Coupled channel dynamics in Λ and Σ production

Transcription

1 Coupled channel dynamics in Λ and Σ production S. Krewald M. Döring, H. Haberzettl, J. Haidenbauer, C. Hanhart, F. Huang, U.-G. Meißner, K. Nakayama, D. Rönchen, Forschungszentrum Jülich, George Washington University, University of Georgia, Universität Bonn EmNN*, Columbia,SC, August 14, 212 1/ 39

2 The model Coupled channel dynamics πn ππn π ηn Nρ σn KΛ KΣ Nω E CM [MeV ] Athens-Bonn-Jülich-Washington collaboration Dynamical coupled-channels model for elastic & inelastic meson-baryon scattering up to and beyond 2 GeV Lagrangian of Wess & Zumino + additional channels(su3) 12 Nucleon and 1 Delta Resonances with total spin up to J = 9/2 Simultaneous fit to πn, ηn, KΣ, and KΛ Used as input for a gauge invariant analysis of pion photoproduction respects unitarity(2-body) and analyticity; Resonance poles and residues EmNN*, Columbia,SC, August 14, 212 2/ 39

3 The scattering equation Coupled channel dynamics T I µν( k, λ, k, λ) = V I µν( k, λ, k, λ) + d 3 qv I µγ( k, λ, q, λ 1 ) Z E γ(q) + iɛ T I γν( q, λ, k, λ) γ,λ EmNN*, Columbia,SC, August 14, 212 3/ 39

4 Analyticity Coupled channel dynamics N*(1535)S11 N*(165)S11 Energy Definition of a resonance: Pole in the complex plane of the scattering energy E ( z W); - Pole position ( mass & width ) - Residues ( branching ratios ) Extractions of resonance parameters in terms of poles and residues Re E Re E Not every bump is a resonance and not every resonance is a bump. Moorhouse 196ties EmNN*, Columbia,SC, August 14, 212 4/ 39

5 πn πn : Partial wave amplitudes I=1/2 (preliminary) Detail P 11:.4 Re S S 11(1535) and S 11(165) Re P Im S 11 Im P dynamical Roper P 11(144) genuine P 11(171) Re P Im P Inclusion of P 11 (171) necessary to improve KΛ Input in the fit: energy-dependent solution (black line) But: Our solution matches single-energy solution (data points) Coupled-channels essential Signal for a N (17XX)P 11 Single-energy solutions are not data Fit to πn observables required EmNN*, Columbia,SC, August 14, 212 5/ 39

6 πn πn : Partial wave amplitudes I=1/2 (preliminary) Detail P 11:.4 Re S S 11(1535) and S 11(165) Re P Im S 11 Im P dynamical Roper P 11(144) genuine P 11(171) Re P Im P Inclusion of P 11 (171) necessary to improve KΛ Input in the fit: energy-dependent solution (black line) But: Our solution matches single-energy solution (data points) Coupled-channels essential Signal for a N (17XX)P 11 Single-energy solutions are not data Fit to πn observables required EmNN*, Columbia,SC, August 14, 212 5/ 39

7 Importance of polarization measurements: The P 33 (192) 12 z = 219 MeV 231 MeV 259 MeV 274 MeV 1 8 dσ/dω [ µb / sr ] MeV 2118 MeV 2147 MeV 2158 MeV z = 219 MeV 231 MeV 259 MeV 274 MeV Orange: Only dσ/dω in fit: no P 33(192) necessary Green: dσ/dω + P in fit: Need for P 33(192)! -.5 P MeV 2118 MeV 2147 MeV 2158 MeV EmNN*, Columbia,SC, August 14, 212 6/ 39

8 πn πn : Partial wave amplitudes I=1/2 (preliminary) Re P Im P Re F Im F Re D Im D 13 Re D 15 Im D Re G 17 Re G Im G 17 Im G Re F 15.8 Im F Re H Im H EmNN*, Columbia,SC, August 14, 212 7/ 39

9 πn πn : Partial wave amplitudes I=3/2 (preliminary) Re S 31.6 Im S 31.5 Re F 35 Im F Re P 31 Re D 33 Re P Im P 31 Im P 33 Im D Re F Re G 37 Im G Re G Im F 37 Im G Re D 35 Im D Re H Im H EmNN*, Columbia,SC, August 14, 212 8/ 39

