Energy-independent PWA of the reaction γp→ K + Λ
A.V. Anisovich, R. Beck, V. Burkert, E. Klempt, M.E. McCracken, V.A. Nikonov, A.V. Sarantsev, R.A. Schumacher, U. Thoma
aa r X i v : . [ nu c l - e x ] A p r EPJ manuscript No. (will be inserted by the editor)
Energy-independent PWA of the reaction γ p → K + Λ A.V. Anisovich , , R. Beck , V. Burkert , E. Klempt , M.E. McCracken , , V.A. Nikonov , , A.V. Sarantsev , ,R.A. Schumacher , U. Thoma Helmholtz-Institut f¨ur Strahlen- und Kernphysik, Universit¨at Bonn, Germany Petersburg Nuclear Physics Institute, Gatchina, Russia Jefferson Lab, 12000 Jefferson Avenue, Newport News, Virginia, USA Washington & Jefferson College, Washington, Pennsylvania, USA Carnegie Mellon University, 5000 Forbes Ave., Pittsburgh, Pennsylvania 15213, USAReceived: April 12, 2018/ Revised version:
Abstract.
Using all recent data on the differential cross sections and spin observables for the reaction γp → K + Λ ,an energy-independent partial-wave analysis is performed. The analysis requires multipoles up to L = 2 ; there is noevidence that the fit requires multipoles with L = 3 . At present the available data allow us to extract the dominantmultipoles only. These are compatible with the multipoles obtained in the energy-dependent fit. This result supports thereliability of the Bonn-Gatchina energy-dependent results. The spectrum of excited nucleons ( N ∗ ) reflects the structureof quantum chromodynamics in the non-perturbative regime.Full elucidation of the properties of the N ∗ states is an impor-tant long-term goal for the hadron physics community. The ma-jority of experimental information has historically come fromnucleon and pion-induced elastic and inelastic reaction chan-nels. Partial-wave analyses of these data have established a richspectrum of N ∗ states, so that the Particle Data Group [1] lists,for example, well-established ( i.e. , “existence is certain” rat-ing) nucleon resonances with spins up to / . In recent years,new experiments have produced high-precision measurementsof photo-production of several hadronic final states due to theavailability of high quality photon and electron beams at fa-cilities including CLAS/Jefferson Lab, ELSA/Bonn, MAMI/Mainz, LEPS/SPring-8, and GRAAL/Grenoble. These have hada significant impact on our understanding of N ∗ properties.The present paper dwells on the production of a strangeness-containing final state. The reaction γp → K + Λ (1)is sensitive to N ∗ states with masses in the range from 1.6 to2.4 GeV, a two-body final state in a domain where pionic re-actions are dominated by more complicated multi-pion finalstates. This makes it attractive to study, since the simple kine-matics gives straightforward access to non-strange excitationsin a mass range that is otherwise not well understood.Theoretical work using a relativized constituent quark modelmade predictions about the spectrum of N ∗ and ∆ excitations[2,3], as well as their couplings to various initial and final statesincluding hyperonic [4] states. Many of the observed N ∗ res-onances, as well as many “missing” resonances that have not been observed coupling to πN were tabulated. One goal of N ∗ spectroscopy programs is thus to search for such missing statesin channels other than πN .All pseudo-scalar meson photoproduction reactions are char-acterized by eight complex amplitudes; parity invariance of thestrong interaction reduces this number to four independent am-plitudes. Full characterization of the reaction at a given kine-matic relies upon measurement of dσ/dΩ and at least sevenof the fifteen single- and double-polarization observables [5].However, for the sake of redundancy and the reduction of ex-perimental ambiguities, as broad a range of observables as fea-sible must be analyzed for a full decomposition of a reaction atthe amplitude level. In most theoretical or phenomenologicalstudies, these amplitudes are constructed for each bin in energyand angle. In a next step, the set of amplitudes at a given energycan be expanded into multipoles which contain the informationon the underlying physical processes.Less demanding