1 / + , because the resonance width and size ofthe photocoupling at the pole are strongly correlated.On the whole, comparing the resonance parameters ofthe N (1535) 1 / − of the older fits to the new ones, theinclusion of eta photoproduction seems to have noticeableimpact for this resonance. The influence of the new T and F data [22], on the other hand, is rather limitedas the parameters in fit A and B do not exhibit majordifferences. This observation is in agreement with thesimilarity of fit A and B for T and F at E ∼ . S partial wave, N ( ) / − , is located in an energyregion where the deficiencies of the prediction of fit A for T and F become more apparent. Accordingly, slightlylarger variations are found in the pole positions of fitsA and B, and also compared to fit A had . Although thewidth is smaller in the new fits A and B, the resonance isstill broader and has a higher mass than in Ref. [88]. In P x d σ / d Ω [ m b / s r ] -1 -0.5 0 0.5 1cos θ P x d σ / d Ω [ m b / s r ] -1 -0.5 0 0.5 1cos θ θ FIG. 7: Polarization of the reaction π − p → ηn . Dashed (blue)line: fit A; solid (red) line: fit B; dash-dotted (green) line: fitA had of Ref. [10]. Data: Ref. [79]. Note that the data situationfor this reaction is problematic, see text. Ref. [89], however, the results from a Laurent-Pietarinen(L+P) expansion of the GW-SAID CM12 solution [58]for pion photoproduction give a pole position of E =1655(11) − i . .
5) MeV, which is closer to our values.As in fit A had , the current fits reveal a strong coupling tothe KY channels. The magnitude of the photocouplingto the N (1650) 1 / − is more than twice as large in thefits including eta photoproduction compared to the olderfit 2 of Ref. [51], where only pion photoproduction wasconsidered. While fit 2 yielded a value smaller than theone of Ref. [89], the photocoupling is now larger in fit Aand B.In summary, the pole positions of the N (1535) 1 / − and the N (1650) 1 / − come closer to the values of theGW-SAID analysis. As resonance widths and sizes ofhelicity couplings are correlated, the values of the lat-ter also approach the values of that analysis. The etaphotoproduction data have a strong influence on the S resonance properties. P : Besides the nucleon pole, we find two resonancesin the P partial wave. One of them, the Roper reso-nance N (1440) 1/2 + is dynamically generated from theinterplay of the t - and u -channel diagrams. As the fit pa-rameters corresponding to these T NP diagrams are notaltered in the present study the extracted resonance pa-rameters do not change much. The situation is differ-ent for the third state, the N (1710) 1/2 + . This explicit s -channel state was introduced in the J¨ulich model inRef. [10] mainly to improve the description of the pion-induced ηN and K Λ channels. Since it couples onlyweakly to the πN channel its resonance parameters arepoorly constrained from πN elastic scattering. As can be2 -16-808 R e ( A ) [ - f m ] -20 -0.8-0.40 -2-101 -0.6-0.300.30.6 00.20.40.6 -16-80 I m ( A ) [ - f m ] -2-101 -0.6-0.4-0.20 -0.200.2 -0.9-0.6-0.30 -0.6-0.4-0.200.2-0.200.2 R e ( A ) [ - f m ] -0.200.2 -0.200.2 -0.100.10.2 -0.0300.03 -0.10 -0.4-0.20 I m ( A ) [ - f m ] -0.2-0.100.10.20.3 -0.2-0.100.1 -0.2-0.100.10.2 -0.08-0.040 -0.0500.05-0.15-0.1-0.0500.05 R e ( A ) [ - f m ] -0.03-0.02-0.0100.01 I m ( A ) [ - f m ] -0.10 -0.0200.020.04 -0.01-0.00500.005 -0.02-0.010E M E M E M E M E M E E M M E M E M E [MeV]
FIG. 8: Electric and magnetic multipoles for the reaction γp → ηp . (Black) dash-dotted line: BG 2014-02 solution [85]. Dashed(blue) line: fit A; solid (red) line: fit B. seen in Table II, the extension of the fit to new, inelasticreaction channels results in noticeable changes in the poleposition. Moreover, also the inclusion of new observablesfor a specific reaction, here T and F in γp → ηp in fit B,leads to significant variations not only in the pole posi-tion but also for the residues and photocouplings. Thelatter observation suggests that additional informationfrom inelastic channels, e.g. in form of new polarizationmeasurements, might help to fix the parameters of the N (1710) 1/2 + .In all our fits, the N (1710) 1/2 + has a lower mass andis narrower than in recent analyses by the ANL-Osaka( E = 1746 − i
