Pion photoproduction in a gauge invariant approach

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1 Pion photoproduction in a gauge invariant approach F. Huang, K. Nakayama (UGA) M. Döring, C. Hanhart, J. Haidenbauer, S. Krewald (FZ-Jülich) Ulf-G. Meißner (FZ-Jülich & Bonn) H. Haberzettl (GWU) Jun., 2 F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 / 9

2 Introduction Study of N is important for understanding of strong interactions A clear understanding of N structure, spectrum and decay will reveal the role of confinement and chiral symmetry in QCD non-perturbative region. Dynamical coupled-channel models are needed N states are unstable and couple strongly with meson-baryon continuum states. In order to extract N parameters and understand N structures, dynamical coupled-channel models are needed to analyze the meson production data. Gauge invariance is important for photo-production Gauge invariance is a fundamental symmetry. Without it results become arbitrary. Note current conservation is necessary but not sufficient for gauge invariance. This work: gauge invariant approach for π photo-production in Jülich dynamical coupled-channel model F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 2 / 9

3 Hadronic scattering Γ = F + X G F S = S + S F G Γ S 8 8 X = U + U G X 7 T = Γ S Γ + X F : bare vertex S : bare propagator X : non-polar part of T matrix T : full T matrix Γ : dressed vertex S : dressed propagator U : driving term of X F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 3 / 9

4 Jülich πn meson-exchange model Dynamical coupled-channel model: Nπ, Nη, π, Nρ, Nσ, ΛK, ΣK Re S3 Re P Re S Re P3 Im P Im S Im S3 Im P Re P Re P3 Re D3 Re D Im P W [MeV] Im P W [MeV] Im D W [MeV] Im D W [MeV] F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 4 / 9

5 Photo-production amplitude H. Haberzettl, PRC6(997)24 F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 / 9

6 Photo-production amplitude H. Haberzettl, PRC6(997)24 γ M = M µ = M µ s + M µ u + M µ t + M µ int F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 / 9

7 Photo-production amplitude H. Haberzettl, PRC6(997)24 γ M = M µ = M µ s + M µ u + M µ t + M µ int - = + U + X U = U F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 / 9

8 Gauge invariance Without gauge invariance results become arbitrary Current conservation k µ M µ = necessary but not sufficient F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 6 / 9

9 Gauge invariance Without gauge invariance results become arbitrary Current conservation k µ M µ = necessary but not sufficient Electromagnetic couplings to driving terms 8 8 F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 6 / 9

10 Gauge invariance Without gauge invariance results become arbitrary Current conservation k µ M µ = necessary but not sufficient Electromagnetic couplings to driving terms 8 8 Amplitude results from a highly non-linear equation F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 6 / 9

11 Gauge invariance Without gauge invariance results become arbitrary Current conservation k µ M µ = necessary but not sufficient Electromagnetic couplings to driving terms 8 8 Amplitude results from a highly non-linear equation Inner consistency requires generalized Ward-Takahashi identity k µ M µ = Γ s τ S p+k Q i S p + S p Q f S p k Γ u τ + p p +k Q π p p Γ t τ Q: charge operator S: nucleon propagator : pion propagator Γ : dressed vertex F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 6 / 9

12 Practical strategy = + U + X U } {{ } M µ a } {{ } M µ a Gauge invariance condition: k µ M µ = Γ s τ S p+k Q i S p M µ int = M µ a + X G (M µ t + M µ u + M µ a ) + S p Q f S p k Γ u τ + p p +k Q π p p Γ t τ k µ M µ a = ( U G)( Γ s e i + Γ u e f + Γ t e π ) k µ U G (M µ tl + Mµ ul ) Approximating M a µ under gauge invariance condition: M a µ = ( U G) M c µ U G (M µ tl + Mµ ul ) + Tµ Interaction current: k µ M µ c = Γ s e i + Γ u e f + Γ t e π, k µ T µ = M µ int = M µ c + T µ + X G [(M µ ut + Mµ tt ) + Tµ ] F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 7 / 9

13 Choosing the contact current M µ c Constraints: gauge invariance contact term crossing symmetry Introduce an auxiliary current C µ : C µ (2q k) µ ( ) (2p = e π t q 2 f t ˆF k) µ ( ) (2p + k) e f u p 2 f u ˆF µ ( ) e i s p 2 f s ˆF ˆF = ĥ( δ s f s ) ( δ u f u )( δ t f t ) k, p, q, p : 4-momenta for incoming γ, N & outgoing π, N ĥ: fit parameter The contact current M c µ can be written as {[ ] M c µ = g q/ βk/ πγ λ + ( λ) m C µ γ µ } ( λ) + m m + m [e πf t βk ρ C ρ ] β: fit parameter Check gauge invariance: M µ c satisfies k µ M µ c = Γ s e i + Γ u e f + Γ t e π F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 8 / 9

