Towards a Unied Description of Meson Production off Nucleons

Átírás

1 Towards a Unied Description of Meson Production off ucleons Helmut Haberzettl, Department of Physics The George Washington University, Washington, DC Collaborators: Kanzo akayama, UGA Siegfried Krewald, FZJ ongseok Oh, UGA 01 H. Haberzettl * JLab, 05ov06 Master file: hhstarjlab06.tex Compiled: 05ov2006

2 What means unified? Consistent treatment of hadronic and electromagnetic sectors Consistent treatment of competing reaction channels Consistency between resonant and non-resonant contributions Requirements: Lorentz covariance Gauge invariance Hadronic final-state interaction 02 H. Haberzettl * JLab, 05ov06 unified.tex

3 What means unified? Consistent treatment of hadronic and electromagnetic sectors Consistent treatment of competing reaction channels Consistency between resonant and non-resonant contributions Requirements: Lorentz covariance Gauge invariance Hadronic final-state interaction Jülich 03 H. Haberzettl * JLab, 05ov06 unified.tex

4 Gauge Invariance k p 1 p 1 p 2 p m 1 M µ p 2 pn 1 p m Time p n Physical condition: Current Conservation: k µ M µ = 0 all hadrons on-shell 04 H. Haberzettl * JLab, 05ov06 gauge1.tex

5 Gauge Invariance k p 1 p 1 p 2 p m 1 M µ p 2 pn 1 p m Time p n Physical condition: Current Conservation: k µ M µ = 0 all hadrons on-shell Recipe for Phenomenological Currents: M µ = M µ a µ k M k a (a µ arbitrary) 05 H. Haberzettl * JLab, 05ov06 gauge2.tex

6 Gauge Invariance k p 1 p 1 p 2 p m 1 M µ p 2 pn 1 p m Time p n Physical condition: Current Conservation: k µ M µ = 0 all hadrons on-shell Recipe for Phenomenological Currents: M µ = M µ a µ k M k a (a µ arbitrary) ot good enough for microscopic approaches! 06 H. Haberzettl * JLab, 05ov06 gauge3.tex

7 Gauge Invariance for the ucleon: consider γ(k) (p) (p ) Simple nucleon current: Γ µ (p, p) = γ µ Q 07 H. Haberzettl * JLab, 05ov06 gaugenucleon.tex

8 Gauge Invariance for the ucleon: consider γ(k) (p) (p ) Simple nucleon current: Four-divergence: Γ µ (p, p) = γ µ Q k µ Γ µ (p, p) = (p/ p/)q = (p/ m)q Q (p/ m) on-shell 0 k = p p 0 H. Haberzettl * JLab, 05ov06 gaugenucleon.tex

9 Gauge Invariance for the ucleon: consider γ(k) (p) (p ) Simple nucleon current: Four-divergence: Γ µ (p, p) = γ µ Q k µ Γ µ (p, p) = (p/ p/)q = (p/ m)q Q (p/ m) on-shell 0 k = p p Ward Takahashi Identity: k µ Γ µ (p, p) = S 1 (p )Q Q S 1 (p) [General result] S(p): nucleon propagator Full electromagnetic information on the LHS Only hadronic information on the RHS 09 H. Haberzettl * JLab, 05ov06 gaugenucleon.tex

10 Gauge Invariance for the ucleon: consider γ(k) (p) (p ) Simple nucleon current: Four-divergence: Γ µ (p, p) = γ µ Q k µ Γ µ (p, p) = (p/ p/)q = (p/ m)q Q (p/ m) on-shell 0 k = p p Ward Takahashi Identity: k µ Γ µ (p, p) = S 1 (p )Q Q S 1 (p) [General result] S(p): nucleon propagator Full electromagnetic information on the LHS Only hadronic information on the RHS Consistency! 10 H. Haberzettl * JLab, 05ov06 gaugenucleon.tex

11 γ π 11 H. Haberzettl * JLab, 05ov06 examplepi.tex

12 Attaching a Photon to the π Vertex π s u t G M µ s M µ u M µ t M µ int s-channel u-channel t-channel Interaction current Total photoproduction amplitude M µ = M µ s M µ u M µ t M µ int 12 H. Haberzettl * JLab, 05ov06 sgampinn.tex

13 Generalized Ward Takahashi Identity for the Pion-Production Current k µ M µ = [F s τ]s pk Q S 1 p S 1 p Q S p k[f u τ] 1 p p k Q π p p [F t τ] s u t G Equivalently: k µ M µ int = [F sτ]q Q [F u τ] Q π [F t τ] ote: Charge conservation: τq Q τ Q π τ = 0 13 H. Haberzettl * JLab, 05ov06 WTI1.tex