177 MeV) [46] and the Bonn-Gatchinagroups ( E = 1687 ± − i (100 ± .
5) MeV) [90]. Inthe L+P analysis of the GW-SAID CM12 solution inRef. [89], a broad state with a higher mass associatedwith this resonance is found that can be alternatively ex-plained as the ρN complex branch point. However, theauthors state that additional information from other de-cay channels beside πN is needed to distinguish betweenthe two options. See also Ref. [88] by the same authors where the same conclusion was drawn. Note that in thepresent study the ρN complex branch point is includedexplicitly.In addition to the N (1440) 1/2 + and the N (1710)1/2 + we find non-conclusive indications for another, verybroad and dynamically generated pole at E ∼ .
75 GeV. P : We include one bare s -channel state in the P partial wave, the N (1720) 3/2 + . Although we observed anoticeable sensitivity of the description of the ηN channelon variations in the P partial wave, the pole positionof the N (1720) 3/2 + is very similar in fits A and B. Theimpact of the new T and F data from MAMI can beseen in the photocouplings of this state (fit B vs. fit A).This is reflected in the discrepancies observed in the M multipole in Fig. 8.In different GW-SAID solutions [88, 89] and in theBonn-Gatchina analysis of Ref. [90] the N (1720) 3/2 + has a pole position with a real part 20 to 80 MeV lowerthan in our fits and an imaginary part more than 50 MeVlarger. By contrast, the ANL-Osaka group [46] finds val-ues closer to ours.3 Re E [MeV] -300-250-200-150-100-500 I m E [ M e V ] Fit AFit B
Fit AFit B π N ππ N π∆ K Σ ρ N N(1720) 3/2 + N(1650) 1/2 - N(1675) 5/2 - N(1520) 3/2 - N(2190) 7/2 - N(1440) 1/2 + N(1680) 5/2 + N(2250) 9/2 - N(1990) 7/2 + N(1710) 1/2 + N(1535) 1/2 - physical axis η N K Λ N(1750) 1/2 + N(2220) 9/2 + σ N N(1710) 1/2 + N(1675) 5/2 - N(1680) 5/2 + N(1650) 1/2 - Re E [MeV] -450-400-350-300-250-200-150-100-500 I m E [ M e V ] π N ππ N π∆ K Σρ N ∆ (1910) 1/2 + ∆ (1920) 3/2 + ∆ (1700) 3/2 - ∆ (2200) 7/2 - ∆ (2400) 9/2 - ∆ (1232) 3/2 + ∆ (1905) 5/2 + ∆ (1950) 7/2 + ∆(1620) 1/2 − physical axis ∆(1930) 5/2 − ∆ (1600) 3/2 + FIG. 9: Pole positions of the isospin I =1/2 (above) and I =3/2 (below) resonances extracted from fit A (filled squares)and from fit B (empty diamonds). For a better differentiation,the names and pole positions of some resonances are markedwith different colors. The green crosses denote the branchpoints of the amplitude. Note that all cuts, starting at thebranch points, are chosen in the negative Im E direction. We tested the influence of a second explicit resonancestate in the P partial wave but observed no significantimprovement of the fit results. In this partial wave, theBonn-Gatchina group finds strong evidence for a statenamed N (1900) 3 / + in the photoproduction of K Λ and K Σ [90, 91]. It has also been confirmed in γp → K + Λin an effective Lagrangian model [92] and in a covariantisobar-model single channel analysis [93]. It remains tobe seen, whether this state is also needed in the J¨ulichapproach once the analysis is extended to kaon photo-production. Note that the N (1900) 3 / + is also in-cluded in the ANL-Osaka analysis [46] and in the Gießenmodel [94]. D : While the real part