K + Λ photoproduction in a dynamical coupled-channel approach

Transcript

1 + Λ photoproduction in a dynamical coupled-channel approach Deborah Rönchen HISP, Bonn University In collaboration with M. Döring, U.-G. Meißner, J. Haidenbauer, H. Haberzettl,. akayama PWA/ATHOS5 July 28, Beijing, China Supported by DFG, SFC HPC support by Jülich Supercomputing Centre Dispersive analysis of D Dispersive analysis of D Franz iecknig

2 Introduction: Baryon spectrum in experiment and theory Theoretical predictions: lattice calculations and quark models Lattice calculation (single hadron approximation): Experimental studies: Photoproduction: e.g. from JLab, ELSA, MAMI, GRAAL, SPring [Edwards et al., Phys.Rev. D84 (2)] above.8 GeV much more states are predicted than observed, Missing resonance problem PDG listing: Until recently major part of the information from elastic source: ELSA; data: ELSA, JLab, MAMI enlarged data base with high quality for different final states (double) polarization observables alternative source of information besides X, e.g. γp Λ 2

3 U. Löring et al.: The light baryon spectrum in a relativistic quark model with instanton-induced quark forces parameters a, b, mn and gnn, λ fixed from the -spectrum and the splitting, all the excited resonances of the Introduction: Baryon spectrum experiment and theory -spectrum are now true predictions. In the subsequent subsection 7.3 we will then illustrate in some more detail, how instanton-induced effects due to t Hooft s quark-quark interaction are in fact responsible for the phenomenology of the -spectrum. 7.2 Discussion of the complete -spectrum Theoretical predictions: lattice calculations and quark models Figures 9 and show the resulting positions of the positive- and negative-parity nucleon resonances with total spins up to J = 3 2 obtained in model A and B, respectively. These are compared with the experimentally observed positions of all presently known resonances of each status taken from the Particle Data Group [37]. Again, the resonances in each column are classified by the total spin J and the parity, where left in each column the results for at most ten excitations in model A or B are shown. In comparison the experimental positions [37] are displayed on the right in each column with the uncertainties of the resonance positions indicated by the shaded boxes and the rating of each resonance denoted by the corresponding number of stars and a different shading of the error box. In addition we also display the determined resonance positions of the three new states that have been recently discovered by the SAPHIR collaboration [54,56,52,53]. These states are indicated by the symbol S. In the following, we turn to Relativistic a shell-by-shell discussion of the quark complete nucleon model: spectrum. According to their assignment to a particular shell, we additionally summarized the explicit positions of the excited model states in tables, 2, 4, 5, 6 and 7. Experimental studies: Photoproduction: e.g. from JLab, ELSA, MAMI, GRAAL, SPring ** *** Mass [MeV] 2 5 * S ** ** *** **** ** **** **** **** * ** S S **** **** *** **** ** **** **** **** J L 2T 2J **** /2+ 3/2+ 5/2+ 7/2+ 9/2+ /2+ 3/2+ /2-3/2-5/2-7/2-9/2- /2-3/2- P P3 F5 F7 H9 H 3 S D3 D5 G7 G9 I I 3 Fig. 9. The calculated positive and negative parity -resonance spectrum (isospin T = and strangeness 2 S = ) in model A (left part of each column) in comparison to the experimental spectrum taken from Particle Data Group [37] (right part of each column). The resonances are classified by the total spin J and parity. The experimental resonance position is indicated by a bar, the corresponding uncertainty by the shaded box, which is darker the better a resonance is established; the status of each resonance is additionally indicated by stars. The states labeled by S belong to new SAPHIR results [54,56,52,53], see text. Löring et al. EPJ A, 395 (2), experimental spectrum: PDG 2 above.8 GeV much more states are predicted than observed, Missing resonance problem PDG listing: Until recently major part of the information from elastic source: ELSA; data: ELSA, JLab, MAMI enlarged data base with high quality for different final states (double) polarization observables alternative source of information besides X, e.g. γp Λ 2