is to use only a finite number of multipolesin a truncated partial-wave expansion of the photoproductionamplitude. For small numbers of contributing partial waves,five observables can already be sufficient to determine the am-plitudes [6,7]. For example, seven observables have been mea-sured for γp → π p , the differential cross section dσ/dΩ , thebeam asymmetry Σ , target asymmetry T , the recoil polariza-tion P , and different correlations between photon and targetpolarization yielding G , E , and H . The seven data sets spana common mass range from 1.462 to 1.662 GeV in which nocontributions with orbital angular momenta L ≥ are ex-pected. A truncated partial-wave analysis returned multipoleswith L = 0 , and [8]. These multipoles lead to the excitationof nucleon and ∆ resonances. In the 1500 MeV region, how-ever, the impact of ∆ resonances is small, and that is why the N (1520)3 / − helicity coupling could be determined in [8]. A.V. Anisovich et al.:
Energy-independent PWA of the reaction γp → K + Λ , The γp → K + Λ reaction profits from the fact that - due toisospin conservation - only isospin-1/2 intermediate states con-tribute and thus all ∆ excitations are excluded. Furthermore,the weak decay of the Λ to πN allows for determination of itsrecoil polarization. Use of polarized photon beams gives accessto other polarization observables, notably the beam asymmetry( Σ ), and beam-recoil double-polarization observables for bothcircular ( C x , C z ) and linear ( O x , O z ) photon polarizations.Recent and forthcoming measurements from experiments withpolarized nucleon targets will give access to the remaining setof polarization observables. For these reasons, the γp → K + Λ reaction is presently the best candidate for full amplitude-levelcharacterization.So far, all partial-wave analyses (PWA) of the reaction γp → K + Λ used energy-dependent representations of the contribut-ing N ∗ states in the reaction. We shortly review these analy-ses in Sec. 2. Energy-dependent analyses benefit from the ana-lytic structure of the amplitudes. However, the dominant partialwaves over a range of energies could tend to mask the contribu-tions of weaker partial waves nearby due to the force of statis-tics. A crucial test of the validity of this approach is to compareit to the results of an energy-independent analysis. The two ap-proaches should agree within their respective limitations. Herewe present application of the Bonn-Gatchina (BnGa) model tothe γp → K + Λ reaction data in eleven independent energybins from threshold up to √ s = 1918 MeV, as itemized inSec. 3. The formalism is reviewed in Sec. 4. Results are dis-cussed in Sec. 5, where we compare the fits with the latestenergy-dependent BnGa multi-channel PWA. We also comparethe energy-independent fits with multipolarity L = 0 , withthose including L = 2 to assess the need for higher J P inter-mediate states in the γp → K + Λ reaction. We show how usingthe energy-independent solution can be used to check the sta-bility of earlier energy-dependent fit results. Sec. 6 summarizesour findings. In light of the several attractive features of the γp → K + Λ reaction, it has been the most suitable candidate for partial-wave analysis (PWA); several analyses have been performedwith varied techniques and results. Early analyses [9] applieda single-channel tree-level resonant isobar model to SAPHIR dσ/dΩ data [10] and found contributions from known N (1650)1 / − , N (1710)1 / + , and N (1720)3 / + states, as well as evidencefor a previously unobserved / − state with a mass of 1894 MeV.Soon after, other work showed features of the SAPHIR datathat had been interpreted as resonant contributions could be de-scribed in a Regge model [11] to describe t -channel exchangeof strange mesons [12]. The group at Ghent [13] used the Regge-plus-resonance approach to analyze forward-angle dσ/dΩ and P data and found evidence for N (1650)1 / − , N (1710)1 / + ,and N (1720)3 / + states near threshold and J P = 3 / + and / + states with masses near 1.9 GeV. Exploratory PWA stud-ies at √ s > . GeV indicate resonance structure near 2.1 GeV,for