May 18-20, 2005 SIR05 JLab 1

PP?? Dependence of Dependence of SSASSA

-- from nonpert. to pert.-- from nonpert. to pert.Feng Yuan

RBRC , , Brookhaven National Laboratory

References: Ji, Ma, Yuan, Phys. Rev. D70, 074021; Phys. Lett. B597, 299; hep-ph/0503015; to be published;

May 18-20, 2005 2SIR05 JLab

OutlineOutline Introduction Introduction Low PLow PTT region: TMD parton region: TMD parton

distributions and QCD distributions and QCD factorizationfactorization

High PHigh PTT region: quark-gluon- region: quark-gluon-quark correlation (ETQS quark correlation (ETQS mechanism)mechanism)

SummarySummary

May 18-20, 2005 3SIR05 JLab

Inclusive and Inclusive and Semi-inclusive DISSemi-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, 2005 4SIR05 JLab

Different Different PPTT Region Region

Integrate out PT

-- similar to inclusive DIS, probe int. PDF Large PT (>>QCD)

-- hard gluon radiation, can be calculated from perturbative QCD,

Polarized ->q-g-q correlations Low PT (~QCD)

-- nonperturbative information (TMD): new factorization formula

May 18-20, 2005 5SIR05 JLab

TMD PhysicsTMD Physics

A way to measure Transversity A way to measure Transversity Distribution, the last Distribution, the last unkownunkown leading leading twisttwist distribution distribution

Collins 1993Collins 1993 The Novel Single Spin AsymmetriesThe Novel Single Spin Asymmetries Connections with GPDs, and Quantum Connections with GPDs, and Quantum

Phase Space Wigner distributionsPhase Space Wigner distributions Quark Orbital Angular Momentum andQuark Orbital Angular Momentum and

Many others …Many others …

May 18-20, 2005 6SIR05 JLab

TMD Distribution: the TMD Distribution: the definitiondefinition

Gauge Invariance requires the Gauge Link

Brodsky,Hwang,Schmidt 02’Collins 02’Belitsky,Ji,Yuan 02’

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This definition is This definition is also consistent also consistent

with the with the QCD factorizationQCD factorization

May 18-20, 2005 8SIR05 JLab

Illustrate the FactorizationIllustrate the Factorization(one-loop order)(one-loop order)

Take an on-shell quark targetTake 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 Separate the cross section into different

piecespieces Show the soft divergence is cancelled outShow the soft divergence is cancelled out

May 18-20, 2005 9SIR05 JLab

TMD Dis. At One-loopTMD Dis. At One-loop

May 18-20, 2005 10SIR05 JLab

No soft divergenceNo soft divergence

Collinear divergence : Collinear divergence : ln(mln(m22))

Double Logarithms: Double Logarithms: lnln22((22 b b22) ) --- Collins-Soper equation--- Collins-Soper equation

May 18-20, 2005 11SIR05 JLab

FactorizationFactorization

Soft Factor

May 18-20, 2005 12SIR05 JLab

The Factorization Applies The Factorization Applies toto

Semi-inclusive DIS (polarized and Semi-inclusive DIS (polarized and unpolarized)unpolarized)

Drell-Yan at Low transverse momentumDrell-Yan at Low transverse momentum Di-hadron production in e+e- Di-hadron production in e+e-

annihilation (extract the Collins annihilation (extract the Collins function)function)

Di-jet and/or di-hadron correlation at Di-jet and/or di-hadron correlation at hadron collider (work in progress)hadron collider (work in progress)

Many others, …Many others, …

May 18-20, 2005 13SIR05 JLab

Large Logarithms Large Logarithms ResummationResummation

At low At low PT, , PT<< Q, we must resum the , we must resum the large logarithms large logarithms ss

nnln2n-1(Q2/PT2)

-- Dokshitzer, Diakonov, Troian, 1978-- Dokshitzer, Diakonov, Troian, 1978

-- Parisi, Petronzio, 1979-- Parisi, Petronzio, 1979 These large logarithms can be These large logarithms can be

resummed by solving the energy resummed by solving the energy evolution equation for the TMD parton evolution equation for the TMD parton dis.dis.

-- Collins-Soper 1981 -- Collins-Soper 1981

May 18-20, 2005 14SIR05 JLab

TMD: the Energy TMD: the Energy DependenceDependence

The TMD distributions depend on the The TMD distributions depend on the energy of the hadron! (or Q in DIS)energy of the hadron! (or Q in DIS)

One can write down an evolution One can write down an evolution equation in equation in ζζ==2(P2(P++))22vv--/v/v++

K and G obey an RG equation, K and G obey an RG equation,

Collins and Soper (81’)

μ independent!

May 18-20, 2005 15SIR05 JLab

Spin-dep. Collins-Soper Spin-dep. Collins-Soper Eq.Eq.