of the pole positions of the N (1520) 3/2 − is unchanged in fits A and B, the imagi-nary part is about 10 MeV smaller in fit B. In the previous fit A had the real and the imaginary part were similar tofit A, -2Im E =110 MeV. In the GWU-SAID solutionsanalyzed in Refs. [88, 89] and the Bonn-Gatchina analy-sis [90], widths of about 110 MeV are also found. With-2Im E =78 MeV, a smaller width was extracted in theANL-Osaka analysis [46].Moderate changes in our three fits can also be observedfor the values of the residues and photocouplings. Al-though the N (1520) 3/2 − is well determined from elastic πN scattering and no new information from this chan-nel was included in the new fits, certain changes in theresonance parameters are not surprising. Due to the well-known SD -wave interference in the pion-induced ηN pro-duction resulting in a u -shape form of the differentialcross section (cf. Fig. 6), the N (1520) 3/2 − shows somesensitivity to the parameterization of the ηN channel. Ascan be seen in Fig. 6 at E = 1509 MeV and especiallyat E = 1576 MeV, the description of the data differs inall three fits. The energy bin at E = 1576 MeV is, onthe other hand, a prime example for the systematic prob-lems in the data. Data at backward angles are availablewith small error bars, but not in agreement with otherdata spanning the entire angular region. Underestimatednormalization problems can obviously change the angu-lar dependence significantly and have a large impact onthe partial-wave content. Better data are called for. D , F : Although the poles of the N (1675) 5/2 − and the N (1680) 5/2 + are located in an energy regionwhere the prediction of fit A for T and F becomes worse(cf. Fig. 5), the pole positions and residues exhibit onlyminor differences in fit A and B. Still, as can be seenin Fig. 10, in the current fit B, the D and F areimportant to achieve a good description of the new T data in eta photoproduction. Whereas for F a qualitativedescription of the data is feasible with the S , P , and D partial waves alone, in case of T all S -, P -, D -, and F -waves are needed at medium and higher energies.In the current form of the approach, only one bare s -channel state is incorporated in the F partial wave.For a discussion of a possible second explicit resonancewe refer the reader to Sec. 4.3 of Ref. [10]. F : The ηN normalized residue of the N (1990) 7/2 + is small. Still, the inclusion of the new polarization datafor the reaction γp → ηp results in a pole position withthe real part 53 MeV and the imaginary part almost60 MeV smaller in fit B than in fit A. The magnitudes ofphotocouplings, on the other hand, show much less vari-ations in the two new fits compared to the previous fit 2.As remarked in Sec. 4.2 of Ref. [10], from an analysis ofelastic πN scattering not much evidence can be claimedfor this resonance. However, in our current fit B the F partial wave seems to play a certain role in the param-eterization of T in γp → ηp , cf. Fig. 10. At energies inthe range of the pole position of the N (1990) 7/2 + , the F alone plus the S -, P -, and D -waves does not yielda qualitative description of the data. However, evidencefor this resonance from the current database is weak