example in [14], but in the present study we only work withmultipoles below about 1.92 GeV.Interpretation of K + Λ production mechanism was com-plicated when dσ/dΩ and recoil polarization data published by the CLAS Collaboration [15,16] showed significant differ-ences when compared to the SAPHIR data. The implicationswere studied and discussed by Mart et al. [17] and others inboth single-channel effective Lagrangian models and multipoleanalyses.The Bonn-Gatchina Group produced several PWA studiesover the years of the γp → K + Λ reaction. A 2005 publicationSarantsev et al. [18] demonstrated that partial-wave analysis of dσ/dΩ , Σ and P data, when coupled with data from photopro-duction of KΣ , πN , and ηN , necessitates a J P = 1 / + statewith mass of approximately 1840 MeV. It also suggested theexistence of four / − states between 1520 and 2170 MeV, butproduces no evidence for / − states of mass above 1650 MeV.They noted that the discrepancy between the then availableSAPHIR and CLAS dσ/dΩ results could lead to ambiguities infitting. With the publication [19], Anisovich et al. incorporatedthe large spin-transfer probability observables C x and C z mea-sured with circularly polarized photons at CLAS [20] into ananalysis coupling several observables from π , η , and K pho-toproduction reactions. This analysis showed that all observ-ables could be reproduced with the further addition of only onestate, a / + resonance with mass of approximately 1900 MeV.The 2010 publication of higher-statistics and independent re-sults from CLAS [21], the discrepancy in the cross section for γp → K + Λ seems to have been resolved. Reduced ambiguityin PWA of the channel was thus expected.The most recent BnGa analyses have applied coupled-channelPWA to a large set of observables for many reactions [22], andextracted transition amplitudes for pion- and photon-inducedproduction of η and K mesons [23]. These analyses show thatadequate description of the data is possible with two separatesets of resonances, distinguished by the presence of either oneor two / + states. A subsequent multi-channel analysis [24]focusing on information from KΣ production, provided anupdated set of resonance contributions (referred to here as BnGa2013),but concluded that further polarization information for Σ pro-duction is needed to unambiguously determine production am-plitudes for these reactions. γp → K + Λ In the region from threshold up to √ s = 1918 MeV the follow-ing data are used: differential cross section dσ/dΩ from [15]and [21], recoil polarization P from [21] and [25], Σ from[25], T , O x ′ and O z ′ from [26], and the spin transfer coef-ficients C x and C z from [20]. In the low energy bin the re-cent data on dσ/dΩ from CB@MAMI [27] are also used. Thefitting region was divided into eleven energy bins in √ s eachcontaining data on at least one double polarization observable: − , − , − , − , − , − , − , − , − , − , − (in MeV). The data are shown in Fig. 1jointly with the BnGa2013 energy-dependent fit and two fur-ther fits described below. The data are organized in eleven blockseach containing the eight observables. .V. Anisovich et al.: Energy-independent PWA of the reaction γp → K + Λ , 3 -1 -0.6 -0.2 0.2 0.6 100.10.20.30.4 [mb/sr] Ω /d σ d W = 1642-1653 MeV z
CLAS_2006CLAS_2010MAMI13 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Σ W = 1642-1653 MeV z
GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 T W = 1642-1653 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 P W = 1642-1653 MeV z
CLAS_2010GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Ox’
W = 1642-1653 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Oz’
W = 1642-1653 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cx W = 1642-1653 MeV z -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cz W = 1642-1653 MeV z -1 -0.6 -0.2 0.2 0.6 100.10.20.30.4 [mb/sr] Ω /d σ d W = 1672-1683 MeV z
CLAS_2010MAMI13 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Σ W = 1672-1683 MeV z
GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 T W = 1672-1683 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 P W = 1672-1683 MeV z