The evolution kennel is the same for all the leading-twist TMD quark distributionsquark distributions

KKtt-even ones: q(x, k-even ones: q(x, k┴┴), ), ΔΔqqLL, , δδqqTT

KKtt-odd ones: -odd ones: qqTT(x, k(x, k┴┴), ), δδq, q, ΔΔqqTT, , δδqqLL

Idilbi, Ji, Ma, Yuan: PRD70 (2004)

May 18-20, 2005 16SIR05 JLab

Large Logarithms Large Logarithms Resumm. (II)Resumm. (II)

After resummation, large logarithms After resummation, large logarithms associated with associated with QQ22 can be factorized can be factorized into the into the Sudakov form factors, e.g.form factors, e.g.

And the Sudakov form factorAnd the Sudakov form factor

17

Double Logarithmic (DL) Double Logarithmic (DL) Approx.Approx.

If If QQ22 is not too large, DL approx. applies. The is not too large, DL approx. applies. The Sudakov suppression form factor then only Sudakov suppression form factor then only depends on depends on QQ22

The The QQ22 dependence of the structure functions dependence of the structure functions can be factorized outcan be factorized out

We can predict the We can predict the PPTT distribution at higher distribution at higher QQ22 from that of lower from that of lower QQ22

The The PT spectrum of the polarization asymmetry spectrum of the polarization asymmetry will be the same for different Qwill be the same for different Q22 at fixed at fixed B and

May 18-20, 2005 18SIR05 JLab

Phenomenogical applications of the QCD Phenomenogical applications of the QCD resummation to the Presummation to the PTT spectrum of EW spectrum of EW bosons production have been very bosons production have been very successfulsuccessfulYuan, Nadolsky, Ladinsky, Landry, Yuan, Nadolsky, Ladinsky, Landry, Qiu, Zhang, Berger, Li, Qiu, Zhang, Berger, Li, Laenen, Sterman, Vogelsang, Kulesza,Laenen, Sterman, Vogelsang, Kulesza,Bozzi, Catani, deFlorian,Bozzi, Catani, deFlorian,Kulesza, Stirling, Kulesza, Stirling, and many others, …and many others, …working even at NNLL level for someworking even at NNLL level for some

Our interest however, is in the region of Our interest however, is in the region of not so large Qnot so large Q22, where DLA (LLA) should , where DLA (LLA) should applyapply

May 18-20, 2005 19SIR05 JLab

Drell-Yan at Fixed TargetDrell-Yan at Fixed TargetQT spectrum from E288, PRD23,604(81)

Normalized to the same spectrum!Consequence of DLA

May 18-20, 2005 20SIR05 JLab

At very large Q2 (e.g., Z0 and W boson), DL Approx. breaks down

May 18-20, 2005 21SIR05 JLab

Power Counting at Large Power Counting at Large PPTT

kktt-even distributions have the same -even distributions have the same dependence on kdependence on ktt

kktt-odd distributions are suppressed -odd distributions are suppressed at large kat large ktt

Power Counting RulePower Counting Rule

kktt-even: -even: 1/k1/ktt22

kktt-odd: -odd: 1/k1/ktt33

May 18-20, 2005 22SIR05 JLab

SIDIS: at Large PSIDIS: at Large PTT

Large Large PPTT SIDIS can be calculated from SIDIS can be calculated from perturbative QCD (same for TMDs)perturbative QCD (same for TMDs)

Polarized scattering depends on the Polarized scattering depends on the quark-gluon correlations in the quark-gluon correlations in the nucleonnucleon

Single Spin Asymmetry (Sivers and Single Spin Asymmetry (Sivers and Collins) at large PCollins) at large P?? is not suppressed is not suppressed by by 1/Q1/Q, but by , but by 1/P1/PTT

May 18-20, 2005 23SIR05 JLab

An example: SSA at An example: SSA at Large PLarge PTT

ETQS mechanismETQS mechanism

It is suppressed by It is suppressed by 1/P1/PTT

Efremov & Teryaev: 1982 & 1984Qiu & Sterman: 1991 & 1999

Quark-gluon correlations

Ji,Ma,Yuan, work in progress

May 18-20, 2005 24SIR05 JLab

Transition from Transition from Perturbative region to Perturbative region to

Nonperturbative region?Nonperturbative region? Compare different region of PCompare different region of PTT

Nonperturbative TMD Perturbative region

May 18-20, 2005 25SIR05 JLab

SummarySummary

QQ22 evolution of the P evolution of the PTT spectrum is spectrum is calculable from perturbative QCD, by calculable from perturbative QCD, by solving the associated Collins-Soper eqssolving the associated Collins-Soper eqs

Large PLarge PTT SSA depends on the quark-gluon- SSA depends on the quark-gluon-quark correlation of nucleon (ETQS), quark correlation of nucleon (ETQS), which is suppressed by 1/Pwhich is suppressed by 1/PTT, not 1/Q, not 1/Q

Detailed study of the PDetailed study of the PTT dependence will dependence will provide us information for the provide us information for the perturbative to nonperturbative transition perturbative to nonperturbative transition of of QQCCDD