4 aon photoproduction Y final states certain (missing) resonances might couple predominantly to strangeness channels measurement of recoil polarization easier due to self-analyzing weak decay of hyperons more recoil and beam-recoil data, which are needed for a complete experiment Photoproduction of pseudoscalar mesons: 6 polarization observables: asymmetries composed of beam, target and recoil polarization measurements Complete Experiment: unambiguous determination of the amplitude 8 carefully selected observables Chiang and Tabakin, PRC 55, 254 (997) e.g. {σ, Σ, T, P, E, G, C x, C z } (unambiguous up to an overall phase and experimental uncertainties) 3

5 Different analyses frameworks: a few examples [Edwards et al., Phys.Rev. D84 (2)] GWU/SAID approach: PWA based on Chew-Mandelstam -matrix parameterization unitary isobar models: unitary amplitudes + Breit-Wigner resonances MAID, Yerevan/JLab, SU multi-channel -matrix: BnGa (mostly phenomenological Bgd, /D approach), Gießen (microscopic Bgd) dynamical coupled-channel (DCC): 3-dim scattering eq., off-shell intermediate states AL-Osaka (EBAC), Dubna-Mainz-Taipeh, Jülich-Bonn 4

6 Different analyses frameworks: a few examples [Edwards et al., Phys.Rev. D84 (2)] GWU/SAID approach: PWA based on Chew-Mandelstam -matrix parameterization unitary isobar models: unitary amplitudes + Breit-Wigner resonances MAID, Yerevan/JLab, SU multi-channel -matrix: BnGa (mostly phenomenological Bgd, /D approach), Gießen (microscopic Bgd) dynamical coupled-channel (DCC): 3-dim scattering eq., off-shell intermediate states AL-Osaka (EBAC), Dubna-Mainz-Taipeh, Jülich-Bonn 4

7 The Jülich-Bonn DCC approach

8 The Jülich-Bonn DCC approach EPJ A 49, 44 (23) Dynamical coupled-channels (DCC): simultaneous analysis of different reactions The scattering equation in partial-wave basis L S p T IJ µν LSp = L S p V IJ µν LSp + γ,l S dq q 2 L S p V IJ µγ L S q E E γ(q) + iɛ L S q T IJ γν LSp potentials V constructed from effective L s-channel diagrams: T P genuine resonance states t- and u-channel: T P dynamical generation of poles partial waves strongly correlated contact terms 6

9 The Jülich-Bonn DCC approach Resonance states: Poles in the T -matrix on the 2 nd Riemann sheet pole position E is the same in all channels residues branching ratios Re(E ) = mass, -2Im(E ) = width Analytic structure (P) 2.3 GeV (2-body) unitarity and analyticity respected 3-body channel: - parameterized effectively as, σ, ρ - / subsystems fit the respective phase shifts branch points move into complex plane Pl(x) from PW decomposition, e.g. Vu = dx u m 2 + iɛ, 7

10 Photoproduction EPJ A 5, (25) Multipole amplitude γ, η γ m, η M IJ µγ = V IJ µγ + T IJ µκ GκV IJ κγ κ (partial wave basis) V γ V κγ B G T m =, η,, B =,, Λ T µκ: Jülich hadronic T -matrix Watson s theorem fulfilled by construction analyticity of T: extraction of resonance parameters Photoproduction potential: approximated by energy-dependent polynomials γ m γ m V µγ = (E, q) P P µ + B P P i, γ a µ B = γa µ (q) P P µ m (E) + i γ a µ;i (q)pp i (E) E m b i γ a µ, γa µ;i : hadronic vertices correct threshold behaviour, cancellation of singularity at E = mb i γ a µ;i affects pion- and photon-induced production of final state mb i: resonance number per multipole; µ: channels, η,, Y Polynomials 8