CLAS_2010GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Ox’
W = 1672-1683 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Oz’
W = 1672-1683 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cx W = 1672-1683 MeV z
CLAS_2006 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cz W = 1672-1683 MeV z
CLAS_2006 -1 -0.6 -0.2 0.2 0.6 100.10.20.30.4 [mb/sr] Ω /d σ d W = 1697-1708 MeV z
CLAS_2006CLAS_2010MAMI13 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Σ W = 1697-1708 MeV z
GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 T W = 1697-1708 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 P W = 1697-1708 MeV z
CLAS_2010GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Ox’
W = 1697-1708 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Oz’
W = 1697-1708 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cx W = 1697-1708 MeV z -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cz W = 1697-1708 MeV z -1 -0.6 -0.2 0.2 0.6 100.10.20.30.4 [mb/sr] Ω /d σ d W = 1727-1738 MeV z
CLAS_2010MAMI13 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Σ W = 1727-1738 MeV z
GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 T W = 1727-1738 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 P W = 1727-1738 MeV z
CLAS_2010GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Ox’
W = 1727-1738 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Oz’
W = 1727-1738 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cx W = 1727-1738 MeV z
CLAS_2006 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cz W = 1727-1738 MeV z
CLAS_2006 -1 -0.6 -0.2 0.2 0.6 100.10.20.30.4 [mb/sr] Ω /d σ d W = 1752-1758 MeV z
CLAS_2006CLAS_2010MAMI13 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Σ W = 1752-1758 MeV z
GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 T W = 1752-1758 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 P W = 1752-1758 MeV z
CLAS_2010GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Ox’
W = 1752-1758 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Oz’
W = 1752-1758 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cx W = 1752-1758 MeV z -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cz W = 1752-1758 MeV z -1 -0.6 -0.2 0.2 0.6 100.10.20.30.4 [mb/sr] Ω /d σ d W = 1777-1788 MeV z
CLAS_2006CLAS_2010MAMI13 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Σ W = 1777-1788 MeV z
GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 T W = 1777-1788 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 P W = 1777-1788 MeV z
CLAS_2010GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Ox’
W = 1777-1788 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Oz’
W = 1777-1788 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cx W = 1777-1788 MeV z
CLAS_2006 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cz W = 1777-1788 MeV z
CLAS_2006
Fig. 1.
Data and fit to data on γp → K + Λ . Black line is the energy-dependent solution BnGa2013, the dashed (blue) line is the truncatedPWA with L = 0 , , and dot-dashed (red) line the truncated PWA with L = 0 , , . A.V. Anisovich et al.: Energy-independent PWA of the reaction γp → K + Λ , -1 -0.6 -0.2 0.2 0.6 100.10.20.30.4 [mb/sr] Ω /d σ d W = 1807-1818 MeV z
CLAS_2010MAMI13 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Σ W = 1807-1818 MeV z
GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 T W = 1807-1818 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 P W = 1807-1818 MeV z
CLAS_2010GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Ox’
W = 1807-1818 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Oz’
W = 1807-1818 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cx W = 1807-1818 MeV z -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cz W = 1807-1818 MeV z -1 -0.6 -0.2 0.2 0.6 100.10.20.30.4 [mb/sr] Ω /d σ d W = 1832-1843 MeV z
CLAS_2006CLAS_2010MAMI13 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Σ W = 1832-1843 MeV z
GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 T W = 1832-1843 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 P W = 1832-1843 MeV z