10 π p ηn : Cross section and Polarization (preliminary) Selected results dσ/dω [ mb/sr ] P x dσ/dω [ mb/sr ] Bayadilov 8, Eur. Phys. J. A. 35, 287 z=1496 MeV MeV MeV MeV 1674 MeV z = 27 MeV 1897 MeV MeV Difficult data situation Inconsistencies among different data sets Data at higher energies questionable Polarization data questionable Full results ηn EmNN*, Columbia,SC, August 14, 212 9/ 39

11 π p K Λ: dσ/dω, Polarization & Spinrotation angle (preliminary) Seclected results dσ/dω [ µb/sr ] P.5-1 z = 1626 MeV z = 1633 MeV Full results KΛ 177 MeV MeV MeV MeV MeV MeV β [rad] β [rad] z = 1852 MeV 216 MeV EmNN*, Columbia,SC, August 14, 212 1/ 39

12 π p K Σ : dσ/dω & Polarization (preliminary) dσ/dω [ µb/sr ] z = 1694 MeV MeV MeV MeV P MeV MeV MeV MeV Full results K Σ EmNN*, Columbia,SC, August 14, / 39

13 π p K + Σ : dσ/dω (preliminary) dσ/dω [ µb/sr ] MeV 1818 MeV z = 225 MeV 245 MeV No polarization data Full results K + Σ EmNN*, Columbia,SC, August 14, / 39

14 π + p K + Σ + (preliminary) Cross section, polarization and spinrotation parameter dσ/dω [ µb/sr ] P z = 1729 MeV z = 1729 MeV Full results K + Σ MeV z = 1822 MeV MeV MeV MeV MeV β [rad] β [rad] z = 221 MeV z = 217 MeV Green dashed line: Jülich model solution from NPA 851, 58 (211) EmNN*, Columbia,SC, August 14, / 39

15 Photoproduction: dσ/dω and Σ γ for γp π + n F. Huang, M. Döring, K. Nakayama et al., Phys. Rev. C85 (212) 543 Differential cross section for γp π + n Photon spin asymmetry for γp π + n Data: CNS Data analysis center [CBELSA/TAPS, JLAB, MAMI,...] EmNN*, Columbia,SC, August 14, / 39

16 Resonance content: I = 1/2 (preliminary) Re(z ), -2Im(z ) r, θ Γ πn /Γ tot (Γ 1/2 πn Γ1/2 ηn )/Γ tot (Γ 1/2 πn Γ1/2 KΛ )/Γ tot (Γ 1/2 πn Γ1/ K N(144)P /2 + **** (a) N(152)D /2 **** N(1535)S /2 **** 66-4 N(165)S /2 **** N(171)P /2 + *** N(1675)D /2 **** N(168)F /2 + **** N(172)P /2 + **** N(199)F /2 + ** N(219)G /2 **** N(222)H /2 + **** N(225)G /2 **** EmNN*, Columbia,SC, August 14, / 39

17 Resonance content: I = 3/2 (preliminary) Re(z ), -2Im(z ) r, θ Γ πn /Γ tot (1232)P (Γ 1/2 πn Γ1/2 KΣ )/Γtot 3/2 + **** (162)S /2 **** (17)D /2 **** (195)F /2 + **** (191)P /2 + **** (192)P /2 + *** (193)D /2 *** (195)F /2 + **** (22X)G /2 * (24)G /2 ** EmNN*, Columbia,SC, August 14, / 39

18 Summary and outlook Coupled channel dynamics Lagrangian based, field theoretical description of meson-baryon interaction Unitarity and analyticity are ensured; branch points in the complex plane included precise, model independent determination of resonance parameters Coupled channel formalism links different reactions in one combined description: πn πn, π + p K + Σ +,..., γn πn,... Extension to kaon photoproduction EmNN*, Columbia,SC, August 14, / 39

19 Matching with lattice Coupled channel dynamics The lattice results depend on a finite volume V and a pion mass >= M exp pion. M lattice pion The dynamical coupled channel approach is based on a Lagrangian and therefore can vary V and M pion. May we hope to interpolate between experiment and the state-of-the-art lattice? EmNN*, Columbia,SC, August 14, / 39