11 Data analysis and fit results

12 Combined analysis of pion- and photon-induced reactions EPJ A 54, (28) Simultaneous fit Fit parameters: p ηn, Λ, Σ, + Σ + p + Σ + s-channel: resonances (T P ) Λ 34 free parameters resonances ( m bare + couplings to, ρ, η,, Λ, Σ) + resonances ( m bare + couplings to, ρ,, Σ) m bare + f mb contact terms: one per partial wave, couplings to, η, (,) Λ, Σ 6 free parameters γp p, + n, ηp, + Λ: couplings of the polynomials 566 free parameters γ P P µ m γ + B P P i, γ a µ B m 76 in total, calculations on the JURECA supercomputer [Jülich Supercomputing Centre, JURECA: General-purpose supercomputer at Jülich Supercomputing Centre, Journal of large-scale research facilities, 2, A62 (26)] t- & u-channel parameters: fixed to values of hadronic DCC analysis (JüBo 23)

13 Combined analysis of pion- and photon-induced reactions EPJ A 54, (28) Data base Reaction Observables (# data points) p./channel PWA GW-SAID WI8 (ED solution) 3,76 p ηn dσ/dω (676), P (79) 755 p Λ dσ/dω (84), P (472), β (72),358 p Σ dσ/dω (47), P (2) 59 p + Σ dσ/dω (5) 5 + p + Σ + dσ/dω (24), P (55), β (7),682 γp p dσ/dω (743), Σ (2927), P (768), T (44), σ 3 (4), G (393), H (225), E (467), F (397), C x (74), C L z (26) L 7,564 γp + n dσ/dω (596), Σ (456), P (265), T (78), σ 3 (23), G (86), H (28), E (93) 9,748 γp ηp dσ/dω (568), Σ (43), P (7), T (44), F (44), E (29) 6,57 γp + Λ dσ/dω (2478), P (62), Σ (459), T (383), C x (2), C z (23), O x (66), O z (66), O x (34), O z (34), 5,936 in total 48,5

14 Combined analysis of pion- and photon-induced reactions EPJ A 54, (28) Data base Reaction Observables (# data points) p./channel PWA GW-SAID WI8 (ED solution) 3,76 p ηn dσ/dω (676), P (79) 755 p Λ dσ/dω (84), P (472), β (72),358 p Σ dσ/dω (47), P (2) 59 p + Σ dσ/dω (5) 5 + p + Σ + dσ/dω (24), P (55), β (7),682 γp p dσ/dω (743), Σ (2927), P (768), T (44), σ 3 (4), G (393), H (225), E (467), F (397), C x (74), C L z (26) L 7,564 γp + n dσ/dω (596), Σ (456), P (265), T (78), σ 3 (23), G (86), H (28), E (93) 9,748 γp ηp dσ/dω (568), Σ (43), P (7), T (44), F (44), E (29) 6,57 γp + Λ dσ/dω (2478), P (62), Σ (459), T (383), C x (2), C z (23), O x (66), O z (66), O x (34), O z (34), 5,936 in total 48,5

15 Combined analysis of pion- and photon-induced reactions EPJ A 54, (28) Data base Reaction Observables (# data points) p./channel PWA GW-SAID WI8 (ED solution) 3,76 p ηn dσ/dω (676), P (79) 755 p Λ dσ/dω (84), P (472), β (72),358 p Σ dσ/dω (47), P (2) 59 p + Σ dσ/dω (5) 5 + p + Σ + dσ/dω (24), P (55), β (7),682 γp p dσ/dω (743), Σ (2927), P (768), T (44), σ 3 (4), G (393), H (225), E (467), F (397), C x (74), C L z (26) L 7,564 γp + n dσ/dω (596), Σ (456), P (265), T (78), σ 3 (23), G (86), H (28), E (93) 9,748 γp ηp dσ/dω (568), Σ (43), P (7), T (44), F (44), E (29) 6,57 γp + Λ dσ/dω (2478), P (62), Σ (459), T (383), C x (2), C z (23), O x (66), O z (66), O x (34), O z (34), 5,936 [ beam-target obs. missing ] in total 48,5