ingeneral.4 -0.500.51 E=1534 MeV 1646 1847 T -1-0.500.51 θ [deg] F FIG. 10: Partial wave contribution to T (upper row) and F (lower row) in fit B. Solid (red) line: fit B; dashed (green)line: S , P , D wave only; dash-dotted (indigo) line: all S -, P -, D -, and F -waves. G , G , H : We include bare s -channel statesidentified with the N (2190) 7/2 − , N (2250) 9/2 − and N (2220) 9/2 + resonance. The parameters of those broadstates are less stable, as was already observed in theJ¨ulich2012 analysis [10]. Isospin I = 3 / resonances : Since the ηN channeldoes not couple to resonances with I = 3 /
2, the polepositions and hadronic residues are very similar in fitsA and B. The inclusion of eta photoproduction data canchange the I = 3 / πN channels in pion photoproduction,and the mixed-isospin channels in the reaction πN → K Σ. Nonetheless, the mass of the ∆(1930) 5/2 − is about40 MeV higher and the widths about 70 MeV larger in fitB. Smaller differences in the parameters from fits A andB can also be seen in the ∆(2200) 7/2 − and the ∆(2400)9/2 − . The photocouplings at the pole are marginally lessstable than the pole positions.Comparing fit A and fit A had of Ref. [10], the influenceof pion photoproduction data included in the former fitbut not in the latter is visible in the results for somestates, as e.g. the ∆(1910) 1/2 + . Note that also the∆(1232) 3/2 + changes its pole position slightly.In the analysis of pion photoproduction within theJ¨ulich framework [51] the uncertainties of the extractedphotocouplings were estimated from re-fits based on dif-ferent re-weighted data sets. In Ref. [51], all data in-cluded in the fit entered with a universal weight of one.In the present study, however, the situation is different.As described in Sec. III A, the quality of the hadronicdata requires a specific weighting of the various data sets.Moreover, in case of the elastic πN channel we fit to theenergy-dependent partial-wave amplitudes of the GW-SAID group for which no errors are provided. As a sideremark, it should be noted that the error bars of the cor- responding single-energy solutions do not provide enoughinformation for correlated χ fits. Furthermore, also foreta photoproduction certain data sets were included witha higher weight. This renders an error estimation as per-formed in Ref. [51] impracticable for the present analysis.A comprehensive statistical error analysis is compli-cated by the large number of data points and free param-eters, typically inherent in the kind of analysis at hand.Such an analysis is, to our knowledge, not pursued in anyof the current DCC approaches and we postpone a rigor-ous error analysis to future work. Without such an erroranalysis, the assessment of the significance of certain lesswell-determined states, like a potential N (1750) 1 / + , isnot possible. Concerning the significance of resonancesignals, the systematic elimination of states as pursuedin Ref. [9] is also a necessary task which we postpone tofuture work. V. SUMMARY