CLAS_2010GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Ox’
W = 1832-1843 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Oz’
W = 1832-1843 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cx W = 1832-1843 MeV z
CLAS_2006 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cz W = 1832-1843 MeV z
CLAS_2006 -1 -0.6 -0.2 0.2 0.6 100.10.20.30.4 [mb/sr] Ω /d σ d W = 1857-1868 MeV z
CLAS_2006CLAS_2010MAMI13 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Σ W = 1857-1868 MeV z
GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 T W = 1857-1868 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 P W = 1857-1868 MeV z
CLAS_2010GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Ox’
W = 1857-1868 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Oz’
W = 1857-1868 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cx W = 1857-1868 MeV z -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cz W = 1857-1868 MeV z -1 -0.6 -0.2 0.2 0.6 100.10.20.30.4 [mb/sr] Ω /d σ d W = 1882-1893 MeV z
CLAS_2006CLAS_2010MAMI13 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Σ W = 1882-1893 MeV z
GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 T W = 1882-1893 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 P W = 1882-1893 MeV z
CLAS_2010GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Ox’
W = 1882-1893 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Oz’
W = 1882-1893 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cx W = 1882-1893 MeV z
CLAS_2006 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cz W = 1882-1893 MeV z
CLAS_2006 -1 -0.6 -0.2 0.2 0.6 100.10.20.30.4 [mb/sr] Ω /d σ d W = 1900-1918 MeV z
CLAS_2006CLAS_2010 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Σ W = 1900-1918 MeV z
GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 T W = 1900-1918 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 P W = 1900-1918 MeV z
CLAS_2010GRAAL_2007 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Ox’
W = 1900-1918 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5
Oz’
W = 1900-1918 MeV z
GRAAL_2009 -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cx W = 1900-1918 MeV z -1 -0.6 -0.2 0.2 0.6 1-1.5-1-0.500.511.5 Cz W = 1900-1918 MeV z
Fig. 1 continued. .V. Anisovich et al.:
Energy-independent PWA of the reaction γp → K + Λ , 5 The amplitude for photoproduction of a single pseudoscalarmeson is well known and can be found in the literature (seefor example [28] and references therein). Here, we consider thecase of a K + meson recoiling against a Λ hyperon. The generalstructure of the amplitude can be written in the form A = ω ∗ J µ ε µ ω ′ , where ω ′ and ω are spinors representing the baryon in the ini-tial and final state, J µ is the electromagnetic current of the elec-tron, and ε µ characterizes the polarization of the photon. Thefull amplitude can be expanded into four invariant (CGLN) am-plitudes F i [5] J µ = (2) i F σ µ + F ( σq ) ε µij σ i k j | k || q | + i F ( σk ) | k || q | q µ + i F ( σq ) q q µ . where q is the momentum of the nucleon in the K + Λ channel, k is the momentum of the nucleon in the γN channel calcu-lated in the center-of-mass system of the reaction, and σ i arethe Pauli matrices.The functions F i have the following angular dependence: F ( W, z ) = ∞ X L =0 [ LM L + + E L + ] P ′ L +1 ( z ) +[( L + 1) M L − + E L − ] P ′ L − ( z ) , F ( W, z ) = ∞ X L =1 [( L + 1) M L + + LM L − ] P ′ L ( z ) , F ( W, z ) = ∞ X L =1 [ E L + − M L + ] P ′′ L +1 ( z ) + (3) [ E L − + M L − ] P ′′ L − ( z ) , F ( W, z ) = ∞ X L =2 [ M L + − E L + − M L − − E L − ] P ′′ L ( z ) . Here, L corresponds to the orbital angular momentum in the K + Λ system, W is the total energy, P L ( z ) are Legendre poly-nomials with z = ( kq ) / ( | k || q | ) , and E L ± and M L ± are elec-tric and magnetic multipoles describing transitions to stateswith J = L ± / . There are no contributions from M , E − ,and E − for spin 1/2 resonances.Differential cross section and polarization observables canbe expressed in terms of the F i functions. The relations can befound, e.g., in [29]. For convenience, we give the expressionsfor the observables