Meson-baryon interaction in the meson exchange picture

Transcription

1 Meson-baryon interaction in the meson exchange picture M. Döring C. Hanhart, F. Huang, S. Krewald, U.-G. Meißner, K. Nakayama, D. Rönchen, Forschungszentrum Jülich, University of Georgia, Universität Bonn Nucl. Phys. A 829, 7 (29), Phys. Lett. B 68, 26 (29)

2 The scattering equation Poles and Residues The Jülich model of pion-nucleon interaction Motivation Hadronic reactions provide insight to QCD in the non-perturbative region. Intense experimental effort at JLab (Clas), ELSA, MAMI,... Theoretical data analysis required (e.g. EBAC/JLab). Chiral Lagrangian of Wess and Zumino. Channels πn, ηn; effective ππn channels σn, ρn, π. Baryonic resonances up to J = 3/2 with derivative couplings as required by chiral symmetry. New: KY channels, J = 5/2, 7/2 resonances, chiral unitary σ meson. Other non-k-matrix meson exchange models: DMT, Surya, Gross, PRC 53 (996); EBAC Juliá Díaz et al., PRC 76 (27); Suzuki, Sato, Lee, PRC 79 (29); Kamano et al., PRC 79 (29).

3 The scattering equation Poles and Residues Scattering equation in the JLS basis L S k T IJ µν LSk = L S k V IJ µν LSk + γ,l S k 2 dk L S k V IJ µγ L S k T: Amplitude V: Pseudopotential G: Propagator Z E γ(k )+iǫ L S k T IJ γν LSk J: total angular momentum L: orbital angular momentum S: total Spin of MB system I: isospin k(k, k ): incoming(outgoing, intermediate) momentum, on- or off-shell µ(ν, γ): incoming(outgoing, intermediate) channel [πn, ηn, π, ρn, σn]

4 The scattering equation Poles and Residues Scattering equation in the JLS basis L S k T IJ µν LSk = L S k V IJ µν LSk + γ,l S k 2 dk L S k V IJ µγ L S k Z E γ(k )+iǫ L S k T IJ γν LSk

5 The scattering equation Poles and Residues Scattering equation in the JLS basis L S k T IJ µν LSk = L S k V IJ µν LSk + Features γ,l S k 2 dk L S k V IJ µγ L S k Z E γ(k )+iǫ L S k T IJ γν LSk Meson exchange provides the correct and relevant degrees of freedom in the second and third resonance region (no contact terms). No on-shell factorization. Full analyticity (dispersive parts). Meson exchange puts strong constraints as all partial waves are linked. Dynamical generation of resonances is possible, but not easy (strong constraints!) Topic of missing resonances: no need to model background with resonances.

6 The scattering equation Poles and Residues Partial waves in πn πn (Solution 22) Re S Im S - - Re P Im P W [MeV] Data : GWU/SAID analysis, single energy solution Re S3 Im S3 Re P3 Im P W [MeV] Re P3 Im P3 Re D3 Im D W [MeV] Re D33 Im D33 Re P33 Im P W [MeV]

7 The scattering equation Poles and Residues Partial waves in πn πn (Solution 22) Re S Im S - - Re P Im P W [MeV] Data : GWU/SAID analysis, single energy solution Re S3 Im S3 Re P3 Im P W [MeV] Re P3 Im P3 Re D3 Im D W [MeV] Re D33 Im D33 Re P33 Im P W [MeV]

8 The scattering equation Poles and Residues Poles and residues on the unphysical sheets, given by the analytic continuation. Re z -2 Im z R θ [deg] [MeV] [MeV] [MeV] [ ] N (44) P ARN HOE CUT 375±3 8±4 52±5 -±35 N (52) D ARN HOE CUT 5±5 4± 35±2-2±5 Re z -2 Im z R θ [deg] [MeV] [MeV] [MeV] [ ] N (535) S ARN HOE 487 CUT 5±5 26±8 2±4 +5±45 N (65) S ARN HOE CUT 64±2 5±3 6± -75±25 N (72) P ARN HOE CUT 68±3 2±4 8±2-6±3 (232) P ARN HOE CUT 2± ±2 53±2-47± (62) S ARN HOE CUT 6±5 2±2 5±2 -±2 (7) D ARN HOE CUT 675±25 22±4 3±3-2±25 (9) P ARN HOE CUT 88±3 2±4 2±4-9±3 [ARN]: Arndt et al., PRC 74 (26), [HOE]: Höhler, πn Newsl. 9 (993), [CUT]: Cutkowski et al., PRD 2 (979). Analytic continuation, residues to ηn, σn, ρn, π. Zeros. Branching ratios to πn, ηn.