Over the last years, measurements of pseudoscalar me-son photoproduction reactions with unprecedented qual-ity at facilities like ELSA, MAMI, and JLab have openeda path towards a more complete picture of the baryonspectrum. The photoproduction of η mesons is isospinselective and allows for an analysis of N ∗ states unaf-fected by contributions from ∆ ∗ states. Furthermore, the ηN final state is physically open for all resonances in thesecond resonance region and beyond. Eta photoproduc-tion is, thus, a prime reaction for resonance analysis andfuture complete experiments. Recently, polarization ob-servables with large angular coverage and high statisticshave emerged. Among them are the target asymmetry T and the beam-target asymmetry F . The latter observ-able has been measured at MAMI for the very first timein γp → ηp .However, even with the measurement of more observ-ables and an improved coverage of the data in anglesand energy, a reliable determination of resonance prop-erties requires a combined analysis of reactions with dif-ferent initial and final states. One of these reactions is π − p → ηn where, however, the data situation is knownto be problematic. To avoid bias, this requires a refit notonly of parameters tied to photoproduction, but also ofhadronic parameters. Baryon resonance analyses wouldgreatly benefit from a re-measurement of the π − p → ηn reaction.Dynamical coupled-channel (DCC) approaches providean especially suited tool to combine different reactionchannels in a global analysis. In the present study, weextended the J¨ulich DCC framework to eta photoproduc-tion. Based on a simultaneous analysis of nearly 30,000data points for pion and eta photoproduction off the pro-ton and the world database on the pion-induced reactions πN → πN, ηN, K Λ , and K Σ, we extracted the spec-trum of nucleon and ∆ resonances in terms of pole posi-tions, residues and photocouplings at the pole in an en-5ergy regime from πN threshold up to E ∼ S resonances N (1535) 1 / − and N (1650) 1 / − whosewidths change when photoproduction data are included.Also, some photocouplings at the pole changed in thepresent analysis compared to the J¨ulich2013 solution [51]in which only pion photoproduction data, but no eta pho-toproduction data were considered.In order to estimate the influence of the recent MAMI T and F measurements, two different fits were performed,including the new data only in the second fit. Changesin the resonance parameters are predominantly observedfor less well established states like the N (1710) 1 / + orhigher lying resonances. Smaller but significant changesappear also for well reputed states and particularly forthe photocouplings at the pole. Moreover, the new dataon T and F have a major influence on the multipoles.In general, the multipole content of eta photoproduc-tion is less well established than for pion photoproduc-tion. This calls for further measurements of single- anddouble polarization observables. Upcoming experimentson polarization observables will have significant impacton the resonance spectrum and will help to identify so-called missing states and determine their resonance pa-rameters. Acknowledgments