used in the fit. The single polarization ob-servables Σ , P and T are given by Σ I = − sin ( θ )2 (4) Re [ F F ∗ + F F ∗ + 2 F F ∗ + 2 F F ∗ + 2 z F F ∗ ] ,P I = sin( θ ) Im [(2 F ∗ + F ∗ + z F ∗ ) F + (5) F ∗ ( z F + F ) + sin ( θ ) F ∗ F ] ,T I = sin( θ ) Im [ F ∗ F − F ∗ F + (6) z ( F ∗ F − F ∗ F ) − sin ( θ ) F ∗ F ] , where I = Re [ F F ∗ + F F ∗ − z F F ∗ + (7) sin ( θ )2 ( F F ∗ + F F ∗ + 2 F F ∗ + 2 F F ∗ + 2 z F F ∗ )] . Here the center of mass (c.m.) scattering angle is θ . The doublepolarization observables O x ′ , O z ′ , C x and C z can be written as O x ′ I = (8) sin( θ ) Im [ F F ∗ − F F ∗ + z ( F F ∗ − F F ∗ )] ,O z ′ I = − sin ( θ ) Im [ F F ∗ + F F ∗ ] , (9) C x = sin( θ ) C z ′ + cos( θ ) C x ′ , (10) C z = cos( θ ) C z ′ − sin( θ ) C x ′ , (11)where C x ′ I = sin( θ ) Re [ F F ∗ − F F ∗ + F F ∗ − F F ∗ + , z ( F F ∗ − F F ∗ )] , (12) C z ′ I = Re [ − F F ∗ + z ( F F ∗ + F F ∗ ) − sin ( θ )( F F ∗ + F F ∗ )] . (13)Let us remind the reader that the z axis defines the directionof the incoming particles in the c.m. system, while the z ′ axisdefines the direction of the outgoing particles (see [29]). Finallythe differential cross section is equal to: dσdΩ = kq I , (14)where q and k are the moduli of the initial and final c.m. mo-menta, respectively. The energy-independent (or single energy) PWA uses the fulldatabase of the Bonn-Gatchina partial-wave analysis [22]. Areasonable description of all data is achieved; the breakdown ofthe χ contribution from various data sets is given in Table 1.In the following, we use the data on the reaction γp → K + Λ only. L = 0 , multipoles It is natural to assume that in the energy region not far above thethreshold only multipoles of low spin play a role. The energydependent PWA [23] supports this assumption: in the region upto 2000 MeV there are four large multipoles, E , E , M and M − which are 5 to 10 times larger than multipoles with L = 2 .The multipole decomposition is shown in Fig. 2. In the fit,it is assumed that only the four multipoles shown in the fig-ure contribute to the reaction γp → K + Λ , all other contribu-tions are set to zero. The errors of the multipoles correspond tochanges in description of the data by one unit in of χ . Let usnote that the phases of the multipoles in a fit are defined up to A.V. Anisovich et al.:
Energy-independent PWA of the reaction γp → K + Λ , E magnitude[mfm] W[GeV] -150-100-50050100150 E phase[deg] W[GeV] E magnitude[mfm] W[GeV] -150-100-50050100150 E phase[deg] W[GeV] M magnitude[mfm] W[GeV] -150-100-50050100150 M phase[deg] W[GeV] M magnitude[mfm] W[GeV] -150-100-50050100150 M phase[deg] W[GeV]
Fig. 2.
Decomposition of the γp → K + Λ amplitude with S and P multipoles. The general phase is not defined so we choose φ ( E +0 ) =0 . The dashed line is the energy dependent solution BnGa2013. one overall phase. Here, we determine the phases relative to thephase of the E multipole. Hence φ ( E +0 ) = 0 holds by con-struction. The comparison with the Bonn-Gatchina PWA showsthat the energy-dependent fit is approximately compatible withthe single-energy fit, at least in the region up to 1750 MeV. Wealso note that a truncated PWA with L = 0 , multipoles givesa good description of the data up to this energy. At W >
MeV the L = 2 multipoles are important, see Fig. 1. The qual-ity of this fit (in terms of χ ) is shown in Table 1. While thedifferential cross sections are described very reasonably, the fitto polarization observables is not convincing: in particular thebeam asymmetry is poorly reproduced and several other polar-ization variables have χ values exceeding 2. A more detailedview reveals that the predicted beam asymmetry Σ and the datahave a different angular dependence; this difference is ratherpronounced in the mass region above 1800 MeV. Obviously, afit with only L = 0 and L = 1 multipoles is not sufficient todescribe the data over the full mass range. L = 0 , , multipoles The multipoles with L = 2 significantly improve the fit quality.The mean χ per data point drops from 1.8 to 0.8 (see Table 1).The improvement is particularly large for the GRAAL