16 Fit result + Λ photoproduction EPJ A 54, (28) simultaneous fit of γp p, + n, ηp, + Λ and, η, Λ, Σ γp + Λ: Differential cross section dσ/dω [µb/sr] E cm =65 MeV JU θ [deg].2 MC MeV θ [deg] JU4: Jude PLB 735 (24), MC: McCracken PRC 8 (2) Beam asymmetry.5 Σ LL7 72 MeV θ [deg] ZE3 274 MeV θ [deg] LL7: Lleres EPJA 3 (27), ZE3: Zegers PRL (23) Recoil polarization.5 P MeV MC θ [deg] MeV MC θ [deg] MC4: Mcabb PRC 69 (24), MC: McCracken PRC 8 (2) Target asymmetry.5 T MeV LL θ [deg] MeV LL θ [deg] LL9: Lleres EPJA 39 (29) 2

17 Fit result + Λ photoproduction: beam-recoil asymmetries EPJ A 54, (28) simultaneous fit of γp p, + n, ηp, + Λ and, η, Λ, Σ γp + Λ: C x C z 987 MeV 269 MeV C x MeV BR θ [deg] BR7: Bradford PRC 75 (27) O x BR MeV θ [deg] C z BR θ [deg] BR7: Bradford PRC 75 (27) O z BR θ [deg] 649 MeV 883 MeV.5 LL O x θ [deg] LL9: Lleres EPJA 39 (29) LL θ [deg] O z MeV LL θ [deg] LL9: Lleres EPJA 39 (29) MeV LL θ [deg] 3

18 Fit result + Λ photoproduction EPJ A 54, (28) simultaneous fit of γp p, + n, ηp, + Λ and, η, Λ, Σ Fit of new CLAS data (Paterson et al. Phys. Rev. C 93, 652 (26)):.5 Data not included Fit JuBo27- Σ E cm =72 MeV Θ [deg] 4

19 Fit result + Λ photoproduction EPJ A 54, (28) simultaneous fit of γp p, + n, ηp, + Λ and, η, Λ, Σ Fit of new CLAS data (Paterson et al. Phys. Rev. C 93, 652 (26)): Data not included Fit JuBo27- E cm =72 MeV T Θ [deg] 5

20 Fit result + Λ photoproduction EPJ A 54, (28) simultaneous fit of γp p, + n, ηp, + Λ and, η, Λ, Σ Fit of new CLAS data (Paterson et al. Phys. Rev. C 93, 652 (26)): O x Data not included Fit JuBo27- E cm =72 MeV Θ [deg] 6

21 Fit result + Λ photoproduction EPJ A 54, (28) simultaneous fit of γp p, + n, ηp, + Λ and, η, Λ, Σ Fit of new CLAS data (Paterson et al. Phys. Rev. C 93, 652 (26)): O z Data not included Fit JuBo27- E cm =72 MeV Θ [deg] 7

22 The resonance spectrum

23 Resonance spectrum Resonance states: Poles in the T -matrix on the 2 nd Riemann sheet Re(E ) = mass, -2Im(E ) = width elastic residue ( r, θ ), normalized residues for inelastic channels ( Γ Γ µ/γ tot, θ µ ) photocouplings at the pole: Ãh pole = Ah pole eiϑh, h = /2, 3/2 In the present analysis: Ã h pole = I F q p k p 2 (2J+)E m r Res A h L± I F : isospin factor qp (kp): meson (photon) momentum at the pole J = L ± /2 total angular momentum E : pole position r : elastic residue A L± h : helicity multipole all 4-star and states up to J = 9/2 are seen (exception: (895)/2 ) + some states rated with less than 4 stars one additional s-channel diagram included: (9)3/2 + hints for new dynamically generated poles 9