The authors gratefully acknowledge the computingtime granted on the supercomputer JUROPA at J¨ulichSupercomputing Centre (JSC). This work is also partiallysupported by the EU Integrated Infrastructure InitiativeHadronPhysics3 (contract number 283286), by the DFG(Deutsche Forschungsgemeinschaft, SFB/TR 16, “Sub-nuclear Structure of Matter”), by the DFG and NSFCthrough the Sino-German CRC 110, by the National Sci-ence Foundation, PIF grant No. PHY 1415459, andby the National Research Foundation of Korea, GrantNo. NRF-2011-220-C00011.
Appendix A: Renormalization of the nucleon massand coupling
The nucleon is included as an s -channel state in the P partial wave. In contrast to the other explicit statesin this partial wave the bare mass m bN and the bare cou-pling f bN of the nucleon are not free parameters but un-dergo a renormalization process such that the nucleonpole position and residue to the πN channel correspond to the physical values, i.e. E = m phys N = 938 MeV and f πNN = 0 .
964 [95]. Note that in the present study thenucleon is only allowed to couple to the πN channel. Ef-fects of the coupling to other channels with significantlyhigher threshold energies are small and can be absorbedin the renormalization process.In Ref. [10] the renormalization of the nucleon in thepresence of two s -channel states was illustrated. In thepresent study, we introduce an additional contact termin the P partial wave. Hence, the renormalization pro-cedure has to be modified.For this purpose we define the following reduced self-energies ˜ΣΣ = ( f bN ) Σ red11 = x f πNN Σ red11 = x ˜Σ Σ = f bN Σ red12 = x f πNN Σ red12 = x ˜Σ Σ = f b Σ red21 = x f πNN Σ red21 = x ˜Σ Σ = f bN Σ red13 = x f πNN Σ red13 = x ˜Σ Σ = f b Σ red31 = x f πNN Σ red31 = x ˜Σ , (A1)with x ∈ R and the bare πN N coupling constant f bN , f bN = x f πNN . (A2)We also define the reduced nucleon resonance vertices˜Γ a,cµ ;1 viaΓ a,cµ ;1 = f bN Γ red; a,cµ ;1 = x f πNN Γ red; a,cµ ;1 = x ˜Γ a,cµ ;1 . (A3)The nucleon pole position at E = m phys N is given bya zero of the determinant of D − of Eq. (10). UsingEqs. (A1) and (A3) we obtain an expression for the barenucleon mass m bN : m bN = m phys N + x " − ˜Σ (A4)+ G − ˜Σ + ˜Σ ( G − ˜Σ + 2 ˜Σ Σ )(Σ − G − G − ) . Here, we introduced the auxiliary quantities G − and G − : G − = G − ( E ) ≡ (cid:0) E − m b − Σ ( E ) (cid:1) | E = m N phys G − = G − ( E ) ≡ m N − Σ ( E ) | E = m N phys . (A5)In Eq. (A4) all quantities Σ( E ), G − ( E ), and G − ( E )are evaluated at E = m phys N .To determine x and thus the bare nucleon coupling f bN ,we exploit that at the nucleon pole the physical residue( a − ) πN → πN has to agree with the residue of T P fromEq. (9). The physical residue is given by( a − ) πN → πN = ˜ γ a ˜ γ c , (A6)where ˜ γ a (˜ γ c ) are the bare nucleon vertices calculated at E = E with the physical nucleon coupling f πNN instead6of the bare coupling (cf. Appendix B.1. of Ref. [50]),˜ γ a = i r f πNN kπ m π E N + ω π + m N p E N ω π ( E N + m N ) , (A7)where k is the particle momentum in the center-of-massframe. The residue of T P of Eq. (9) can be calculated asRes E = m phys N T P = det D − ∂ E det D − T P (cid:12)(cid:12)(cid:12)(cid:12) E = m phys N (A8) with ∂ E := ∂∂E . We obtain˜ γ a ˜ γ c = det D − ∂ E det D − ( x ˜Γ aµ ;1 , Γ aµ ;2 , Γ aµ ;3 ) E − m b − x ˜Σ − x ˜Σ − x ˜Σ − x ˜Σ E − m b − Σ − Σ − x ˜Σ − Σ m N − Σ − x ˜Γ cµ ;1 Γ cµ ;2 Γ cµ ;3 . (A9)Both sides of Eq. (A9) are evaluated at E = m phys N . Solv-ing Eq. (A9) we arrive at an expression for x which de-pends only on known or fitted quantities. We can calcu-late the bare mass and coupling of the nucleon by insert-ing this expression for x in Eqs. (A2) and (A4). Setting Σ = Σ = Σ = 0 we recover the two resonance case,cf. Sec. 2.2 of Ref. [10].The renormalization procedure is performed for eachstep in the fitting process. [1] S. Capstick and W. Roberts, Phys. Rev. 