20 Matching with quark dynamics Question: Do the towers of excited baryons predicted by quark models merge into a continuum after coupling to meson-baryon dynamics is considered? Why not a pragmatic approach? Quark model M N couplings N N π, N η, ΛK, ΣK. Match: N H quark N π = ψ N L int ψ N π for q on shell = MN 2 (1 ( m N +m π M N ) 2 ) 1 2 (1 ( m N m π M N ) 2 ) 1 2. Use as input in time-ordered perturbation theory! This might be the first step for a generalization to electroproduction. Analogy: quasielastic bump in nuclear physics = superposition of nuclear modes coupled to continuum. EmNN*, Columbia,SC, August 14, / 39

21 π p K Λ: Total cross section (preliminary) σ [mb] EmNN*, Columbia,SC, August 14, 212 2/ 39

22 π p K Λ: Total cross section (preliminary) σ [mb] Partial wave content: P 11 S 11 P 13 σ [mb].1 D 15 H 19 D 13.1 F 17 G 17 G 19 F EmNN*, Columbia,SC, August 14, 212 2/ 39

23 π p K Σ : Total cross section (preliminary).25.2 σ [mb] EmNN*, Columbia,SC, August 14, / 39

24 π p K Σ : Total cross section (preliminary).25 σ [mb] Partial wave content I = 1/2: S 11.1 σ [mb].1 D 15 F 17 P 13 D 13 P 11.1 F 15 G 17 1e H 19 G 19 EmNN*, Columbia,SC, August 14, / 39

25 π p K Σ : Total cross section (preliminary).25 σ [mb] Partial wave content I = 3/2: P 33.1 S 31 G 39 σ [mb].1 D 35 F 37 D 33 P 31 F 35.1 G 37 1e EmNN*, Columbia,SC, August 14, / 39

26 π p K + Σ : Total cross section (preliminary).25.2 σ [mb] EmNN*, Columbia,SC, August 14, / 39

27 π p K + Σ : Total cross section (preliminary).25.2 σ [mb].15.1 Partial wave content I = 1/2: S 11.1 D 15 F 17 σ [mb].1 P D P 11 F 15.1 G 17 H 19 G 19 1e EmNN*, Columbia,SC, August 14, / 39

28 π p K + Σ : Total cross section (preliminary).25.2 σ [mb].15.1 Partial wave content I = 3/2: P 33.1 S 31 F 37 σ [mb].1 D 33 D 35 P 31.1 G 39 F 35 G 37 1e EmNN*, Columbia,SC, August 14, / 39

29 π + p K + Σ + : Total cross section (preliminary) σ [mb] EmNN*, Columbia,SC, August 14, / 39

30 π + p K + Σ + : Total cross section (preliminary) σ [mb] Partial wave content: S 31 P 33 F 37.1 D 33 G 39 σ [mb].1 D 35 P 31 F 35.1 G 37 1e EmNN*, Columbia,SC, August 14, / 39

31 π p ηn : Total cross section (preliminary) σ [mb] EmNN*, Columbia,SC, August 14, / 39

32 π p ηn : Total cross section (preliminary) 3.5 σ [mb] Partial wave content: S P11 11 D P σ [mb].1.1 F 17 D 13 G17.1 F 15 1e-5 G 19 H EmNN*, Columbia,SC, August 14, / 39

33 Photoproduction: Coupled channels and gauge invariance Haberzettl, PRC56 (1997), Haberzettl, Nakayama, Krewald, PRC74 (26) Gauge invariance: Generalized Ward-Takahashi identity (WTI) k µm µ = F sτ S p+k Q i S 1 p + S 1 p Q f S p k F uτ + 1 p p +k Q pi p p F t τ Photoproduction amplitude: M µ = M µ s + M µ u + M µ t }{{} coupling to external legs + M µ int }{{} coupling inside hadronic vertex Strategy: Replace generalized WTI is satisfied by phenomenological contact term such that the EmNN*, Columbia,SC, August 14, / 39

34 Photoproduction: dσ/dω and Σ γ for γn π p F. Huang, M. Döring, K. Nakayama et al., Phys. Rev. C85 (212) 543 Differential cross section for γn π p Photon spin asymmetry for γn π p Data: CNS Data analysis center [CBELSA/TAPS, JLAB, MAMI,...] EmNN*, Columbia,SC, August 14, / 39

