May 18-20, 2005 SIR05 JLab 1 P ? Dependence of SSA -- from nonpert. to pert. Feng Yuan, RBRC, Brookhaven National Laboratory References: Ji, Ma, Yuan, Phys. Rev. D70, ; Phys. Lett. B597, 299; hep-ph/ ; to be published;

May 18-20, SIR05 JLab Outline Introduction Introduction Low P T region: TMD parton distributions and QCD factorization Low P T region: TMD parton distributions and QCD factorization High P T region: quark-gluon-quark correlation (ETQS mechanism) High P T region: quark-gluon-quark correlation (ETQS mechanism) Summary Summary

May 18-20, SIR05 JLab Inclusive and Semi-inclusive DIS Inclusive DIS: Partonic Distribution depending on the longitudinal momentum fraction Semi-inclusive DIS: Probe additional information for partons’ transverse distribution in nucleon

May 18-20, SIR05 JLab Different P T Region Integrate out P T -- similar to inclusive DIS, probe int. PDF Large P T (>>  QCD ) -- hard gluon radiation, can be calculated from perturbative QCD, Polarized ->q-g-q correlations Low P T (~  QCD ) -- nonperturbative information (TMD): new factorization formula

May 18-20, SIR05 JLab TMD Physics A way to measure Transversity Distribution, the last unkown leading twist distribution A way to measure Transversity Distribution, the last unkown leading twist distribution Collins 1993 Collins 1993 The Novel Single Spin Asymmetries The Novel Single Spin Asymmetries Connections with GPDs, and Quantum Phase Space Wigner distributions Connections with GPDs, and Quantum Phase Space Wigner distributions Quark Orbital Angular Momentum and Quark Orbital Angular Momentum and Many others …

May 18-20, SIR05 JLab TMD Distribution: the definition Gauge Invariance requires the Gauge Link Brodsky,Hwang,Schmidt 02’ Collins 02’ Belitsky,Ji,Yuan 02’

May 18-20, 2005 SIR05 JLab 7 This definition is also consistent with the QCD factorization

May 18-20, SIR05 JLab Illustrate the Factorization (one-loop order) Take an on-shell quark target Take an on-shell quark target Calculate the TMD Dis. and F.F. Calculate the TMD Dis. and F.F. Separate the cross section into different pieces Separate the cross section into different pieces Show the soft divergence is cancelled out Show the soft divergence is cancelled out

May 18-20, SIR05 JLab TMD Dis. At One-loop

May 18-20, SIR05 JLab No soft divergence No soft divergence Collinear divergence : ln(m 2 ) Collinear divergence : ln(m 2 ) Double Logarithms: ln 2 (  2 b 2 ) Double Logarithms: ln 2 (  2 b 2 ) --- Collins-Soper equation

May 18-20, SIR05 JLab Factorization Soft Factor

May 18-20, SIR05 JLab The Factorization Applies to Semi-inclusive DIS (polarized and unpolarized) Semi-inclusive DIS (polarized and unpolarized) Drell-Yan at Low transverse momentum Drell-Yan at Low transverse momentum Di-hadron production in e+e- annihilation (extract the Collins function) Di-hadron production in e+e- annihilation (extract the Collins function) Di-jet and/or di-hadron correlation at hadron collider (work in progress) Di-jet and/or di-hadron correlation at hadron collider (work in progress) Many others, … Many others, …

May 18-20, SIR05 JLab Large Logarithms Resummation At low,, we must resum the large logarithms  s n At low P T, P T << Q, we must resum the large logarithms  s n ln 2n-1 (Q 2 /P T 2 ) -- Dokshitzer, Diakonov, Troian, Dokshitzer, Diakonov, Troian, Parisi, Petronzio, Parisi, Petronzio, 1979 These large logarithms can be resummed by solving the energy evolution equation for the TMD parton dis. These large logarithms can be resummed by solving the energy evolution equation for the TMD parton dis. -- Collins-Soper Collins-Soper 1981

May 18-20, SIR05 JLab TMD: the Energy Dependence  The TMD distributions depend on the energy of the hadron! (or Q in DIS)  One can write down an evolution equation in ζ = 2(P + ) 2 v - /v +  K and G obey an RG equation, Collins and Soper (81’) μ independent!