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G , TABLE II: Properties of the I = 1 / E (Γ tot defined as -2Im E ), elastic πN residues( | r πN | , θ πN → πN ), and the normalized residues ( p Γ πN Γ µ / Γ tot , θ πN → µ ) of the inelastic reactions πN → µ with µ = ηN , K Λ, K Σ. (*): not identified with PDG name; (a): dynamically generated. The resonance parameters of fit A had were extractedfrom fit A of Ref. [10], where only hadronic data were considered.Re E -2Im E | r πN | θ πN → πN Γ / πN Γ / ηN Γ tot θ πN → ηN Γ / πN Γ / K Λ Γ tot θ πN → K Λ Γ / πN Γ / K Σ Γ tot θ πN → K Σ [MeV] [MeV] [MeV] [deg] [%] [deg] [%] [deg] [%] [deg]fit N (1535) 1/2 − A 1497 105 23 −
48 51 110 5.6 −
26 8.9 −
78B 1499 104 22 −
46 51 112 5.0 32 5.0 − had −
37 51 120 7.7 68 15 − N (1650) 1/2 − A 1664 126 31 −
59 20 47 21 −
56 26 −
76B 1672 137 37 −
59 21 48 20 −
54 26 − had −
43 15 57 25 −
46 26 − N (1440) 1/2 +( a ) A 1349 221 32 −
69 5.0 11 4.5 −
178 1.1 140B 1355 215 62 −
98 7.8 −
27 16 145 2.7 113A had −
103 2 −
40 2 156 1 67 N (1710) 1/2 + A 1611 140 2.7 −
40 6.1 175 3.7 −
49 0.9 −
58B 1651 121 3.2 55 16 −
180 12 −
32 0.4 − had −
30 24 130 9.4 −
83 3.9 − N ( ) 1/2 +( ∗ ,a ) A 1746 317 10 −
133 0.7 −
85 0.3 127 1.8 −
48B 1747 323 14 −
144 0.2 138 0.4 86 1.6 − had −
140 0.8 −
170 2.2 4 N (1720) 3/2 + A 1711 209 5.3 −
45 1.0 132 0.3 −
97 2.2 96B 1710 219 4.2 −
47 0.7 106 1.1 −
70 0.2 79A had −
76 1.2 98 3.1 −
89 1.7 64 N (1520) 3/2 − A 1512 107 42 −
15 4.6 82 7.8 155 8.9 140B 1512 89 37 − − had −
16 3.5 87 5.8 158 0.8 163 N (1675) 5/2 − A 1649 129 25 −
21 6.1 −
43 0.3 90 2.0 − −
22 4.4 −
43 0.1 100 3.1 − had −
19 6.0 −
40 0.3 −
93 3.3 − N (1680) 5/2 + A 1668 107 37 −
23 1.0 130 0.7 −
96 0.1 145B 1669 100 34 −
19 2.7 136 0.1 90 0.4 148A had −
24 0.4 −
47 0.2 −
99 0.1 141 N (1990) 7/2 + A 1791 305 5.9 −
93 0.6 −
106 1.6 −
131 1.2 21B 1738 188 4.3 −
70 1.3 −
82 2.2 −
111 0.5 24A had −
84 0.4 −
99 1.7 −
123 0.8 28 N (2190) 7/2 − A 2081 359 45 −
27 0.4 137 2.0 −
48 1.2 −
59B 2074 327 35 −
40 1.6 129 0.5 −
51 1.3 − had −
31 0.1 −
28 1.9 −
51 1.3 − N (2250) 9/2 − A 2165 487 21 −
67 0.7 −
94 0.6 −
105 0.4 − −
64 1.7 −
89 0.6 −
101 0.2 70A had −
67 0.6 −
92 1.1 −
103 0.3 − N (2220) 9/2 + A 2173 611 70 −
62 0.3 −
103 1.0 60 0.1 − −
59 0.4 −
101 0.7 62 0.9 − had −
67 0.1 63 0.9 53 0.8 − , 406 (1981).[30] Y. Wunderlich, R. Beck and L. Tiator,Phys. Rev. C , 055203 (2014).[31] B. C. Jackson, Y. Oh, H. Haberzettl andK. Nakayama, Phys. Rev. C , 025206 (2014) [arXiv:1311.2836 [hep-ph]].[32] B. C. Jackson, Y. Oh, H. Haberzettl and K. Nakayama,[arXiv:1503.00845 [nucl-th]].[33] W. J. Briscoe, M. D¨oring, H. Haberzettl, D. M. Man-ley, M. Naruki, I. I. Strakovsky and E. S. Swanson,arXiv:1503.07763 [hep-ph]. TABLE III: Properties of the I = 3 / E (Γ tot defined as -2Im E ), elastic πN residues( | r πN | , θ πN → πN ), and the normalized residues ( p Γ πN Γ µ / Γ tot , θ πN → µ ) of the inelastic reactions πN → K Σ and πN → π ∆.