9 The scattering equation Poles and Residues An additional state in P * New pole in P found at z= i MeV. full (T P + T NP ) Roper: a _ / (z-z ) + a N(62): a _ / (z-z ) * Very weak branching to πn. -. * Very large branching to π. -.5 * Resonance transition amplitude πn --> π : Manley (πn --> ππn): -.5 here: -5 * Roper: generated from σn. * N(62): generated from π..5 Re τ πn P --> πn P πn P --> π P πn P --> σn S -.5 * Dynamics of the nucleon pole: Pole repulsion N <--> Roper z [MeV] Im τ

10 The scattering equation Poles and Residues The analytic structure of the P partial wave

11 The chiral unitary σ meson Inclusion of KΛ and KΣ Chiral unitary approach to ππ scattering

12 The chiral unitary σ meson Inclusion of KΛ and KΣ Implementation of the chiral σ

13 The chiral unitary σ meson Inclusion of KΛ and KΣ Result for the Roper resonance Readjustment of cut-offs and g σσσ coupling Red lines: Chiral σ Blue lines: Fit Gasparyan 22 Re P Im P.6 Analytic structure of the amplitude exactly the same as before (3 branchpoints from σn) Dynamical generation of Roper and second P does not depend on details of the model Chiral σ provides better description z [MeV]

14 The chiral unitary σ meson Inclusion of KΛ and KΣ The reaction π + p K + Σ + Pure isospin 3/2; relatively few resonances. Strong constraints on the amplitude of the Jülich model! Good, simple data situation (Candlin ). Simultaneous fit to πn πn partial waves plus π + p K + Σ + observables. Requires extended fitting efforts: Parallelization of the code in energies, implementation of Minuit done. Code runs on Juropa@Jülich. First tasks: Inclusion of KY in t and u channel exchange processes. Inclusion of Spin 5/2 and Spin 7/2 resonances.

15 The chiral unitary σ meson Inclusion of KΛ and KΣ Inclusion of the KY channels Inclusion via SU(3) symmetry [no additional freedom]

16 The chiral unitary σ meson Inclusion of KΛ and KΣ Differential cross section of π + p K + Σ + 8 Candlin et al., NPB 226, (983) Final 3 Z=822 MeV 845 MeV 87 MeV 89 MeV 6 dσ/dω [ µb / cosθ ] MeV 939 MeV 97 MeV 985 MeV cosθ

17 The chiral unitary σ meson Inclusion of KΛ and KΣ Total cross section of π + p K + Σ + σ [mb].8.6 Final

18 The chiral unitary σ meson Inclusion of KΛ and KΣ Polarization of π + p K + Σ +.5 Candlin et al., NPB 226, (983) Final 3 Z=822 MeV 845 MeV 87 MeV 89 MeV.5 polarization MeV 939 MeV 97 MeV 985 MeV cosθ

19 The chiral unitary σ meson Inclusion of KΛ and KΣ Spin rotation parameter β of π + p K + Σ + Definitions Observabbles 2 Final 3 z = 22 MeV β [rad] -2 2 z = 27 MeV β [rad] -2 Data: Candlin NPB 3, 63 (988) cos θ

20 The chiral unitary σ meson Inclusion of KΛ and KΣ πn πn phase shifts Simultaneous fit to π + p K + Σ + Re τ.8 Im τ S 3.6 Gasparyan 22 This fit P s /2 [MeV] s /2 [MeV]

21 The chiral unitary σ meson Inclusion of KΛ and KΣ πn πn phase shifts Simultaneous fit to π + p K + Σ P 33 Re τ.8.6 Fit 22 This fit P 33 Im τ D 33.3 D z [MeV] z [MeV]