35 Photoproduction: dσ/dω and Σ γ for γp π p F. Huang, M. Döring., K. Nakayama et al., Phys. Rev. C85 (212) 543 Differential cross section for γp π p Photon spin asymmetry for γp π p Data: CNS Data analysis center [CBELSA/TAPS, JLAB, MAMI,...] EmNN*, Columbia,SC, August 14, / 39

36 Photoproduction: dσ/dω and Σ γ for γp π + n F. Huang, M. Döring, K. Nakayama et al., Phys. Rev. C85 (212) 543 Differential cross section for γp π + n Photon spin asymmetry for γp π + n Data: CNS Data analysis center [CBELSA/TAPS, JLAB, MAMI,...] EmNN*, Columbia,SC, August 14, / 39

37 /appendix Coupled channel dynamics EmNN*, Columbia,SC, August 14, / 39

38 Error estimates for masses: (195)F 35 Table: Error estimates of bare mass m b and bare coupling f for the (195)F 35 resonance. m b [MeV] πn ρn π ΣK EmNN*, Columbia,SC, August 14, / 39

39 Motivation: Baryon spectrum, πn scattering Higher energies: more states are predicted than seen in elastic πn scattering ( misssing resonance problem ) coupling to other channels like multi-pion and KY Predicted resonances from recent lattice calculations [Edwards et al., Phys.Rev. D84 (211)]: m π = 396MeV EmNN*, Columbia,SC, August 14, 212 3/ 39

40 The scattering equation Coupled channel dynamics T I µν( k, λ, k, λ) = V I µν( k, λ, k, λ) + d 3 qv I µγ( k, λ, q, λ 1 ) Z E γ(q) + iɛ T I γν( q, λ, k, λ) γ,λ EmNN*, Columbia,SC, August 14, / 39

41 The scattering equation Coupled channel dynamics T I µν ( k, λ, k, λ) = V I µν ( k, λ, k, λ) + d 3 qv I µγ ( k, λ, q, λ 1 ) Z E γ(q) + iɛ T I γν ( q, λ, k, λ) γ,λ Features: Coupled channels πn, ηn, KΛ, KΣ, ππn [π, σn, ρn] Hadron exchange: relevant degrees of freedom in 2nd and 3rd resonance region t- and u-channel processes: background, all channels are linked s-channel processes: genuine resonances (only a minimum) Channels/reactions linked (SU(3) symmetry in Lagrangian framework) No on-shell factorization, full analyticity (dispersive parts) EmNN*, Columbia,SC, August 14, / 39

42 s-, t- and u-channel exchanges s-channel states coupling to πn, ηn, KΛ, KΣ, π, ρn. t- and u-channel exchanges: πn ρn ηn πδ σn KΛ KΣ πn N,Δ,(ππ) σ, N, Δ, Ct., N, a N, Δ, ρ N, π Σ, Σ*, K* Λ, Σ, Σ*, (ππ) ρ π, ω, a 1 K* ρn N, Δ, Ct., ρ - N, π ηn N, f - - K*, Λ Σ, Σ*, K* πδ N, Δ, ρ π - - σn N, σ - - KΛ Ξ, Ξ*, f, Ξ, Ξ*, ρ ω, φ KΣ Ξ, Ξ*, f, ω, φ, ρ EmNN*, Columbia,SC, August 14, / 39

43 ππn states: π, σn, ρn Coupled channel dynamics π π J,M l,s,m Δ π Δ ππ/πn subsystems fit the respective phase shifts. σ, ρ π Towards a consistent inclusion of 3-body cuts. l,s,m J,M N σ, ρ π σ, ρ π π Consistent 3-body cuts N Allow for a-priori 3-body unitarity per construction [Aaron, Almado, Young, PR 174 (1968) 222]. EmNN*, Columbia,SC, August 14, / 39

44 Crossing symmetry at the level of the potential (not the amplitude) N N t (NN ππ) π π σ, ρ π π s (πn πn) π π For σ(6) and ρ(77) quantum numbers: πn t-channel interaction from N N ππ (analytically continued) data. Use of crossing symmetry and dispersion techniques [Schütz et al. PRC 49 (1994) 2671]. EmNN*, Columbia,SC, August 14, / 39