24 Uncertainties of extracted resonance parameters Challenges in determining resonance uncertainties, e.g.: elastic channel: not data but GWU SAID PWA correlated χ 2 fit including the covariance matrix ˆΣ (available on SAID webpage!) PRC 93, 6525 (26) χ 2 (A) = χ 2 (Â) + (A Â)T ˆΣ (A Â) A vector of fitted PWs, Â vector of SAID SE PWs same χ 2 as fitting to data up to nonlinear and normalization corrections error propagation data fit parameters derived quantities: bootstrap method: generate pseudo data around actual data, repeat fit model selection, significance of resonance signals: determine minimal resonance content using Bayesian evidence [PRL 8, 822; PRC 86, 522 (22)] or the LASSO method [PRC 95, 523 (27); J. R. Stat. Soc. B 58, 267 (996)]: λ penalty factor, a i fit parameter χ 2 T = i max χ2 + λ a i i= talk by R. Molina on Wednesday 2

25 Uncertainties of extracted resonance parameters In JüBo framework: such methods are numerically challenging, but planned for the (near) future Estimation of uncertainties of extracted resonance parameters in the present study: from 9 re-fits to re-weighted data sets individually increase the weight in each reaction channel extract resonance parameters from refits maximal deviation of resonance parameters of the refits = error only a qualitative estimation of relative uncertainties, absolute size not well determined 2

26 Resonance spectrum: selected results I = /2, J P = 3/2 + EPJ A 54, (28) (9) 3/2 + Re E 2Im E r θ [MeV] [MeV] [MeV] [deg] (2) 27(23).6(.2) 6(2) PDG ± 2 5 ± 5 4 ± 2 2 ± 3 Γ /2 Γ/2 /2 η Γ Γ [%] θ Γ/2 /2 Γ Λ η tot Γ [%] θ Γ/2 Σ Λ tot Γ [%] tot θ Σ 27.(.7) (79) 2.(.4).7(86) (7) 34(74) PDG (BnGa) 5 ± 2 7 ± 6 3 ± 2 9 ± 4 4 ± 2 ± 3 (72) 3/2 + Re E 2Im E r θ [MeV] [MeV] [MeV] [deg] (4) 9(3) 2.3(.5) 57(22) 25-B PDG ± ± 3 Γ /2 Γ/2 /2 η Γ Γ [%] θ Γ/2 /2 Γ Λ η tot Γ [%] θ Γ/2 Σ Λ tot Γ [%] θ Σ tot 27.3(.2) 39(35).5(.9) 66(3).6(.4) 26(58) 25-B PDG (BnGa) 3 ± 2 6 ± 4 5 ± 45 2

27 Resonance spectrum: selected results I = /2, J P = 3/2 + EPJ A 54, (28) (9) 3/2 + Re E 2Im E r θ [MeV] [MeV] [MeV] [deg] (2) 27(23).6(.2) 6(2) PDG ± 2 5 ± 5 4 ± 2 2 ± 3 Γ /2 Γ/2 /2 η Γ Γ [%] θ Γ/2 /2 Γ Λ η tot Γ [%] θ Γ/2 Σ Λ tot Γ [%] tot θ Σ 27.(.7) (79) 2.(.4).7(86) (7) 34(74) PDG (BnGa) 5 ± 2 7 ± 6 3 ± 2 9 ± 4 4 ± 2 ± 3 E +, M + multipoles in γp + Λ: Im(A) [mfm] Re(A) [mfm] E + -2 M E cm [MeV] E cm [MeV] Red lines: JüBo27 Black (dashed) lines: BnGa24-2 Gutz et al. EPJ A 5, 74 (24) (9) 3/2 + : seen by several other groups included ( by hand ) to improve fit result for γp + Λ 2