May 18-20, SIR05 JLab Spin-dep. Collins-Soper Eq. quark distributions The evolution kennel is the same for all the leading-twist TMD quark distributions K t -even ones: q(x, k ┴ ), Δq L, δq T K t -even ones: q(x, k ┴ ), Δq L, δq T K t -odd ones: q T (x, k ┴ ), δq, Δq T, δq L K t -odd ones: q T (x, k ┴ ), δq, Δq T, δq L Idilbi, Ji, Ma, Yuan: PRD70 (2004)

May 18-20, SIR05 JLab Large Logarithms Resumm. (II) After resummation, large logarithms associated with Q 2 can be factorized into the form factors, e.g. After resummation, large logarithms associated with Q 2 can be factorized into the Sudakov form factors, e.g. And the Sudakov form factor And the Sudakov form factor

17 Double Logarithmic (DL) Approx. If Q 2 is not too large, DL approx. applies. The Sudakov suppression form factor then only depends on Q 2 If Q 2 is not too large, DL approx. applies. The Sudakov suppression form factor then only depends on Q 2 The Q 2 dependence of the structure functions can be factorized out The Q 2 dependence of the structure functions can be factorized out We can predict the P T distribution at higher Q 2 from that of lower Q 2 We can predict the P T distribution at higher Q 2 from that of lower Q 2 The spectrum of the polarization asymmetry will be the same for different Q 2 at fixed The P T spectrum of the polarization asymmetry will be the same for different Q 2 at fixed  B and 

May 18-20, SIR05 JLab Phenomenogical applications of the QCD resummation to the P T spectrum of EW bosons production have been very successful Phenomenogical applications of the QCD resummation to the P T spectrum of EW bosons production have been very successful Yuan, Nadolsky, Ladinsky, Landry, Qiu, Zhang, Berger, Li, Laenen, Sterman, Vogelsang, Kulesza, Bozzi, Catani, deFlorian, Kulesza, Stirling, and many others, … working even at NNLL level for some Our interest however, is in the region of not so large Q 2, where DLA (LLA) should apply Our interest however, is in the region of not so large Q 2, where DLA (LLA) should apply

May 18-20, SIR05 JLab Drell-Yan at Fixed Target Q T spectrum from E288, PRD23,604(81) Normalized to the same spectrum! Consequence of DLA

May 18-20, SIR05 JLab At very large Q 2 (e.g., Z 0 and W boson), DL Approx. breaks down

May 18-20, SIR05 JLab Power Counting at Large P T k t -even distributions have the same dependence on k t k t -even distributions have the same dependence on k t k t -odd distributions are suppressed at large k t k t -odd distributions are suppressed at large k t Power Counting Rule Power Counting Rule k t -even: 1/k t 2 k t -even: 1/k t 2 k t -odd: 1/k t 3 k t -odd: 1/k t 3

May 18-20, SIR05 JLab SIDIS: at Large P T Large P T SIDIS can be calculated from perturbative QCD (same for TMDs) Large P T SIDIS can be calculated from perturbative QCD (same for TMDs) Polarized scattering depends on the quark-gluon correlations in the nucleon Polarized scattering depends on the quark-gluon correlations in the nucleon Single Spin Asymmetry (Sivers and Collins) at large P ? is not suppressed by 1/Q, but by 1/P T Single Spin Asymmetry (Sivers and Collins) at large P ? is not suppressed by 1/Q, but by 1/P T

May 18-20, SIR05 JLab An example: SSA at Large P T ETQS mechanism ETQS mechanism It is suppressed by 1/P T It is suppressed by 1/P T Efremov & Teryaev: 1982 & 1984 Qiu & Sterman: 1991 & 1999 Quark-gluon correlations Ji,Ma,Yuan, work in progress

May 18-20, SIR05 JLab Transition from Perturbative region to Nonperturbative region? Compare different region of P T Compare different region of P T Nonperturbative TMDPerturbative region

May 18-20, SIR05 JLab Summary Q 2 evolution of the P T spectrum is calculable from perturbative QCD, by solving the associated Collins-Soper eqs Q 2 evolution of the P T spectrum is calculable from perturbative QCD, by solving the associated Collins-Soper eqs Large P T SSA depends on the quark-gluon-quark correlation of nucleon (ETQS), which is suppressed by 1/P T, not 1/Q Large P T SSA depends on the quark-gluon-quark correlation of nucleon (ETQS), which is suppressed by 1/P T, not 1/Q Detailed study of the P T dependence will provide us information for the perturbative to nonperturbative transition of QCD Detailed study of the P T dependence will provide us information for the perturbative to nonperturbative transition of QCD