14 Strategy Start from eld theory Cook it down to something manageable Be aware where the approximations are made, what physics they destroy, and make sure that you can live with it! 14 H. Haberzettl * JLab, 05ov06 strategypi.tex

15 Pions, ucleons, and Photons Hadronic scattering: π π = X Photoproduction: γ π Amplitude: = 0 time X = U U X s-channel u-channel t-channel M µ int T = X M = Theory: PRC 56, 2041 (1997) = U X U I Interaction Current M µ int 15 H. Haberzettl * JLab, 05ov06 sfieldth0.tex

16 Hadronic Driving Terms π σ,ρ σ,ρ π Electromagnetic Couplings to Driving Terms Dressed ucleon Current E 16 H. Haberzettl * JLab, 05ov06 fieldth2.tex

17 Ward Takahashi Identities for the ucleon and the Pion Currents p',m' k p,m k µ Γ µ (p, p) = S 1 (p )Q Q S 1 (p) nucleon Form factors: [ ] [ ] Γ ν (p, p) = γ ν Q t ν f1 00 iσνµ k µ 2m f 2 00 p/ m t ν f1 10 iσνµ k µ 2m 2m f 2 10 f ij n = f ij n (p 2, p 2 k 2 ) [ ] [ ] t ν f1 01 iσνµ k µ p/ m 2m f m p/ m t ν f1 11 iσνµ k µ p/ m 2m 2m f m t ν = γ ν k 2 k ν k/ k q, µ q, µ k µ Γ µ π(q, q) = 1 π (q )Q π Q π 1 π (q) pion [ ] Γ ν π(q, q) = (q q) ν Q π (q q) ν k 2 k ν k (q q) Q π f(q 2, q 2 k 2 ) 17 H. Haberzettl * JLab, 05ov06 WTI.tex

18 Lowest-Order Strategy M = = U X U Replace U is satisfied. U by phenomenological contact term such that WTI for interaction current 1 H. Haberzettl * JLab, 05ov06 strategy.tex

19 Lowest-Order Strategy M = = U X U Replace U is satisfied. U Result by phenomenological contact term such that WTI for interaction current M µ [ int = M c µ T µ XG 0 (M u µ m µ }{{ u) (M µ t m µ t } ) T µ] }{{} transverse transverse T µ : undetermined transverse current with k µ M µ c = F s e i F u e f F t e π. 19 H. Haberzettl * JLab, 05ov06 result1.tex

20 Phenomenological Choice for M µ c M µ c { [ ] q/ βk/ = g π γ 5 λ (1 λ) m C µ γ µ [ (1 λ) m m e π f t β k ρ C ρ]} m λ, β: free parameters on-singular auxiliary current: with C µ = e π (2q k) µ t q 2 (f t ˆF ) e f (2p k) µ u p 2 (f u ˆF ) e i (2p k) µ s p 2 (f s ˆF ), k µ C µ = e π f t e f f u e i f s Subtraction function: ˆF = 1 ĥ ( 1 δ s f s )( 1 δu f u )( 1 δt f t ) δ x = 0, 1 The function ĥ = ĥ(s, u) is free fit function. [Ohta: ĥ = 0, X = 0, T µ = 0] 20 H. Haberzettl * JLab, 05ov06 choice1.tex

21 First Application (straight out of the box not optimized) H.H., K. akayama, S. Krewald, nucl-th/ dσ/dω (µb/sr) γp-->π o p γp-->π n γn-->π - -p T γ =390 MeV T γ =340 MeV T γ =220 MeV T γ =10 MeV θ (deg) σ (µb) γp > π o p Born FSI total T γ (MeV) Includes ρ, ω, and a 1 exchanges in the t-channel in s- and u-chanels FSI with Jülich π T -matrix. 21 H. Haberzettl * JLab, 05ov06 application.tex

22 ext, two mesons: γ M 1 M 2 22 H. Haberzettl * JLab, 05ov06 next2pions.tex

23 Producing Two Mesons Start from underlying hadronic processes: M 2 M R M 1 M V M 2 23 H. Haberzettl * JLab, 05ov06 twopion0.tex

24 Producing Two Mesons Start from underlying hadronic processes: M 2 M R M 1 M V M 2 = X = X = U U X time T = X 24 H. Haberzettl * JLab, 05ov06 twopion0a.tex

25 Basic Two-meson Production Mechanisms q q p 2 s u 1 q q p s k t 2 t t 1 u u 2 C C 1 C C 2 1 time time µ µ M M 2 1 (a) M (b) 25 H. Haberzettl * JLab, 05ov06 twopion1.tex