28 Resonance spectrum: selected results I = /2, J P = 3/2 + EPJ A 54, (28) (9) 3/2 + Re E 2Im E r θ [MeV] [MeV] [MeV] [deg] (2) 27(23).6(.2) 6(2) PDG ± 2 5 ± 5 4 ± 2 2 ± 3 Γ /2 Γ/2 /2 η Γ Γ [%] θ Γ/2 /2 Γ Λ η tot Γ [%] θ Γ/2 Σ Λ tot Γ [%] tot θ Σ 27.(.7) (79) 2.(.4).7(86) (7) 34(74) PDG (BnGa) 5 ± 2 7 ± 6 3 ± 2 9 ± 4 4 ± 2 ± 3 P 3 S D 33. P3 D 35 σ tot [mb] P 33 D 5 Total cross section p + Σ. D 3 P S E cm [MeV] 2

29 Resonance spectrum: selected results I = /2, J P = 5/2 EPJ A 54, (28) (675) 5/2 Re E 2Im E r θ [MeV] [MeV] [MeV] [deg] (8) 35(9) 28(2) 22(3) 25-B PDG ± ± 5 25 ± 5 Γ /2 Γ/2 /2 η Γ Γ [%] θ Γ/2 /2 Γ Λ η tot Γ [%] θ Γ/2 Σ Λ tot Γ [%] θ Σ tot 27 9.(.8) 45(3).7(.2) 9(6) 2.3(.2) 75() 25-B PDG 28 (26) 5/2 Re E 2Im E r θ dynamically generated [MeV] [MeV] [MeV] [deg] (2) 2(3).4(.) 72(2) PDG ± 2 Γ /2 Γ/2 /2 η Γ Γ [%] θ Γ/2 /2 Γ Λ η tot Γ [%] θ Γ/2 Σ Λ tot Γ [%] θ Σ tot 27.2(.2) 9(2) 2.2(.2) 86(3) 3.(.3) 86(3) PDG (BnGa) 5 ± 3 4 ± 25 ±.5 4 ± 2 7 ± 3 22

30 Resonance spectrum: selected results I = /2, J P = 5/2 EPJ A 54, (28) E 2+, M 2+ multipoles in γp + Λ: Re(A) [mfm] Im(A) [mfm] E cm [MeV] E 2+ M E cm [MeV] Red lines: JüBo27 Black (dashed) lines: BnGa24-2 Gutz et al. EPJ A 5, 74 (24) (26) 5/2 : seen by several other groups inconclusive indications already in previous JüBo fits (-2Im E > 6 MeV) (26) 5/2 Re E 2Im E r θ dynamically generated [MeV] [MeV] [MeV] [deg] (2) 2(3).4(.) 72(2) PDG ± 2 Γ /2 Γ/2 /2 η Γ Γ [%] θ Γ/2 /2 Γ Λ η tot Γ [%] θ Γ/2 Σ Λ tot Γ [%] θ Σ tot 27.2(.2) 9(2) 2.2(.2) 86(3) 3.(.3) 86(3) PDG (BnGa) 5 ± 3 4 ± 25 ±.5 4 ± 2 7 ± 3 22

31 Resonance spectrum: states EPJ A 54, (28) Λ pure I = /2 mixed isospin, Σ channels changes in I = 3/2 spectrum most of the well established s similar to previous JüBo results in general larger uncertainties than for I = /2 ( extension to γ Σ!) Example: (6) 3/2 + dynamically generated (6) 3/2 + Γ /2 Re E 2Im E r θ Γ/2 Σ Γ θ Σ tot [MeV] [MeV] [MeV] [deg] [%] [deg] (7) 8(3) (6) 62(4) 3(7) 2(4) 25-B PDG 28 5 ± 5 27 ± 7 25 ± 5 8 ± 3 A /2 pole ϑ /2 A 3/2 pole ϑ 3/2 [ 3 GeV 2 ] [deg] [ 3 GeV 2 ] [deg] 27 54(25) 44(3) 46(9) 72(36) 25-B BnGa [] 53 ± 3 ± 5 55 ± 52 ± 5 [] V. Sokhoyan et al. EPJ A5 95 (25) 23

