Pion photoproduction in a gauge invariant approach

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

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

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

Jülich πn meson-exchange model Dynamical coupled-channel model: Nπ, Nη, π, Nρ, Nσ, ΛK, ΣK 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8.4.2 Re S3 Re P33.4.2 -.2 -.4.8 Re S Re P3 Im P33 -.2 -.4.8.6.6.4.2.4 Im S Im S3 Im P3.4.2.4.2.2 -.2 -.4 Re P Re P3 Re D3 Re D33 -.2 -.4.8.6.4.2 Im P 2 4 6 8 W [MeV] Im P3 2 4 6 8 W [MeV] Im D3 2 4 6 8 W [MeV] Im D33 2 4 6 8 W [MeV].8.6.4.2 F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 4 / 9

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

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

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

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

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

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

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

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

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

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

Results: dσ/dω for γ p π + n + n) ( b/sr) d /d ( p 36 27 8 9 24 6 8 24 6 8 8 2 6 (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) 6 2 6 2 (deg.) (36, 247) 6 2 8 dσ/dω (mb/sr) 3 3 2 2 3 3 2 2 EBAC 3 3 3 W=4 MeV 3 W=62 MeV 3 W=78 MeV 3 W=86 MeV 2 2 2 2 2 2 6 2 8 6 2 8 6 2 8 6 2 8 3 3 3 W=29 MeV 3 W=27 MeV 3 W=232 MeV 3 W=263 MeV 2 2 2 2 2 2 6 2 8 6 2 8 6 2 8 6 2 8 W=28 MeV W=299 MeV W=47 MeV W=48 MeV 6 2 8 6 2 8 W=63 MeV W=632 MeV 6 2 8 6 2 8 6 2 8 6 2 8 W=496 MeV W=3 MeV W=44 MeV W=74 MeV 6 2 8 6 2 8 θ (deg.) 6 2 8 6 2 8 F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 / 9

Results: Σ γ for γ p π + n + ( p n) 6 2 Jülich-Georgia.6 (244, 7) (27, 82) (322, 29) (3, 24).8. -.8.6 (4, 277) (4, 33) (, 349) (, 383).8. -.8.6 (6, 46) (6, 449) (688, 474) (7, 3).8. -.8.6 (8, 43) (848, 72) (9, 63) (92, 633).8. -.8 6 2 6 2 (deg.) 6 2 8 Σ.... -. -. -. -. EBAC W=4 MeV W=62 MeV W=78 MeV W=86 MeV.. 6 2 8 6 2 8 6 2 8 6 2 8 W=29 MeV W=27 MeV W=232 MeV W=263 MeV.. 6 2 8 6 2 8 6 2 8 6 2 8 W=28 MeV W=299 MeV W=47 MeV W=48 MeV. 6 2 8 6 2 8 6 2 8 6 2 8 W=496 MeV W=3 MeV W=44 MeV W=74 MeV... 6 2 8 -. - 6 2 8 6 2 8 W=63 MeV W=632 MeV. -. - -. 6 2 8 θ (deg.) - 6 2 8... -. - 6 2 8 F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 / 9

Results: dσ/dω for γ p π p p) ( b/sr) d /d ( p 2 39 26 3 2 4 7 6 4 2 6 4 2 (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) 6 2 6 2 (deg.) (36, 247) 6 2 8 dσ/dω (mb/sr) 3 2 4 3 2 2 EBAC W=4 MeV 3 W=62 MeV 3 W=78 MeV 3 W=86 MeV 2 2 6 2 8 6 2 8 6 2 8 6 2 8 4 4 4 W=29 MeV W=27 MeV W=232 MeV W=263 MeV 3 3 3 2 6 2 8 6 2 8 6 2 8 6 2 8 W=28 MeV W=299 MeV W=47 MeV W=48 MeV 6 2 8 6 2 8 W=63 MeV W=632 MeV 6 2 8 6 2 8 6 2 8 6 2 8 W=496 MeV W=3 MeV W=44 MeV W=74 MeV 6 2 8 6 2 8 θ (deg.) 6 2 8 2 2 6 2 8 F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 2 / 9

Results: Σ γ for γ p π p ( p p) 6 2 Jülich-Georgia.6 (244, 7) (27, 82) (322, 29) (3, 24).8. -.8.6 (4, 277) (4, 33) (, 349) (, 384).8. -.8.6 (6, 426) (666, 49) (73, 483) (769, 24).8. -.8.6 (8, 44) (862, 8) (89, 6) (96, 638).8. -.8 6 2 6 2 (deg.) 6 2 8 Σ. -.. -.. -.. -.. -.. EBAC W=4 MeV W=62 MeV W=78 MeV W=86 MeV. -. -. -. 8 6 2 8 6 2 8 6 2 6 2 8 W=29 MeV W=27 MeV W=232 MeV W=263 MeV.. -. -. -. 8 6 2 8 6 2 8 6 2 6 2 8 W=28 MeV W=299 MeV W=47 MeV W=48 MeV.. -. -. -. 8 6 2 8 6 2 8 6 2 6 2 8 W=496 MeV W=3 MeV W=44 MeV W=74 MeV. -. -. 6 2 8 6 2 8 W=63 MeV W=632 MeV. -. 6 2 8. 6 2 8 θ (deg.)... -. 6 2 8. 6 2 8 F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 3 / 9