22 The chiral unitary σ meson Inclusion of KΛ and KΣ πn πn phase shifts (higher partial waves; selection) Simultaneous fit to π + p K + Σ + Re τ.5 Im τ. -. F SAID solution This fit including π + p --> K + Σ +. G z [MeV] z [MeV]

23 The chiral unitary σ meson Inclusion of KΛ and KΣ The π + p K + Σ + partial wave amplitudes of Candlin 984 (Comparable quality of the fit) Black solid lines: partial waves π + p --> K + Σ +, Candlin et al., NPB 238, 477 (984) Final 3 Re S3 Re P3 Re P33 Re D33 Re, Im τ Im S3 Re D35 Re F35 Im P3 Im P33 Re F37 Re G37 Im D33 Very different partial wave content! Additional constraints from πn πn indeed necessary. Im D35 Im F37 Im G37 Im F z [MeV]

24 The chiral unitary σ meson Inclusion of KΛ and KΣ New results on photoproduction (Talk of Fei Huang, S. Krewald) Full gauge invariance respected! p) ( b/sr) d /d ( p (245, 58) (597, 44) (652, 45) (75, 484) (758, 57) (88, 548) (859, 579) (99, 68) (957, 636) 6 2 (278, 84) (32, 27) (4, 277) (45, 33) (5, 356) (555, 386) (deg.) Differential cross section for γp π p (36, 247) ( p p) (244, 57) (275, 82) (322, 29) (35, 24) (4, 277) (45, 33) (5, 349) (55, 384) (65, 426) (666, 459) (73, 483) (769, 524) (8, 544) (862, 58) (895, 6) (96, 638) (deg.) Photon spin asymmetry for γp π p 6 2 8

25 The chiral unitary σ meson Inclusion of KΛ and KΣ Conclusions Lagrangian based, field theoretical description of meson-baryon interaction. (Some) chiral constraints are fulfilled (σ meson). Unitarity and full analyticity are ensured. Analytic continuation: precise, model independent determination of resonance parameters (poles). Roper dynamically generated; hints for additionally generated structures. Examples: P(7) (π ), D3(7) (ρn from hidden gauge). Meson and baryon exchange are the relevant degrees of freedom in the 2 nd and 3 rd resonance region. Exchange provides heavy constraints, because all partial waves are linked. With this background, only a few bare states are needed (so far). Coupled channel formalism links different reactions in one combined description: πn πn, π + p K + Σ +,..., γn πn,...

26 Analytic continuation via Contour deformation...enables access to all Riemann sheets Π σ(z) = q 2 (v σππ (q)) 2 dq z E E 2 + iǫ z E E 2 = q = q c.m. q c.m. = 2z [z2 (m m 2) 2 ][z 2 (m + m 2) 2 ] back Plot q c.m.(z) in the q plane of integration (X: Pole positions). case (a), Im z > : straight integration from q = to q =. case (b), Im z = : Pole is on real q axis. case (c), Im z < : Deformation gives analytic continuation. Special case: Pole at q = branch point at z = m + m 2 (= threshold).

27 Effective ππn channels: Analytic structure back Three branch points and four sheets for each of the σn, ρn, and π propagators. The cut along Im z = is induced by the cut of the self energy of the unstable particle. The poles of the unstable particle (σ) induce branch points in the σn propagator at z b2 = m N + z, z b 2 = m N + z

28 Effective ππn channels: Analytic structure back Three branch points and four sheets for each of the σn, ρn, and π propagators. The cut along Im z = is induced by the cut of the self energy of the unstable particle. The poles of the unstable particle (σ) induce branch points in the σn propagator at z b2 = m N + z, z b 2 = m N + z