32 Summary and outlook Impact of + Λ photoproduction on the resonance spectrum in the JüBo DCC approach: confirmation of (9)3/2 + and (26)5/2 many resonances move closer to PDG values hints for additional new states Outlook: extension to Σ photoproduction (already in progress) include covariance matrices in fit model selection methods (LASSO): find the minimal model, reduce number of fit parameters 24

33 Thank you for your attention!

34 Details of the formalism Polynomials: n ( ) E j Pi P (E) = gi,j P E e gp i,n+ (E E ) m j= n Pµ P (E) = gµ,j P j= ( E E m ) j e gp µ,n+ (E E ) - E = 77 MeV - gi,j P, gp µ,j : fit parameter - e g(e E ) : appropriate high energy behavior - n = 3 back

35 The scattering potential: s-channel resonances n V P γµ;i a = γc ν;i z m b i= i - i: resonance number per PW - γ c ν;i (γ a µ;i ): creation (annihilation) vertex function with bare coupling f (free parameter) - z: center-of-mass energy - m b i : bare mass (free parameter) J 3/2: γ c ν;i (γ a µ;i ) from effective L Vertex (S ) (S )η (S )ρ (S ) L int f Ψ m γ µ τ µ Ψ + h.c. f Ψ m γ µ µη Ψ + h.c. f Ψ γ 5 γ µ τ ρ µ Ψ + h.c. f Ψ m γ 5 S µ µ + h.c. 5/2 J 9/2: correct dependence on L (centrifugal barrier) (γ a,c ) 52 = k M (γa,c ) 32 + (γ a,c ) 52 + = k M (γa,c ) 32 (γ a,c ) 72 = k2 M 2 (γa,c ) 32 (γ a,c ) 72 + = k 2 M 2 (γa,c ) 32 + (γ a,c ) 92 = k3 M 3 (γa,c ) 32 + (γ a,c ) 92 + = k 3 M 3 (γa,c ) 32 2

36 The scattering potential: t- and u-channel exchanges ρ η Δ σ Λ Σ,Δ,() σ,, Δ, Ct.,, a, Δ, ρ, Σ, Σ*, * Λ, Σ, Σ*, () ρ, ω, a * ρ, Δ, Ct., ρ -, η, f - - *, Λ Σ, Σ*, * Δ, Δ, ρ - - σ, σ - - Λ Ξ, Ξ*, f, Ξ, Ξ*, ρ ω, φ Σ Ξ, Ξ*, f, ω, φ, ρ ( ) n Λ Free parameters: cutoffs Λ in the form factors: F(q) = 2 mx 2, n =, 2 Λ 2 + q 2 3

37 Δ Δ σ ρ ρ ρ ρ ρ ρ ρ ω a a η η Δ Δ Δ Δ ρ σ σ * Λ 4

38 Λ Λ Λ Σ κ Σ* * Σ Σ Σ Σ Λ κ Σ Σ Σ* ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ Δ Δ Δ η η η η f Δ Δ Δ Δ ρ 4

39 Δ Δ Δ Δ σ σ σ σ σ σ Λ Λ Λ Λ f ω Λ Λ φ Λ Σ ρ f Σ Σ ω φ ρ Σ Σ Σ Σ Σ Σ η * Λ Λ η Λ * Σ η η Σ Σ Σ* η Σ Λ Ξ Λ 4

