# Single (transverse) Spin Asymmetry & QCD Factorization

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Single (transverse) Spin Single (transverse) Spin Asymmetry & QCD Asymmetry & QCD

FactorizationFactorizationXiangdong JiXiangdong Ji

University of MarylandUniversity of Maryland

— Workshop on SSA, BNL, June 1, 2005 —

Outline Outline 1.1. General RemarksGeneral Remarks

2.2. DIS/Drell Yan processesDIS/Drell Yan processes

3.3. ppp p ππX & friendsX & friends

4.4. SummarySummary

Single (transverse) Spin AsymmetrySingle (transverse) Spin Asymmetry

SSA is a general phenomenon in physics, and it SSA is a general phenomenon in physics, and it exists so long as there areexists so long as there are– A single transverse spin A single transverse spin

– A mechanism for helicity flipA mechanism for helicity flip

– Initial and/or state interactionsInitial and/or state interactions

““Ok, SSA is an interesting phenomenon, but what do Ok, SSA is an interesting phenomenon, but what do you learn about QCD from it?” or “Why do we have you learn about QCD from it?” or “Why do we have to spend to spend $$ & time to measure it?” & time to measure it?”

• We have some models that fit the dataWe have some models that fit the data (who cares about models? we (who cares about models? we have QCD) have QCD)

• We learn something about the nucleon spin structureWe learn something about the nucleon spin structure (what exactly do (what exactly do you learn? And why that is interesting? can you check it in lattice you learn? And why that is interesting? can you check it in lattice QCD? What is it missing if we don’t measure it?)QCD? What is it missing if we don’t measure it?)

• ……..

Pertubative & Nonperturbative Pertubative & Nonperturbative MechanismsMechanisms

In general, however, the physics mechanism In general, however, the physics mechanism for SSA in strong interactions can be either be for SSA in strong interactions can be either be perturbative & non-perturbative,perturbative & non-perturbative,– pppp to pp at low energy: to pp at low energy: non-perturbativenon-perturbative– What one would like to understand is the SSA in What one would like to understand is the SSA in

perturbative region=> we hope to learn something perturbative region=> we hope to learn something simple, maybe!simple, maybe!

There must be some hard momentum:There must be some hard momentum:Perturbative description of the cross section must be Perturbative description of the cross section must be

valid valid • Factorization Factorization • A good description of spin-averaged cross sectionsA good description of spin-averaged cross sections

SSA & processesSSA & processes

DIS & Drell-Yanpp -> πX & friends

Hard scale Q2 PT

Small PT~ΛQCD

QCD factorizationIn TMD’s

Non-perturbative

Q2,s» PT» ΛQCDQCD factorizationIn TMD’sTwist-3 effects

QCD factorizationIn TMD’s ?Twist-3 effects

SIDIS at low pSIDIS at low pT T

Single-jet production Single-jet production

If the target is transversely polarized, the If the target is transversely polarized, the current jet with a transverse momentum kcurrent jet with a transverse momentum kTT has a SSA which allows a QCD factorization has a SSA which allows a QCD factorization theorem even when ktheorem even when kT T is on the order is on the order ΛΛQCDQCD

The SSA is of order 1 in the scaling limit, i.e. a The SSA is of order 1 in the scaling limit, i.e. a twist-2 effect!twist-2 effect!

qk’

P

k

X

Factorization for SIDIS with PFactorization for SIDIS with P┴┴

Must consider generic Feynman diagrams with Must consider generic Feynman diagrams with partons having transverse momentum, and partons having transverse momentum, and gluon loops.gluon loops.

We have We have two observable scales, Q and Ptwo observable scales, Q and P┴┴

((softsoft). We consider leading order effects in ). We consider leading order effects in PP┴┴ /Q./Q.

The gluons can be hard, soft and collinear. The gluons can be hard, soft and collinear. Can one absorb these contributions into Can one absorb these contributions into different factors in the cross sections. different factors in the cross sections. – X. Ji, F. Yuan, and J. P. Ma, X. Ji, F. Yuan, and J. P. Ma, PRD71:034005,2005 PRD71:034005,2005

Example at one-loopExample at one-loop

Vertex correctionsVertex corrections

Four possible regions of gluon momentum k: 1) k is collinear to p (parton dis) 2) k is collinear to p′ (fragmentation) 3) k is soft (wilson lines) 4) k is hard (pQCD correction)

p

p′q

k

A general reduced diagramA general reduced diagram

Leading contribution in pLeading contribution in p┴┴ /Q./Q.