14 Reaction theory: all together γ M = Full photo-production current: M µ = M µ s + Mµ u + Mµ t + M µ int M µ int = M µ c + Tµ + X G [(M µ ut + Mµ tt ) + Tµ ] M µ satisfies the generalized Ward-Takahashi identity (gauge invariant) k µ M µ = Γ s τ S p+k Q i S p + S p Q f S p k Γ u τ + p p +k Q π p p Γ t τ T µ : undetermined transverse contact current (set to be zero in this work) X: non-polar part of hadronic scattering (Jülich coupled-channel πn model) F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 9 / 9

15 Results: dσ/dω for γ p π + n + n) ( b/sr) d /d ( p (2, 62) 6 2 Jülich-Georgia (28, 86) (32, 27) (4, 277) (4, 33) (, 349) (, 383) (6, 46) (6, 449) (72, 497) (77, 28) (82, 8) (87, 88) (92, 67) (97, 646) (deg.) (36, 247) dσ/dω (mb/sr) EBAC W=4 MeV 3 W=62 MeV 3 W=78 MeV 3 W=86 MeV W=29 MeV 3 W=27 MeV 3 W=232 MeV 3 W=263 MeV W=28 MeV W=299 MeV W=47 MeV W=48 MeV W=63 MeV W=632 MeV W=496 MeV W=3 MeV W=44 MeV W=74 MeV θ (deg.) F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 / 9

16 Results: Σ γ for γ p π + n + ( p n) 6 2 Jülich-Georgia.6 (244, 7) (27, 82) (322, 29) (3, 24) (4, 277) (4, 33) (, 349) (, 383) (6, 46) (6, 449) (688, 474) (7, 3) (8, 43) (848, 72) (9, 63) (92, 633) (deg.) Σ EBAC W=4 MeV W=62 MeV W=78 MeV W=86 MeV W=29 MeV W=27 MeV W=232 MeV W=263 MeV W=28 MeV W=299 MeV W=47 MeV W=48 MeV W=496 MeV W=3 MeV W=44 MeV W=74 MeV W=63 MeV W=632 MeV θ (deg.) F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 / 9

17 Results: dσ/dω for γ p π p p) ( b/sr) d /d ( p (24, 8) 6 2 Jülich-Georgia (278, 84) (32, 27) (4, 277) (4, 33) (, 36) (, 386) (97, 44) (62, 4) (7, 484) (78, 7) (88, 48) (89, 79) (99, 68) (97, 636) (deg.) (36, 247) dσ/dω (mb/sr) EBAC W=4 MeV 3 W=62 MeV 3 W=78 MeV 3 W=86 MeV W=29 MeV W=27 MeV W=232 MeV W=263 MeV W=28 MeV W=299 MeV W=47 MeV W=48 MeV W=63 MeV W=632 MeV W=496 MeV W=3 MeV W=44 MeV W=74 MeV θ (deg.) F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 2 / 9

18 Results: Σ γ for γ p π p ( p p) 6 2 Jülich-Georgia.6 (244, 7) (27, 82) (322, 29) (3, 24) (4, 277) (4, 33) (, 349) (, 384) (6, 426) (666, 49) (73, 483) (769, 24) (8, 44) (862, 8) (89, 6) (96, 638) (deg.) Σ EBAC W=4 MeV W=62 MeV W=78 MeV W=86 MeV W=29 MeV W=27 MeV W=232 MeV W=263 MeV W=28 MeV W=299 MeV W=47 MeV W=48 MeV W=496 MeV W=3 MeV W=44 MeV W=74 MeV W=63 MeV W=632 MeV θ (deg.) F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 3 / 9

19 - Results: dσ/dω & Σ γ for γ n π p - p) ( b/sr) d /d ( n (24, ) 6 2 Jülich-Georgia (27, 79) (33, 23) (39, 27) (436, 3) (, 3) (6, 48) (64, 444) 6 2 (36, 249) (4, 378) (7, 483) (748, 3) (8, 4) (8, 7) (9, 64) (9, 633) (deg.) Differential cross section for γ n π p p) ( n (2, 63) 6 2 Jülich-Georgia (3, 23) (33, 226) (4, 278) (4, 3) (, 3) (63, 438) (67, 463) (7, 489) (7, 4) (79, 39) (89, 7) (8, 7) (9, 64) (947, 632) (deg.) (3, 24) Photon spin asymmetry for γ n π p F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 4 / 9

20 Results: γ N πn total cross sections (mb) p n p p Full results Contact term off n p data not included in the fit W (GeV) F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 / 9

21 Resonance & background Splitting T = T P + T NP arbitrary and highly model-dependent e.g. Jülich model and EBAC model both get the data but they have quite different T NP Identifying T NP as T BG problematic T = T P + T NP T = a + a + O(z z ) z z a = a P + TNP T = T R + T BG a T R = z z T BG = T T R Pole position & residue are less model-dependent Re P 33 Im P T P +T NP T NP T NP T P +T NP z [MeV] T R & T BG more meaningful than T P & T NP to compare in various models F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 6 / 9