- Results: dσ/dω & Σ γ for γ n π p - p) ( b/sr) d /d ( n 48 36 24 2 27 8 9 27 8 9 8 2 6 (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) 6 2 6 2 8 (deg.) Differential cross section for γ n π p p) ( n.6.8. -.8.6.8. -.8.6.8. -.8.6.8. -.8 (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) 6 2 6 2 (deg.) (3, 24) 6 2 8 Photon spin asymmetry for γ n π p F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 4 / 9

Results: γ N πn total cross sections (mb).4.3.2...3.2...3.2.. p n p p Full results - - - - Contact term off n p data not included in the fit..2.3.4..6 W (GeV) F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 / 9

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 33.6.4.2 -.2 -.4.8.6.4.2 T P +T NP T NP T NP T P +T NP 2 3 4 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

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 33.6.4.2 -.2 -.4.8.6.4.2 T NP + a - /(z-z ) T NP T NP T NP + a - /(z-z ) 2 3 4 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

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 33.6.4.2 -.2 -.4.8.6.4.2 T NP T NP a + a - /(z-z ) a + a - /(z-z ) 2 3 4 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

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 4.9 +.3 i P (44) 38 72 i.7.26 i.92 +.8 i D 3 (2) 3 47 i 3. + 2.4 i 2.8.34 i.99.84 i 4. +.68 i S (3) 2 64 i.2 +.87 i 4. 2. i S 3 (62) 92 37 i.43.9 i S (6) 666 7 i 3.3 + 2.67 i.22 2. i D 33 (7) 638 22 i.37 8.98 i 3.9 +.8 i P 3 (72) 66 i..4 i.4 +. i. + 2.7 i.28 3.23 i P 3 (9) 833 i 9.29 3. i g γ = a γ /gπ g π = a π F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 7 / 9

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

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 ] 2. 2..... =8 o =6 o =2 o Mart.I Mart.II Nak d / [ b/rad sr 2 ] 3. 2... =8 o, =6 o =8 o, =9 o =6 o, =9 o.. A y. A y. -. -. -. 6 9 2 6 9 2 [deg] 6 9 2 8 -. 6 9 2 6 9 2 [deg] 6 9 2 8 F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 9 / 9

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 ] 2. 2..... =8 o =6 o =2 o Mart.I Mart.II Nak d / [ b/rad sr 2 ] 3. 2... =8 o, =6 o =8 o, =9 o =6 o, =9 o.. A y. A y. -. -. -. 6 9 2 6 9 2 [deg] 6 9 2 8 -. 6 9 2 6 9 2 [deg] 6 9 2 8 Construct the contact current full amplitude obeys WTI (gauge invariant) [K. Nakayama & H. Haberzettl, PRC 8, (29)] d / [ b/rad sr 2 ] 2. 2..... =8 o =6 o =2 o d / [ b/rad sr 2 ] 3. 2... =8 o, =6 o =8 o, =9 o =6 o, =9 o.. A y. A y. -. -. -. 6 9 2 6 9 2 [deg] 6 9 2 8 (c) -. 6 9 2 6 9 2 [deg] 6 9 2 8 (d) F. Huang, MENU2, Williamsburg Pion photo-production Jun., 2 9 / 9

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 ] 2. 2..... =8 o =6 o =2 o Mart.I Mart.II Nak d / [ b/rad sr 2 ] 3. 2... =8 o, =6 o =8 o, =9 o =6 o, =9 o.. A y. A y. -. -. -. 6 9 2 6 9 2 [deg] 6 9 2 8 -. 6 9 2 6 9 2 [deg] 6 9 2 8 Construct the contact current full amplitude obeys WTI (gauge invariant) [K. Nakayama & H. Haberzettl, PRC 8, (29)] d / [ b/rad sr 2 ] 2. 2..... =8 o =6 o =2 o d / [ b/rad sr 2 ] 3. 2... =8 o, =6 o =8 o, =9 o =6 o, =9 o.. A y. A y. -. -. -. 6 9 2 6 9 2 [deg] 6 9 2 8 (c) -. 6 9 2 6 9 2 [deg] 6 9 2 8 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)