beamasymmetry where the χ goes down from 6.77 to 0.57. Mostobservables are now fitted with a χ per data point of less than1. The number of fit parameters (moduli of 8 amplitudes and7 phases at 11 energies) is 165. It is likely that the system-atic errors given in the publications are slightly overestimated.The improvement of the fit can also be seen when Fig. 1 isinspected.The resulting multipole decomposition is shown in the twoleft columns of Fig. 3. We observe that the multipoles scatterfrom bin to bin. Moreover, for some energy bins there are no C x and C z data. The solution is no longer uniquely defined: twodifferent solutions are found which differ less than δχ < .Two conclusions follow from these observations: i) at energies W >
MeV the L = 2 multipoles are definitely needed.ii) the lack of experimental data and the data quality does notallow extraction of multipoles with the desired precision in acompletely free fit. L = 0 , , multipoles and penalty function In a next step we guide the fit with L = 0 , , multipolesso it is not totally free. We assume that the large multipoleswith L = 0 , are reasonably well defined by the fit using L = 0 , multipoles only. Thus we impose a penalty func-tion which sanctions solutions which deviate strongly from thefit with L = 0 , multipoles. More precisely, we introduce apenalty function defined as χ pen = X α ( M α − M , α ) ( δM , α ) + X α ( E α − E , α ) ( δE , α ) , (15)where E , α and M , α are the electric and magnetic multipolesfrom solution with L = 0 , multipoles only; δE , α , δM , α arethe multipole uncertainties.The quality of the fit to the differential cross sections hardlychanges while most polarization observables are now describedwith lesser accuracy (see Table 1).The resulting multipoles are shown in the two center columnsof Fig. 3. There is now a unique solution but the solution stillscatters significantly for the small multipoles, and the E + and M + waves deviate significantly from the the energy-dependentfit. It has to be stressed that so far, the solution has no bias atall; the solution is constructed from the experimental data with-out any input from the energy-dependent solution. Even thoughthere are discrepancies between the energy-dependent and in-dependent solution in detail, the overall agreement is very sat-isfactory. In particular there is no hint that an additional narrowresonance may be hidden or that too many resonances havebeen used to fit the data. .V. Anisovich et al.: Energy-independent PWA of the reaction γp → K + Λ , 7 E magnitude[mfm] W[GeV] -150-100-50050100150 E phase[deg] W[GeV] E magnitude[mfm] W[GeV] -150-100-50050100150 E phase[deg] W[GeV] E magnitude[mfm] W[GeV] -150-100-50050100150 E phase[deg] W[GeV] E magnitude[mfm] W[GeV] -150-100-50050100150 E phase[deg] W[GeV] E magnitude[mfm] W[GeV] -150-100-50050100150 E phase[deg] W[GeV] E magnitude[mfm] W[GeV] -150-100-50050100150 E phase[deg] W[GeV] M magnitude[mfm] W[GeV] -150-100-50050100150 M phase[deg] W[GeV] M magnitude[mfm] W[GeV] -150-100-50050100150 M phase[deg] W[GeV] M magnitude[mfm] W[GeV] -150-100-50050100150 M phase[deg] W[GeV] M magnitude[mfm] W[GeV] -150-100-50050100150 M phase[deg] W[GeV] M magnitude[mfm] W[GeV] -150-100-50050100150 M phase[deg] W[GeV] M magnitude[mfm] W[GeV] -150-100-50050100150 M phase[deg] W[GeV] E magnitude[mfm] W[GeV] -150-100-50050100150 E phase[deg] W[GeV] E magnitude[mfm] W[GeV] -150-100-50050100150 E phase[deg] W[GeV] E magnitude[mfm] W[GeV] -150-100-50050100150 E phase[deg] W[GeV] E magnitude[mfm] W[GeV] -150-100-50050100150 E phase[deg] W[GeV] E magnitude[mfm] W[GeV] -200-150-100-50050100150 E phase[deg] W[GeV] E magnitude[mfm] W[GeV] -200-150-100-50050100150 E phase[deg] W[GeV] M magnitude[mfm] W[GeV] -100-50050100150200250300350 M phase[deg] W[GeV] M magnitude[mfm] W[GeV] -150-100-50050100150 M phase[deg] W[GeV] M magnitude[mfm] W[GeV] -150-100-50050100150 M phase[deg] W[GeV] M magnitude[mfm] W[GeV] -150-100-50050100150 M phase[deg] W[GeV] M magnitude[mfm] W[GeV] -150-100-50050100150 M phase[deg] W[GeV] M magnitude[mfm] W[GeV] -150-100-50050100150 M phase[deg] W[GeV]
Fig. 3.