40 Interaction potential from effective Lagrangian J. Wess and B. Zumino, Phys. Rev. 63, 727 (967); U.-G. Meißner, Phys. Rept. 6, 23 (988); B. Borasoy and U.-G. Meißner, Int. J. Mod. Phys. A, 583 (996). consistent with the approximate (broken) chiral SU(2) SU(2) symmetry of QCD Vertex L int Vertex L int g m Ψγ5 γ µ τ µ Ψ ω g ω Ψ[γ µ κω 2m σ µν ν]ω µψ g µ gωρ S m µ Ψ + h.c. ωρ mω ɛ αβµν α ρ β µ ω ν ρ g ρ( µ ) ρ µ ρ i g ρ µ γ 5 γ µ S ρ mρ µνψ + h.c. ρ g ρ Ψ[γ µ κρ 2m σ µν ν] τ ρ µψ ρρρ g ρ ( ρ µ ρ ν) ρ µν σ g σ ΨΨσ ρρ κρg 2 ρ 2m Ψσ µν τψ( ρ µ ρ ν) σ gσ 2m µ µ σ g m µγ 5 γ ν T µ ν σσσ g σσσm σσσσ ρ g ρ τ (γ µ i κ ρ 2m σ µν ν) ρ a a ρ ρ µ T τ g m 2g ρ Ψγ 5 γ µ τψ( ρ µ ) η g η Ψγ 5 γ µ m µηψ g m ma Ψγ 5 γ µ τψ a µ a g a m Ψ τψ a 2ga ρ ma [ µ a ν ν a µ] [ µ ρ ν ν ρ µ ] ηa g ηa m η a + 2ga ρ 2ma [ ( µ ρ ν ν ρ µ)] [ µ a ν ν a µ ] 5

41 Generalization to SU(3) t- and u-channel exchange: T P coupling constants fixed from SU(3) symmetry e.g. g Λ = 3 3 g ( + 2α BBP ) J. J. de Swart, Rev. Mod. Phys. 35, 96 (963) [Erratum-ibid. 37, 326 (965)]. Λ, Σ Λ, Σ Σ, Σ Σ Λ s-channel: resonances Λ Σ Λ Σ Σ, σ, ω, φ ρ σ, ω φ, ρ Λ ew free parameters: cutoffs Λ Λ Σ ew free parameters: bare couplings g Y and g Σ 6

42 Theoretical constraints of the S-matrix Unitarity: probability conservation 2-body unitarity 3-body unitarity: discontinuities from t-channel exchanges Meson exchange from requirements of the S-matrix [Aaron, Almado, Young, Phys. Rev. 74, 222 (968)] Analyticity: from unitarity and causality correct structure of branch point, right-hand cut (real, dispersive parts) to approximate left-hand cut Baryon u-channel exchange p3, λ3 p4, λ4 p4, λ4 p3, λ3 p3, λ3 p4, λ4 q, λ q, λ q, λ Resonances p, λ p2, λ2 p, λ p2, λ2 p, λ p2, λ2 q = p p3 q = q p4 q = p + p2 = 7

43 The SAID, BnGa and JüBo approaches All three approaches: coupled channel effects unitarity (2 body) amplitudes are analytic functions of the invariant mass SAID PWA based on Chew-Mandelstam -matrix -matrix elements parameterized as energy-dependent polynomials resonance poles are dynamically generated (except for the (232)) masses, width and hadronic couplings from fits to pion-induced and η production Bonn-Gatchina (BnGa) PWA Multi-channel PWA based on -matrix (/D) mostly phenomenological model resonances added by hand resonance parameters determined from large experimental data base: pion-, photon-induced reactions, 3-body final states Jülich-Bonn (JüBo) DCC model based on a Lippmann-Schwinger equation formulated in TOPT hadronic potential from effective Lagrangians photoproduction parameterized by energy-dependent polynomials resonances as s-channel states (dynamical generation possible) resonance parameters determined from pionand photon-induced data 8

44 Construction of the multipole amplitude M IJ µγ Field theoretical approaches : DMT, AL-Osaka, Jülich-Athens-Washington,... Example: Gauge invariant formulation by Haberzettl, Huang and akayama Phys. Rev. C56 (997), Phys. Rrev. C74 (26), Phys. Rev. C85 (22) satisfies the generalized off-shell Ward-Takahashi identity earlier version of the Jülich-Bonn model as FSI Photoproduction amplitude: M µ = M µ s + M µ u + Mµ t }{{} coupling to external legs + M µ int }{{} coupling inside hadronic vertex Strategy: Replace by phenomenological contact term such that the generalized WTI is satisfied Alternative gauge invariant chiral unitary method: Borasoy et al., Eur. Phys. J. A 34 (27) 6 9