(a): dynamically generated. The resonance parameters of fit A had were extracted from fit A of Ref. [10], where only hadronicdata were considered. Pole position πN Residue K Σ channel π ∆, channel (6) π ∆, channel (7)Re E -2Im E | r πN | θ πN → πN Γ / πN Γ / K Σ Γ tot θ πN → K Σ Γ / πN Γ / π ∆ Γ tot θ πN → π ∆ Γ / πN Γ / π ∆ Γ tot θ πN → π ∆ [MeV] [MeV] [MeV] [deg] [%] [deg] [%] [deg] [%] [deg]fit∆(1620) 1/2 − A 1599 69 17 −
106 21 −
106 - - 57 103B 1600 65 16 −
104 22 −
105 - - 57 105A had −
107 22 −
107 - - 57 102∆(1910) 1/2 + A 1799 651 83 −
83 2.1 −
129 54 130 - -B 1799 648 90 −
83 1.9 −
123 58 131 - -A had −
140 4.7 −
144 41 71 - -∆(1232) 3/2 + A 1218 91 45 −
36B 1218 92 46 − had − +( a ) A 1552 348 24 −
156 14 − . −
155 13 − . had −
158 11 − + A 1714 882 36 147 15 −
34 6.3 132 1.3 − −
35 6.9 131 1.3 − had −
21 7 144 1 − − A 1676 305 24 − . −
148 5.4 166 39 150B 1677 305 24 − . −
147 5.4 166 39 151A had − −
150 5 166 39 149∆(1930) 5/2 − A 1797 655 22 −
155 3.3 0.3 14 30 0.6 131B 1836 724 34 −
155 4.3 − . had −
159 3.1 − < + A 1795 245 5.3 −
82 0.2 −
148 0.8 85 9.7 80B 1795 247 5.3 −
89 0.1 −
155 0.9 64 8.7 72A had −
59 0.5 −
142 4 130 34 105∆(1950) 7/2 + A 1872 238 57 −
32 3.4 −
87 55 131 3.7 −
91B 1874 239 56 −
33 3.1 −
87 54 131 3.3 − had −
25 4.0 −
78 55 139 3 − − A 2156 474 17 −
48 0.1 −
92 2.5 −
137 25 115B 2142 486 17 −
56 0.5 −
103 2.2 −
151 23 107A had −
52 0.6 −
98 2 −
145 24 111∆(2400) 9/2 − A 1943 507 15 −
91 1.0 28 19 −
105 1.5 −
33B 1931 442 13 −
96 0.9 25 18 −
110 1.2 − . had −
80 1.3 40 24 −
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19 ∆(1910) 1/2 + A 307 37B 59 −
14 B 321 392 23 +3 − +28 − +24 − − +9 − N (1440) 1/2 +( a ) A −
49 3.2 ∆(1232) 3/2 + A − − . −
225 4.0B − −
23 B − − . −
226 2.82 − +4 − +2 − − +10 − − . +4 − − +3 − +0 . − . N (1710) 1/2 + A 7.1 −
177 ∆(1600) 3/2 +( a ) A − −
43 345 −
64B 20 −
83 B − −
42 332 −
712 28 +9 − +20 − − +23 − − +9 − +85 − − +10 − N ( ) 1/2 +( ∗ ,a ) A 7.1 −
129 ∆(1920) 3/2 + A −
188 47 508 69B 5.0 −
36 B −
192 46 522 672 10 +3 − − +12 − − +50 − +24 − +70 − +4 − N (1720) 3/2 + A 55 21 38 67 ∆(1700) 3/2 − A 128 4.8 110 17B 39 5.3 32 66 B 123 1.1 124 222 51 +5 − +9 − +9 − +29 − +10 − − +12 − +27 − +9 − N (1520) 3/2 − A − − . − A − − . − −
17 75 1.7 B −
270 33 153 812 − +8 − − +16 − +6 − +2 − − +73 − +77 − +3 − − +72 − N (1675) 5/2 − A 21 33 52 −
14 ∆(1905) 5/2 + A 41 91 -9.4 31B 32 36 51 − . +4 − +5 − +4 − − +4 − +13 − +72 − − +16 − − +13 − N (1680) 5/2 + A − −
29 105 −
11 ∆(1950) 7/2 + A − − − − − −
28 102 −
11 B − − − − − +2 − − +9 − +1 − − . +3 − − +4 − − +2 − − +8 − − +3 − N (1990) 7/2 + A 26 8.8 30 126 ∆(2200) 7/2 − A 118 −
22 171 −
54B 29 67 33 39 B 106 −
23 157 −
602 10 +11 − − +108 − +23 − +17 − +11 − − +5 − +24 − − +9 − N (2190) 7/2 − A − −
21 98 −
15 ∆(2400) 9/2 − A −
40 92 79 − − −
21 85 −
22 B −
34 63 54 − − +7 − − +6 − +13 − − . +3 − − +46 − +24 − +42 − − +17 − N (2250) 9/2 − A 130 149 89 − .
6B 26 −
26 119 −
422 90 +25 − +17 − +31 − +36 − N (2220) 9/2 + A 141 105 79 −
47B 135 114 82 −
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