45 Scattering equation: partial wave decomposition Expand T ( k, λ, k, λ) in terms of the eigenstates of the total angular momentum J: λ k T λ k = 1 (2J + 1)D J λλ 4π (Ω k k, ) λ k T J λ k J Scattering equation: 1 4π J + 1 4π (2J + 1)D J λλ (Ω k k, ) λ 3λ 4 k T J λ 1λ 2 k = λ3λ 4 k V J λ 1λ 2 k + γ 1,γ 2 J,J dqdω qk q 2 (2J + 1)(2J + 1)D J γλ (Ω k q, ) D J λγ (Ω qk, ) λ 3λ 4 k V J γ 1γ 2 q G(q) γ 1γ 2 q T J λ 1λ 1 k dω qk D J γλ (Ω k q, ) D J λγ (Ω qk, ) = D J λλ (Ω k k, 4π ) 2J + 1 δ J J EmNN*, Columbia,SC, August 14, / 39

46 Scattering equation: partial wave decomposition Expand T ( k, λ, k, λ) in terms of the eigenstates of the total angular momentum J: λ k T λ k = 1 (2J + 1)D J λλ 4π (Ω k k, ) λ k T J λ k J Scattering equation: 1 4π J + 1 4π (2J + 1)D J λλ (Ω k k, ) λ 3λ 4 k T J λ 1λ 2 k = λ3λ 4 k V J λ 1λ 2 k + γ 1,γ 2 J,J dqdω qk q 2 (2J + 1)(2J + 1)D J γλ (Ω k q, ) D J λγ (Ω qk, ) λ 3λ 4 k V J γ 1γ 2 q G(q) γ 1γ 2 q T J λ 1λ 1 k dω qk D J γλ (Ω k q, ) D J λγ (Ω qk, ) = D J λλ (Ω k k, 4π ) 2J + 1 δ J J EmNN*, Columbia,SC, August 14, / 39

47 Scattering equation: partial wave decomposition Expand T ( k, λ, k, λ) in terms of the eigenstates of the total angular momentum J: λ k T λ k = 1 (2J + 1)D J λλ 4π (Ω k k, ) λ k T J λ k J 1 4π J + 1 4π (2J + 1)D J λλ (Ω k k, ) λ 3λ 4 k T J λ 1λ 2 k = λ3λ 4 k V J λ 1λ 2 k + γ 1,γ 2 J,J dqdω qk q 2 (2J + 1)(2J + 1)D J γλ (Ω k q, ) D J λγ (Ω qk, ) λ 3λ 4 k V J γ 1γ 2 q G(q) γ 1γ 2 q T J λ 1λ 1 k Scattering equation: λ 3λ 4 k T J λ 1λ 2 k = λ3λ 4 k V J λ 1λ 2 k + γ 1,γ 2 dqq 2 λ 3λ 4 k V J γ 1γ 2 q G(q) γ 1γ 2 q T J λ 1λ 1 k. 1-dim. integral equation sizable reduction of numerical effort EmNN*, Columbia,SC, August 14, / 39

48 Scattering equation in JLS basis experimental data (partial wave analyses) usually in JLS basis switch from helicity to JLS basis: JMλ 1 λ 2 k = JMLS JMλ 1 λ 2 JMLS LS J: total angular momentum M: z-projection of J L: orbital angular momentum S: total spin λ i: helicity, λ := λ 1 λ 2 JMLS JMλ 1 λ 2 = L S k V J LSk = Scattering equation in JLS basis: ( ) 1 2L LSλ Jλ S1 λ 1 S 2 λ 2 Sλ 2J + 1 }{{} Clebsch-Gordan coefficients λ 1 λ 2 λ 3 λ 4 JML S JMλ 3 λ 4 λ 3 λ 4 k V J λ 1 λ 2 k JMλ 1 λ 2 JMLS L S k T IJ µν LSk = L S k V IJ µν LSk + γ,l S q 2 dq L S k V IJ µγ L S q G(q) L S q T IJ γν LSk EmNN*, Columbia,SC, August 14, / 39