FactorizationFactorization

Factoring into parton distribution, Factoring into parton distribution, fragmentation function, and soft factor:fragmentation function, and soft factor:

TMD parton distributionsTMD parton distributions

The unintegrated parton distributions is The unintegrated parton distributions is defined asdefined as

where the “light-cone” gauge link is where the “light-cone” gauge link is

the usual parton distribution may be regarded the usual parton distribution may be regarded asas

ClassificationClassification

The leading-twist ones are classified by Boer, The leading-twist ones are classified by Boer, Mulders, and Tangerman (1996,1998)Mulders, and Tangerman (1996,1998)– There are 8 of them, corresponding to the number of There are 8 of them, corresponding to the number of

quark-quark scattering amplitudes without T-constraintquark-quark scattering amplitudes without T-constraint

q(x, kq(x, k┴┴), ), qqTT(x, k(x, k┴┴) (sivers),) (sivers),

ΔΔqqLL(x, (x, kk┴┴), ), ΔΔqqTT(x, (x, kk┴┴),),

δδq(x, q(x, kk┴┴)),, δδLLq(x, q(x, kk┴┴),), δδTTq(x, q(x, kk┴┴), ), δδT’T’q(x, q(x, kk┴┴))

– Similarly, one can define fragmentation functionsSimilarly, one can define fragmentation functions

Sivers’ FunctionSivers’ Function

A transverse-momentum-dependent parton A transverse-momentum-dependent parton distribution which distribution which builds in the physics of SSAbuilds in the physics of SSA!!

S

k

P

The distribution of the parton transverse momentum is not symmetric in azimuth, it has a distribution in S ·(p × k). Since kT is small, the distribution comes fromnon-perturbative structure physics.

Physics of a Sivers FunctionPhysics of a Sivers Function

Hadron helicity flipHadron helicity flip– This can be accomplished through non-perturbative This can be accomplished through non-perturbative

mechanics (chiral symmetric breaking) in hadron mechanics (chiral symmetric breaking) in hadron structure.structure.

– The quarks can be in both s and p waves in relativistic The quarks can be in both s and p waves in relativistic quark models (MIT bag).quark models (MIT bag).

FSI (phase)FSI (phase)– The hadron structure has no FSI phase, therefore Sivers The hadron structure has no FSI phase, therefore Sivers

function vanish by time-reversal (Collins, 1993)function vanish by time-reversal (Collins, 1993)

– FSI can arise from the scattering of jet with background FSI can arise from the scattering of jet with background gluon field in the nucleon (collins, 2002)gluon field in the nucleon (collins, 2002)

– The resulting gauge link is part of the parton dis. The resulting gauge link is part of the parton dis.

Light-Cone Gauge PitfallsLight-Cone Gauge Pitfalls

It seems that if one choose the light-cone It seems that if one choose the light-cone gauge, the gauge link effect disappears. gauge, the gauge link effect disappears. – FSI can be shifted ENTIRELY to the initial state FSI can be shifted ENTIRELY to the initial state

((advanced boundary conditionadvanced boundary condition). Hence the FSI effects ). Hence the FSI effects must come from the LC wave functions. must come from the LC wave functions.

– LCWF components are not real, they have nontrivial LCWF components are not real, they have nontrivial phase factors!phase factors!

A complete gauge-independent TMD PD A complete gauge-independent TMD PD contains a additional FSI gauge link at contains a additional FSI gauge link at ξξ± ± = = ∞∞ which does not vanish in the light-cone gaugewhich does not vanish in the light-cone gauge– Conjectured by Ji & Yuan (2002)Conjectured by Ji & Yuan (2002)

– Proved by Belitsky, Ji & Yuan (2002)Proved by Belitsky, Ji & Yuan (2002)

The extra FSI gauge linkThe extra FSI gauge link

Through an explicit calculation, one can show Through an explicit calculation, one can show that the standard definition of TMD PD is that the standard definition of TMD PD is modified by an additional gauge link modified by an additional gauge link

– Gauge link arises from the eikonal phase accumulation of final Gauge link arises from the eikonal phase accumulation of final state particle traveling in its trajectory. Although the dominant state particle traveling in its trajectory. Although the dominant phase accumulation is in the light-cone direction, however, the phase accumulation is in the light-cone direction, however, the phase accumulation happens also in the transverse direction. phase accumulation happens also in the transverse direction.

SSA in A Simple ModelSSA in A Simple Model

A proton consists of a scalar diquark and a A proton consists of a scalar diquark and a quark, interacting through U(1) gauge boson quark, interacting through U(1) gauge boson ((Brodsky, Hwang, and Schmidt, PLB, 2002Brodsky, Hwang, and Schmidt, PLB, 2002). ).