29 Couplings g = a to other channels back Nπ Nρ () (S = /2) Nρ (2) (S = 3/2) Nρ (3) (S = 3/2) N (535) S S i S i D.5.3i N (65) S S i S.9 9.i D.3 2.i N (44) P P.2 5.i P i P i (62) S 3 S i S 3..i D i (9) P 3 P i P 3 i P 3 i N (72) P 3 P i P i P 3.+.i F 3. + i N (52) D 3 D i D i D i S i (232) P 33 P i P i P i F 33..i (7) D 33 D i D i D 33 i S 33..9i Nη π () π (2) Nσ N (535) S S.9 2.3i D i P.4.2i N (65) S S i D i P 2..i N (44) P P. +.i P 4.6.7i S i (62) S 3 D 3. 4.i (9) P 3 P i N (72) P 3 P i P i F 3..3i D i N (52) D 3 D 3.6.6i D 3. + i S i P 3.6.6i (232) P 33 P 33 (4 to 5) + i( to.5) F 33 (7) D 33 D i S i Resonance couplings g i [ 3 MeV /2 ] to the coupled channels i. Also, the LJS type of each coupling is indicated. For the ρn channels, the total spin S is also indicated.

30 Zeros and branching ratio to πn, ηn back first sheet second sheet [FA2] P 235 i S i i D i S i i P i i P i i P i D i i Position of zeros of the full amplitude T in [MeV]. [FA2]: Arndt et al., PRC 69 (24). Γ πn /Γ Tot [%] Γ ηn /Γ Tot [%] N (535) S 48 [33 to 55] 38 [45 to 6] N (65) S 79 [6 to 95] 6 [3 to ] N (44) P 64 [55 to 75] [±] (62) S 3 34 [2 to 3] (9) P 3 [5 to 3] N (72) P 3 3 [ to 2] 38 [4±] N (52) D 3 67 [55 to 65]. [3±.4] (232) P 33 [] (7) D 33 3 [ to 2] Branching ratios into πn and ηn. The values in brackets are from the PDG, [Amsler et al., PLB 667 (28)].

31 Couplings and dressed vertices Residue a vs. dressed vertex Γ vs. bare vertex γ. back Γ ( ) D Σ S D = = = T P = γ ( ) B γ ( ) B G S B + + Γ D γ ( ) B S B Σ Γ D S D Γ ( ) D G S D T NP a = g r Γd Γ( ) d Z Σ = a = (Γ D γ B)/Γ D, r = Σ, Dressed Γ depends on T NP. a Γ γ γ C Γ C r [%] r [%] N (52) D i i 53 6 N (72) P i.9+4.8i (232) P i i 45 4 (62) S i i 3 66 (7) D i 6.7+.i (9) P i.9 3.2i

32 Pole repulsion in P 33 back Re P T NP a + a - /(z-z ) Poles in T NP may occur pole repulsion in T = T NP + T P! Im P T NP a + a - /(z-z ) z [MeV]

33 The D 3 partial wave back The N (52) and a dynamically generated pole in T NP. Re D Im D z [MeV] T NP : no s-channel N (52). Pole in T NP on 3rd ρn sheet. On physical axis visible through branch point b 2. Pole invisible in full solution. We do not identify it with a dynamically generated N (7). [Ramos, Oset, arxiv: [hep-ph]]

34 Observables back g fi und h fi in JLS-Basis: g fi = 2 [ (2j+)d j (θ) k f k i 2 2 j + 2 [ (2j+)d j k f k i 2 2(θ) j τ j(j 2 ) 2 +τ j(j+ 2 ) 2 τ j(j 2 ) 2 τ j(j+ 2 ) 2 ] cos θ 2 ] sin θ 2 h fi = i 2 [ (2j+)d j (θ) k f k i 2 2 j + i 2 [ (2j+)d j k f k i 2 2(θ) j τ j(j 2 ) 2 +τ j(j+ 2 ) 2 τ j(j 2 ) 2 τ j(j+ 2 ) 2 ] sin θ 2 ] cos θ 2

35 Observables back dσ dω = kf ( g fi 2 + h fi 2 ) k i ( = 2ki 2 2 (2j+)(τ j(j 2 ) 2 +τ j(j+ 2 ) 2) d j (Θ) 2 2 j 2) + (2j+)(τ j(j 2 ) 2 τ j(j+ 2 ) 2) d j 2 2(Θ) j 2 P f = 2Re(gfih fi) g fi 2 + h fi 2 ˆn ( 2Im(h ) fi g fi) β = arctan g fi 2 h fi 2

36