49 Scattering equation in JLS basis experimental data (partial wave analyses) usually in JLS basis switch from helicity to JLS basis: JMλ 1 λ 2 k = JMLS JMλ 1 λ 2 JMLS LS J: total angular momentum M: z-projection of J L: orbital angular momentum S: total spin λ i: helicity, λ := λ 1 λ 2 JMLS JMλ 1 λ 2 = L S k V J LSk = Scattering equation in JLS basis: ( ) 1 2L LSλ Jλ S1 λ 1 S 2 λ 2 Sλ 2J + 1 }{{} Clebsch-Gordan coefficients λ 1 λ 2 λ 3 λ 4 JML S JMλ 3 λ 4 λ 3 λ 4 k V J λ 1 λ 2 k JMλ 1 λ 2 JMLS L S k T IJ µν LSk = L S k V IJ µν LSk + γ,l S q 2 dq L S k V IJ µγ L S q G(q) L S q T IJ γν LSk EmNN*, Columbia,SC, August 14, / 39

50 Scattering equation in JLS basis experimental data (partial wave analyses) usually in JLS basis switch from helicity to JLS basis: JMλ 1 λ 2 k = JMLS JMλ 1 λ 2 JMLS LS J: total angular momentum M: z-projection of J L: orbital angular momentum S: total spin λ i: helicity, λ := λ 1 λ 2 JMLS JMλ 1 λ 2 = L S k V J LSk = Scattering equation in JLS basis: ( ) 1 2L LSλ Jλ S1 λ 1 S 2 λ 2 Sλ 2J + 1 }{{} Clebsch-Gordan coefficients λ 1 λ 2 λ 3 λ 4 JML S JMλ 3 λ 4 λ 3 λ 4 k V J λ 1 λ 2 k JMλ 1 λ 2 JMLS L S k T IJ µν LSk = L S k V IJ µν LSk + γ,l S q 2 dq L S k V IJ µγ L S q G(q) L S q T IJ γν LSk EmNN*, Columbia,SC, August 14, / 39

51 Analytic structure of the scattering amplitude Analytic properties of the amplitude important information: cuts, poles and zeros on different Riemann sheets determine global behaviour of the amplitude on the physical axis parameterization of resonances in a well defined way poles and residues: relevant quantities for comparison of different experiments Jülich model: derived within a field theoretical approach analyticity is respected reliable extraction of resonance properties Extraction of resonance parameters, pole search on 2nd sheet: Analytic continuation EmNN*, Columbia,SC, August 14, / 39

52 Analytic continuation via Contour deformation...enables access to all Riemann sheets [Nucl.Phys.A829:17-29,29] 1 Propagator for a particle with width: G σ(z, k) = z k 2 + (mσ )2 Π σ(z, k) Example: Selfenergy righthand cut along positive real axis, starting at z thresh = 2m π Π σ(z) = q 2 (v σππ (q, k)) 2 analytic continuation along the cut to the 2nd sheet: dq z 2 q 2 + mπ 2 + iɛ Π (2) σ = Π (1) σ 2ImΠ(1) σ = Π (1) σ + 2πiqonE(1) E(2) on v 2 (q on, k) z 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 1 + m 2 (= threshold). EmNN*, Columbia,SC, August 14, / 39

53 Analytic continuation via Contour deformation...enables access to all Riemann sheets [Nucl.Phys.A829:17-29,29] 1 Propagator for a particle with width: G σ(z, k) = z k 2 + (mσ )2 Π σ(z, k) Example: Selfenergy righthand cut along positive real axis, starting at z thresh = 2m π Π σ(z) = q 2 (v σππ (q, k)) 2 analytic continuation along the cut to the 2nd sheet: dq z 2 q 2 + mπ 2 + iɛ Π (2) σ = Π (1) σ 2ImΠ(1) σ = Π (1) σ + 2πiqonE(1) E(2) on v 2 (q on, k) z Scattering equation on the 2nd sheet: with q cd T (2) V q ab = δg + dq mnq 2 q cd V q mn q mn T (2) q ab mn z E mn + iɛ δg = 2πiqonE(1) E(2) on q cd V q on mn z qon mn T (2) q ab EmNN*, Columbia,SC, August 14, / 39

54 Effective ππn channels: Analytic structure new structure, induced by addititonal branch points π σ N σ π = dk k 2 1 z m 2 N + k2 (mσ )2 + k 2 Π σ(z σ(z, k), k) }{{} b 1 }{{} b 2,b 2 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 (b 2, b 2 ) in the σn propagator at z b2 = m N + z, z b 2 = m N + z 3 branch points and 4 sheets for each of the σn, ρn, and π propagators. EmNN*, Columbia,SC, August 14, / 39