The parton distribution asymmetry can be The parton distribution asymmetry can be obtained from calculating Sivers’ function in obtained from calculating Sivers’ function in light-cone gauge (Ji & Yuan)light-cone gauge (Ji & Yuan)

Factorization theoremFactorization theorem

For semi-inclusive DIS with small pFor semi-inclusive DIS with small pTT

~

• Hadron transverse-momentum is generated from multiple sources.• The soft factor is universal matrix elements of Wilson lines and spin-independent.• One-loop corrections to the hard-factor has been calculated

Spin-Dependent processesSpin-Dependent processes

Ji, Ma, Yuan, PLB597, 299 (2004); Ji, Ma, Yuan, PLB597, 299 (2004); PRPRD70:074021(2004)

Additional Structure FunctionsAdditional Structure Functions

Sivers effect

Collins effect

As PAs PTT becomes large… becomes large…

If PIf PTT become hard ( become hard (PT» ΛQCD), so long as Q» PT

the above factorization formula still works!

On the other hand, in this region one can On the other hand, in this region one can calculate the Pcalculate the PTT dependence perturbatively, dependence perturbatively,

– The pThe pTT dependence in the soft factor is easily to dependence in the soft factor is easily to

calculate.. calculate..

– Expanding in parton momentum, one leads to the Expanding in parton momentum, one leads to the followingfollowing

2 2( , ) ( , )T Fq k x k d k T x x

As PAs PTT becomes large… becomes large…

– The pThe pTT dependence in the TMDs can also be calculated dependence in the TMDs can also be calculated

through one-gluon exchange…through one-gluon exchange…

The soft matrix element is the twist-3 matrix The soft matrix element is the twist-3 matrix elements Telements TDD

( , ) ( , )sT Fq k x T x x

k

ˆ( , ) ( , )sFq k x E x x

k

Putting all togetherPutting all together

One should obtain a SSA calculated in Qiu-One should obtain a SSA calculated in Qiu-Sterman approach (H. Eguchi & Y. Koike)Sterman approach (H. Eguchi & Y. Koike)

Therefore, SSA becomes twist-3, JI, Ma, Yuan (to be published)

Relation between TMDs & Twist-3?Relation between TMDs & Twist-3?

The TMD approach for DIS/DY works for both The TMD approach for DIS/DY works for both small and perturbative, but moderate Psmall and perturbative, but moderate PTT. .

– At small PAt small PTT, it is a twist-two effect, it is a twist-two effect

– At moderate PAt moderate PTT, it is a twist-three effect., it is a twist-three effect.

The TMD approach is more general, but not The TMD approach is more general, but not necessary at moderate Pnecessary at moderate PTT

The twist-3 approach works only at large PThe twist-3 approach works only at large PTT, , but is the most economical there!but is the most economical there!

SSA & processesSSA & processes

DIS & Drell-Yanpp -> πX & friends

Hard scale Q2 PT

Small PT~ΛQCD

QCD factorizationIn TMD’s

Non-perturbative

Q2,s» PT» ΛQCDQCD factorizationIn TMD’sTwist-3 effects

QCD factorizationIn TMD’s ?Twist-3 effects

ppp p ππX & friendsX & friends

PPTT must be large so that perturbative QCD must be large so that perturbative QCD works.works.

In this region, it is not need to use the TMD In this region, it is not need to use the TMD formalism. The twist-3 approach is sufficient.formalism. The twist-3 approach is sufficient.

Phases are generated perturbatively.Phases are generated perturbatively.

Perturbative Way to Generate PhasePerturbative Way to Generate Phase

Some propagators in the tree diagrams go on-shell

)(11 22

2222mki

mkP

imk

No loop is needed to generate the phase!

Coulombgluon

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

A possible exception A possible exception

Is it possible that at moderate pIs it possible that at moderate pTT, the intrinsic , the intrinsic transverse-momentum effect is so large that it transverse-momentum effect is so large that it cannot be expanded? cannot be expanded? – Soft function is still perturbative...Soft function is still perturbative...

– One could include the Sudakov form factorsOne could include the Sudakov form factors

I don’t know yet an argument to rule this out. I don’t know yet an argument to rule this out. However, I don’t know an example where this is However, I don’t know an example where this is true.true.

Difficulty:Difficulty:– No proof of factorization (may be it will work!)No proof of factorization (may be it will work!)

– The gauge links on the TMDs might be very complicated The gauge links on the TMDs might be very complicated (both initial and final state interactions are present). (both initial and final state interactions are present).

ConclusionConclusion

For SIDIS/DY with small and moderate For SIDIS/DY with small and moderate transverse momentum, there is a QCD transverse momentum, there is a QCD factorization theorems involving TMDs.factorization theorems involving TMDs.

At moderate PAt moderate P┴┴, one recovers the twist-3 , one recovers the twist-3 mechanism (ETQS).mechanism (ETQS).

For pp->For pp->ππX at perturbative PX at perturbative P┴┴, twist-3 , twist-3 mechanism seems to be complete.mechanism seems to be complete.

One has yet to find a TMD type of factorization One has yet to find a TMD type of factorization for pp->for pp->ππX at perturbative PX at perturbative P┴┴; and the TMD ; and the TMD distributions might not be related to those in distributions might not be related to those in SIDIS/DY.SIDIS/DY.