26 Application: γ KKΞ (w/o hadronic FSI) K. akayama,. Oh, H.H., hep-ph/ Total cross-sections: 20 γp Κ Κ Ξ 20 γp Κ Κ Ξ γp Κ Κ 0 Ξ γp Κ Κ 0 Ξ σ (nb) 0 0 γn Κ Κ 0 Ξ σ (nb) 0 0 γn Κ Κ 0 Ξ γn Κ 0 Κ 0 Ξ γn Κ 0 Κ 0 Ξ T γ (GeV) T γ (GeV) λ = 1 (pv) 26 H. Haberzettl * JLab, 05ov06 twomesonapplication.tex λ = 0 (ps)

27 Summary Gauge invariance is a fundamental symmetry without it results become arbitrary. Current conservation k µ M µ = 0 necessary, but not sufficient, for gauge invariance of microscopic theories. Generalized Ward Takahashi identities are necessary and sufficient. Real-world calculations require truncations of dynamical mechanisms that destroy gauge invariance. Prescriptions required to restore gauge invariance. Formalism allows inclusion of final-state interaction. FSI dynamics can be refined step-bystep in controlled manner. ext application: More complete calculation of pseudoscalar meson production with FSI. Application to γ KKΞ production provides good agreement with the data already at the (dressed) tree level. Applying formalism to two-pion production is underway (FSI necessary: DWA). Application to electroproduction possible (manpower required!!). 27 H. Haberzettl * JLab, 05ov06 summary.tex

28 Transition Currents γ Gauge invariance demands that transition currents be transverse. B 2 B1 Outline of proof: M Coupling the photon to MB 1 B 2 vertex, B1 B 2, produces First line produces correct off-shell relation. Terms in second line must vanish individually. QED In a consistent microscopic approach, this should be ensured dynamically. There should be no need to adjust the transversality manually. 2 H. Haberzettl * JLab, 05ov06 transition.tex

29 Example: γ γ = Transversality of this current is ensured by the fact that is not possible as a physical process, i.e. = = = 0. With consistent dynamics, this current satisfies ) ( ) Γ βµ = G 1 γ 5 (k β γ µ g βµ k/ G 2 γ 5 k β P µ g βµ k P G 3 γ 5 (k β k µ g βµ k 2) as a matter of course, where P = (p p )/2 = (2p k)/2. There is no need for subtractions. 29 H. Haberzettl * JLab, 05ov06 transition.tex

30 ext Order M = = U X U = = Instead of replacing all of U X U, take into account explicitly. } {{ } =E µ exchange current 30 H. Haberzettl * JLab, 05ov06 strategy1.tex

31 ext Order M = = U X U = = Instead of replacing all of U X U General gauge-invariance condition for U µ, take into account explicitly. } {{ } =E µ exchange current k µ U µ (p, q, p, q) = Q πu(p, q k, p, q) Q U(p k, q, p, q) U(p, q, p, q k)q π U(p, q, p k, q)q Holds also true for every subset of U µ that originates from attaching a photon in all possible ways to a single two-body irreducible hadron graph = also true for E µ 31 H. Haberzettl * JLab, 05ov06 strategy1a.tex

32 ext-order Result M = = U X U Previously M µ [ int = M c µ T µ XG 0 (M u µ m µ }{{ u) (M µ t m µ t } ) T µ] }{{} transverse transverse T µ : undetermined transverse current ow: T µ is replaced by T µ = [ E µ Ẽµ] G 0 [F τ] T µ T µ : undetermined transverse current 32 H. Haberzettl * JLab, 05ov06 result2.tex

33 Phenomenological Subtraction Current [ Ẽ µ = gπγ 2 q/ q/ 5 [λ (1 λ) ]S ] m γ 5 λ (1 λ) D µ m m m (1 λ)gπf 2 γ 5 γ µ [ q/ ] π f 1 e π m S γ 5 λ (1 λ) m m m ] (1 λ)gπf 2 q/ 2 f π γ 5 [λ (1 λ) m S e γ 5 γ µ π m m m on-singular auxiliary current: D µ = e (2q k) µ π (q k) 2 q 2 (f 2f π Ĝ) (2p k) µ e (p k) 2 p 2 (f f 1 Ĝ) (2q k) µ e π (q k) 2 q 2 (f πf 1 Ĝ) e (2p k) µ (p k) 2 p 2 (f 2f Ĝ), with Ĝ = 1 ĝ ( )( )( )( ) 1 f π f 1 1 f2 f 1 f2 f π 1 f f 1 ĝ: free fit function ote: k µ D µ = e π f π f 1 e f 2 f e πf 2 f π e f f 1 33 H. Haberzettl * JLab, 05ov06 sgampinn.tex