Decomposition of the γp → K + Λ amplitude with S , P , and D multipoles. In the low-energy region, two solutions (red and blue) existwhich give identical fits to the data. The dashed line is the energy-dependent solution BnGa2013. The left two columns represent a free fit. Inthe two center columns, the penalty function (Eq. 15) is used. For the two columns on the right, (Eq. 16) is used. A.V. Anisovich et al.: Energy-independent PWA of the reaction γp → K + Λ , Table 1.
Quality of the energy-independent fit: χ /N data and number of data points (in brackets).Data BnGa EI PWA EI PWA EI PWA BnGa20132013 L = 0 , L = 0 , , penalty penalty dσ/dΩ (CLAS+GRAAL) 1.85 (316) 1.15 0.81 0.82 0.85 dσ/dΩ (MAMI) 1.55 (510) 1.05 0.84 0.87 0.87 Σ (GRAAL) 2.44 (66) 6.77 0.57 2.08 0.81 P (CLAS) 1.2 (184) 3.02 0.80 1.03 0.86 P (GRAAL) 0.65 (66) 2.49 0.68 1.27 0.64 T O x ′ O z ′ C x C z The energy-independent solution can be used as a test of theenergy-dependent solution BnGa2013 PWA [23]. The goals areto to check the stability of energy-dependent multipoles L =0 , , and to search for any missing structures. Thus the penaltyfunction, Eq. 16, is included in the fit to control the deviationof L = 0 , , multipoles from the BnGa2013 solution: χ pen = X α ( M α − M , , α ) ( δM , , α ) + X α ( E α − E , , α ) ( δE , , α ) , (16)where E , , α and M , , α are the multipoles for BnGa2013 L =0 , , solution and δE , , α , δM , , α are the multipole uncer-tainties for the fit without penalty. In this approach, multipoleswith L ≥ are fixed by the energy-dependent solution. Theerror in the multipoles from the BnGa2013 energy-dependentsolution is not included in the definition of the penalty func-tion. The result of the fit is shown in the two columns on theright in Fig. 3.The fit is only marginally worse than the unconstrained fit.This proves the quality of the energy-dependent fit. We have performed an energy-independent partial-wave analy-sis for the reaction γp → K + Λ in the region up to an invariantmass W = 1918 MeV. Although not yet complete, a data setof differential cross section values and polarization observableswas available that allowed an energy-independent extraction ofthe dominant electromagnetic multipoles that underlie the pro-duction process. The analysis requires multipoles up to L = 2 ,and there is no evidence that the fit requires multipoles with L ≥ .At present the available data allow for the extraction ofmultipoles E , E , M and M − only, without using fur-ther constraints. They are compatible with multipoles obtainedin the energy-dependent fit. Multipoles with L = 2 could notbe extracted unambiguously without imposing further, albeitrather mild constraints. The multipoles from the energy-dependent PWA BnGa2013were checked for stability in the single-energy fit constrainedby BnGa2013 solution. The resulting multipoles are very closeto the original energy-dependent solution. There is no evidencefor any additional structures which may have escaped in theenergy-dependent fit.These results demonstrate that using cross section and po-larization observables for the photoproduction of pseudoscalarmesons can be successfully employed in energy-independentPWA without additional constraints, and that the complex mul-tipoles underlying the production process can be determinedwith good accuracy. It is also demonstrated that the multipolesdetermined in this manner are consistent with those determinedin more strongly constrained energy-dependent PWA fits.These results mark an essential step in the ongoing devel-opment of sound procedures in the search for yet-to-be discov-ered excited states of the nucleon. Using data from major singleproduction channels only, the method enables an independentverification of discovery claims of new excited states in com-plex and highly constrained coupled-channel analyses. References
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