1. SCALE LAWS AT LARGE TRANSVERSE MOMENTUM GENERALIZED COUNTING RULE FOR HARD EXCLUSIVE PROCESS Feb. 23, 2006 @USC Kijun Park

3. WHAT PHYSICS IS IN ? Mapping the transition from the nucleon-meson degrees of freedom to the quark-gluon degrees of freedom of QCD. Exclusive Processes on Nucleons (1) Diff. CRS measurements : Quark Counting Rule, Form Factors, Oscillations, GPDs, Generalized Counting Rule (2) Polarization measurement : Hadron Helicity conservation Nucleon-meson degrees of freedom effective at low energies Quark-gluon degrees of freedom at high energies

4.  First derived based on dimensional analysis (Matveev et al., Brodsky, Farrar,….)  Confirmed within short distance pQCD framework (Brodsky, LePage)  Recently, derived from anti-de Sitter/Conformal Field Theory correspondence or string/gauge duality ( Polchinski, Strassler …..) Exclusive two body reactions (A+B  C+D) at large momentum transfers should scale as: s = c.m. energy sq. n = # of constituent fields Many exclusive processes exhibit global scaling behavior elastic pp scattering @ 90 deg CM angle ( dσ/dt ~ S -10 ) deuteron photo-disintegration at large angles ( dσ/dt ~ S -11 ) C. Bochna et al., PRL 81, 4576 (1998), E.C. Schulte, et al., PRL 87, 102302 (2001), M. Mirazita et al., PRC 70, 014005 (2004), P. Rossi, PRL 94,012301 (2005). Photopion production from nucleon at large angles L.Y. Zhu et al., PRL 91, 022003 (2003) CONSTITUENT COUNTING RULE PRL V31. Num18 p1153 (1973)

7. From data compilation of Landshoff and Polkinghorn Deuteron Photodisintegration Scaling onset: P t ~1.2 GeV/c E. Schulte et al. PRL 87, 102302 (2001) Bochna et al., PRL 81, 4576 (1998) p-p Elastic Scattering

8. Pion Photoproduction Quark Counting Rule predicts (n=9)

9. Transition observed (90 degree)? L.Y. Zhu, PRL 91, 022003 (2003) Quark Counting Rule predicts (n=9)

10. L.Y. Zhu et al., Nucl-exp/0409018, PRC 71, 044603 (2005) Limited Angular Distribution

11. PRL vol.90 num.24 p.241601

12.  What is the light-cone wave function ?  Why it is useful formalism ?  What’s the fock state ?  How is it used here ?  What is the main point of Ji paper ?  What’s the generalization they claim to have obtained ?

13. What is the Light Cone Wave Function (LCWF) ?  Infinite momentum frame - Snap-shots of hadrons when the latter is moving with the speed of light  It is solution of the eigen-equation of light cone H  LQCD, QCD sum-rule can calculate its moments in momentum space  Parametrization to fit experimental data : Possibility of the Phenomenological approach  Light cone gauge : A + = 0  Bethe-Salpeter amplitude calculates LCWF by integrating out parton momenta ( k- ) n-parton state Wave function Orbital angular momentum projection Parton helicity Transverse momentum Longitudinal momentum

14.  Light cone H H QCD = P + P - - P 2 Where, momentum operators P ± = P 0 ± P Z  H QCD |Ψ h > = M 2 h |Ψ h > : wave function for hadron h What is the Light Cone Wave Function (LCWF) ?

15. What is the Fock state ?

19.  Global scaling in good agreement with constituent quark counting  Detailed investigation reveals oscillatory scaling behavior What is the origin for such oscillatory scaling behavior? R = s 10 ds/dt Oscillatory Scaling Behavior The large spin correlation and oscillations in the scaled cross-section explained as:  Resonance state production near the charm threshold (Brodsky, Schmidt, etc).  Interference between short distance (Born) and long distance (Landshoff) amplitudes, (Ralston & Pire and Carlson, Myhrer, …) p-p elastic scattering at 90 0 Not unique in p-p

20. Exciting New Developments  a generalized counting rule based on pQCD analysis, by systematically numerating the components of a hadronic light-cone wave function. (Ji, Ma & Yuan and also Brodsky et al.)  The generalized counting rule includes parton orbital angular momentum and hadron helicity flip.  They provide the scaling behavior of the helicity flipping amplitudes If = 0; reduces to constituent quark counting rule of Brodsky-Farrar. number of partons orbital angular momentum projection  ``restricted locality'' of quark-hadron duality results in oscillations (Zhao & Close).

21. How is it used here ? 1)useful tool to describe hadron physics in high energy scattering 2)It can be solved the eigenequation of the light cone H in L-C gauge A + = 0. 3)It is parameterized to fit the experiment data Light cone wave function General Amplitude Angular momentum projection Parton helicity Transverse momentum Longitudinal momentum Amplitude mixed terms under scale evolution PRL vol.90 num.24 p.241601

22. 1)Light cone power counting focuses on transverse dimension because …… mass dimension in the amplitude Light cone wave function PRL vol.90 num.24 p.241601 2) Leading behavior from equation : smallest “soft mass dimension” “soft scale factor ” contains Using factorization formula How is it used here ?

23. PRL vol.90 num.24 p.241601 Mass dimension contains H = 1 … N : hadrons Traditional counting rule PRL vol.31 num.18 p.1153 Generalized form : fixed angle scattering CRS How is it used here ?

24. PRL vol.31 num.18 p.1153 What is the main point in this paper ?

25. Generalized Counting Rule & pp Elastic Scattering Using their Generalized Counting Rule Ji et al. predict the s dependence of the helicity flipping amplitudes in pp elastic scattering as ~S -4.5 and ~S -5 no hel. flipwith hel. flip D.Dutta and H. Gao, hep-ph/0411267 (accepted for publication in PRC as rapid comm. )

26. What is the generalization ?

27. Cautionary Notes PRL vol.90 num.24 p.241601 1)Ignored Lanshoff contribution in hadron-hadron scattering PRD 10,1024(1974) 2) Integrated over all partons’ light cone function - divergence at x i = 0,1 Nucl. Phys. Proc. Suppl. B111, 62 (2002) 3)Light cone wave function has singularities Adv. Ser. High-Energy Phys. 5, 573 (1989) What is the generalization but constraint ?