Quark Orbital Angular Momentum in the Model

Transcription

1 Quark Orbital Angular Momentum in the Model Barbara Pasquini, Feng Yuan Pavia, INFN, Italy LBNL and RBRC-BNL, USA Ref: Pasquini, Yuan, work in progress 9/22/2010 1

2 Proton Spin Sum Quark spin ~30% DIS, and pp coll. Gluon spin~ 0-70% RHIC, EIC, q G L q L G Deeply Virtual Compton Scattering, FF, Transverse spin physics, in DIS, pp coll. 2

3 Hunting for L q : Generalised Parton Distributions (GPDs) ( H + E) x dx = J q = 1/2 S + L z30%(dis) Ji,96 A new type of parton distributions contains much more information Can be measured in deeply virtual compton scattering and other hard exclusive processes Related to form factors and parton distributions 9/22/2010 DIS Parton Distributions DVCS Generalized Parton Distributions Mueller et al., 94; 3Ji, 96; Radyushkin, 96

4 However, there have been arguments that the Jaffe-Manohar Orbital angular momentum is more intuitive Of course, in the quark model these two should agree with each other BC-Burkardt, 2009 The purpose of this work is to show that the Jaffe-Manohar orbital angular momentum is equally complicated in terms of light-cone wave functions 9/22/2010 4

5 Quark-diquark model L Jaffe-Manohar =L GPD Pauli-Villars regularization BC-Burkardt /22/2010 5

6 Outline Introduction on light-cone wave functions Calculate the Jaffe-Manohar orbital angular momentum Compare to the GPD calculation in the same quark model 9/22/2010 6

7 Light-cone Wave Functions They are building blocks for the hadron structure Which can be used to calculate the integrated parton distributions, GPDs, and hard exclusive scattering amplitudes, including the Compton scattering amplitudes 9/22/2010 7

8 Hard Exclusive Process Probes the light-cone wave functions of hadrons at large transverse momentum Their contributions are determined by their asymptotic behavior: power counting rule If the hadron helicity is conserved, these power counting rules are the conventional ones Brodsky-Farrar 1973 Matveev-Muradian-Tavkhelidze /22/2010 8

9 General Structure Starting from any general structure for a Fock state, l z + =, with l z = i=1 n-1 l zi We will get 9/22/2010 9

10 Asymptotic Behavior The asymptotic behavior for the lightcone wave function can be studied from hard diagrams 9/22/

11 Nucleon s 3-quarks WF According to the general structure, there are six independent light-cone wave functions for three quarks component: L z =0 (2), Lz=1 (3), Lz=2 (1) The power counting rule gives, asymptotically, lz =0~ 1/k T 4 lz =1~ 1/k T 6 lz =2~1/k T 8 9/22/

12 Three Quark Light Cone Amplitudes classification of LCWFs in angular momentum components [Ji, J.P. Ma, Yuan, 03; Burkardt, Ji, Yuan, 02] J z = J zq + L z q total quark helicity J q L zq = -1 L zq =0 L zq =1 L zq =2 parity time reversal isospin symmetry 6 independent wave function amplitudes: q L zq z =

13 Light-Cone Quark Model Phenomenological LCWF for the valence (qqq) component: momentum-space component: S wave parameters fitted to anomalous magnetic moments of the nucleon : normalization constant Schlumpf, Ph.D. Thesis, hep-ph/ spin and isospin component in the rest frame: SU(6) symmetric J z = J z q ) Melosh rotation to convert the rest-frame spins of quarks in LF spins J z = J zq +L z q Six independent wave function amplitudes : eigenstates of the total orbital angular momentum operator in Light-Front dynamics L zq = -1 L zq =0 L zq =1 L zq =2 The six independent wave function amplitudes obtained from the Melosh rotations satisfy the model independent classification scheme in four orbital angular momentum components

14 Quark OAM (Jaffe-Manohar) Definition Using light-cone quantization 9/22/

15 Features Helicity conserved Wave function square Introduce off-forward matrix element: p 1 p 1 P P 9/22/

16 for the calculation: Partial integral 9/22/

17 Final expression 9/22/

18 B -dependence give the OAM contribution (p 1 - ) lz1 (1-y 1 )l z1 (p 1 - ) lz1 (p i - ) lzi -y i l zi (p i - ) lzi-1 (p 1 - ) n (p i ) i(p 1 X n (p i )) (1-y 1 ) p 1 p i -y i p 1 p i y i p 1 p j -y j p 1 p i 9/22/

19 Final results (general form) 9/22/

20 Three-quark Fock state Lz=0 9/22/

21 Lz=1 9/22/

22 Lz=2 9/22/

23 Lz=-1 9/22/

24 GPD E is the interference between different Lz states 9/22/

25 Comments Jaffe-Manohar OAM is the wave function square GPD OAM depends on the interferece between different OAM Fock states It s nontrivial to have the same OAM in general 9/22/

26 Distribution in x of Orbital Angular Momentum Definition of Jaffe and Manohar: contribution from different partial waves TOT up down L z =0 L z =-1 L z =+1 L z =+2 Comparison between the results with the Jaffe-Manohar definiton and the results with the Ji definition (total results for the sum of up and down quark contribution) Jaffe-Manohar Ji

27 Orbital Angular Momentum Definition of Jaffe and Manohar: contribution from different partial waves = 0 x (-1) x (+1) x (+2) x = Definition of Ji:

28 Conclusion Jaffe-Manohar s quark Orbital Angular Momentum is studied in terms of the light-cone wave functions They agree with the OAM from GPD calculations Further developments are needed to build more useful relations between them 9/22/

29 Light-Cone Fock Expansion in the light-cone gauge A + =0, the total angular momentum is conserved Fock state by Fock state ) each Fock-state component can be expanded in terms of eigenfunction of the light-front orbital angular momentum operator fixed light-cone time internal variables: frame INdependent probability amplitude to find the N parton configuration with the complex of quantum number in the nucleon with helicity