22 Resonance & background Splitting T = T P + T NP arbitrary and highly model-dependent e.g. Jülich model and EBAC model both get the data but they have quite different T NP Identifying T NP as T BG problematic T = T P + T NP T = a + a + O(z z ) z z a = a P + TNP T = T R + T BG a T R = z z T BG = T T R Pole position & residue are less model-dependent Re P 33 Im P T NP + a - /(z-z ) T NP T NP T NP + a - /(z-z ) z [MeV] T R & T BG more meaningful than T P & T NP to compare in various models F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 6 / 9

23 Resonance & background Splitting T = T P + T NP arbitrary and highly model-dependent e.g. Jülich model and EBAC model both get the data but they have quite different T NP Identifying T NP as T BG problematic T = T P + T NP T = a + a + O(z z ) z z a = a P + TNP T = T R + T BG a T R = z z T BG = T T R Pole position & residue are less model-dependent Re P 33 Im P T NP T NP a + a - /(z-z ) a + a - /(z-z ) z [MeV] T R & T BG more meaningful than T P & T NP to compare in various models F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 6 / 9

24 Poles & residues Table: Effective electromagnetic couplings g γ in helicity basis (PRELIMINARY). The st & 2nd lines for isospin /2 resonance correspond to results for p & n, respectively. Pole position [MeV] g γ /2 [MeV/2 ] g γ 3/2 [MeV/2 ] P 33 (232) 27 4 i.42. i i P (44) i.7.26 i i D 3 (2) 3 47 i i i i i S (3) 2 64 i i i S 3 (62) i.43.9 i S (6) i i i D 33 (7) i i i P 3 (72) 66 i..4 i.4 +. i i i P 3 (9) 833 i i g γ = a γ /gπ g π = a π F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 7 / 9

25 Summary & perspectives Reaction theory is based on Jülich dynamical coupled-channel model which describes πn scattering successfully Full photoproduction amplitudes satisfy the generalized Ward-Takahashi identity and thus are gauge invariant Both the differential cross sections and photon spin asymmetries for π photoproduction are described quite well up to s =.6 GeV Effective electromagnetic couplings (preliminary) are extracted by analytic continuation of the full amplitudes to un-physical Riemann sheet Resonance and background contributions will be studied Future plan: η- and K-photoproduction Electro-production F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 8 / 9

26 Importance of gauge invariance: p p p p γ None of the existing models can describe high-precision KVI data for coplanar geometries involving small p scattering angles. Lines: Martinus, Scholten, Tjon, PRC 8, 686 (998) PRC 6, 294 (997) Herrmann, Nakayama, Scholten, Arellano, NPA 82, 68 (99) d / [ b/rad sr 2 ] =8 o =6 o =2 o Mart.I Mart.II Nak d / [ b/rad sr 2 ] =8 o, =6 o =8 o, =9 o =6 o, =9 o.. A y. A y [deg] [deg] F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 9 / 9

27 Importance of gauge invariance: p p p p γ None of the existing models can describe high-precision KVI data for coplanar geometries involving small p scattering angles. Lines: Martinus, Scholten, Tjon, PRC 8, 686 (998) PRC 6, 294 (997) Herrmann, Nakayama, Scholten, Arellano, NPA 82, 68 (99) d / [ b/rad sr 2 ] =8 o =6 o =2 o Mart.I Mart.II Nak d / [ b/rad sr 2 ] =8 o, =6 o =8 o, =9 o =6 o, =9 o.. A y. A y [deg] [deg] Construct the contact current full amplitude obeys WTI (gauge invariant) [K. Nakayama & H. Haberzettl, PRC 8, (29)] d / [ b/rad sr 2 ] =8 o =6 o =2 o d / [ b/rad sr 2 ] =8 o, =6 o =8 o, =9 o =6 o, =9 o.. A y. A y [deg] (c) [deg] (d) F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 9 / 9

28 Importance of gauge invariance: p p p p γ None of the existing models can describe high-precision KVI data for coplanar geometries involving small p scattering angles. Lines: Martinus, Scholten, Tjon, PRC 8, 686 (998) PRC 6, 294 (997) Herrmann, Nakayama, Scholten, Arellano, NPA 82, 68 (99) d / [ b/rad sr 2 ] =8 o =6 o =2 o Mart.I Mart.II Nak d / [ b/rad sr 2 ] =8 o, =6 o =8 o, =9 o =6 o, =9 o.. A y. A y [deg] [deg] Construct the contact current full amplitude obeys WTI (gauge invariant) [K. Nakayama & H. Haberzettl, PRC 8, (29)] d / [ b/rad sr 2 ] =8 o =6 o =2 o d / [ b/rad sr 2 ] =8 o, =6 o =8 o, =9 o =6 o, =9 o.. A y. A y [deg] (c) [deg] It is important to properly take into account the interaction current for NN bremsstrahlung reaction! F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 9 / 9 (d)

Meson-baryon interaction in the meson exchange picture

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