Single (transverse) Spin Asymmetry & QCD Factorization
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Single (transverse) Spin Asymmetry & QCD FactorizationXiangdong JiUniversity of Maryland Workshop on SSA, BNL, June 1, 2005

Outline General RemarksDIS/Drell Yan processes pp X & friendsSummary

Single (transverse) Spin AsymmetrySSA is a general phenomenon in physics, and it exists so long as there areA single transverse spin A mechanism for helicity flipInitial and/or state interactionsOk, SSA is an interesting phenomenon, but what do you learn about QCD from it? or Why do we have to spend $ & time to measure it?We have some models that fit the data (who cares about models? we have QCD) We learn something about the nucleon spin structure (what exactly do you learn? And why that is interesting? can you check it in lattice QCD? What is it missing if we dont measure it?).

Pertubative & Nonperturbative MechanismsIn general, however, the physics mechanism for SSA in strong interactions can be either be perturbative & nonperturbative,pp to pp at low energy: nonperturbativeWhat one would like to understand is the SSA in perturbative region=> we hope to learn something simple, maybe!There must be some hard momentum:Perturbative description of the cross section must be valid Factorization A good description of spinaveraged cross sections

SSA & processesDIS & DrellYanpp > X & friendsHard scaleQ2PTSmall PT~QCDQCD factorizationIn TMDsNonperturbativeQ2,s PT QCDQCD factorizationIn TMDsTwist3 effectsQCD factorizationIn TMDs ?Twist3 effects

SIDIS at low pT Singlejet production
If the target is transversely polarized, the current jet with a transverse momentum kT has a SSA which allows a QCD factorization theorem even when kT is on the order QCDThe SSA is of order 1 in the scaling limit, i.e. a twist2 effect!
q

Factorization for SIDIS with PMust consider generic Feynman diagrams with partons having transverse momentum, and gluon loops.We have two observable scales, Q and P (soft). We consider leading order effects in P /Q. The gluons can be hard, soft and collinear. Can one absorb these contributions into different factors in the cross sections. X. Ji, F. Yuan, and J. P. Ma, PRD71:034005,2005

Example at oneloopVertex 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)ppqk

A general reduced diagramLeading contribution in p /Q.

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

TMD parton distributionsThe unintegrated parton distributions is defined as
where the lightcone gauge link is
the usual parton distribution may be regarded as

ClassificationThe leadingtwist ones are classified by Boer, Mulders, and Tangerman (1996,1998)There are 8 of them, corresponding to the number of quarkquark scattering amplitudes without Tconstraint
q(x, k), qT(x, k) (sivers), qL(x, k), qT(x, k), q(x, k), Lq(x, k), Tq(x, k), Tq(x, k)
Similarly, one can define fragmentation functions

Sivers FunctionA transversemomentumdependent parton distribution which builds in the physics of SSA!
SkPThe 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 fromnonperturbative structure physics.

Physics of a Sivers FunctionHadron helicity flipThis can be accomplished through nonperturbative mechanics (chiral symmetric breaking) in hadron structure.The quarks can be in both s and p waves in relativistic quark models (MIT bag).FSI (phase)The hadron structure has no FSI phase, therefore Sivers function vanish by timereversal (Collins, 1993)FSI can arise from the scattering of jet with background gluon field in the nucleon (collins, 2002)The resulting gauge link is part of the parton dis.

LightCone Gauge PitfallsIt seems that if one choose the lightcone gauge, the gauge link effect disappears. FSI can be shifted ENTIRELY to the initial state (advanced boundary condition). Hence the FSI effects must come from the LC wave functions. LCWF components are not real, they have nontrivial phase factors!A complete gaugeindependent TMD PD contains a additional FSI gauge link at = which does not vanish in the lightcone gaugeConjectured by Ji & Yuan (2002)Proved by Belitsky, Ji & Yuan (2002)

The extra FSI gauge linkThrough an explicit calculation, one can show that the standard definition of TMD PD is modified by an additional gauge link
Gauge link arises from the eikonal phase accumulation of final state particle traveling in its trajectory. Although the dominant phase accumulation is in the lightcone direction, however, the phase accumulation happens also in the transverse direction.

SSA in A Simple ModelA proton consists of a scalar diquark and a quark, interacting through U(1) gauge boson (Brodsky, Hwang, and Schmidt, PLB, 2002). The parton distribution asymmetry can be obtained from calculating Sivers function in lightcone gauge (Ji & Yuan)

Factorization theoremFor semiinclusive DIS with small pT
~ Hadron transversemomentum is generated from multiple sources. The soft factor is universal matrix elements of Wilson lines and spinindependent. Oneloop corrections to the hardfactor has been calculated

SpinDependent processesJi, Ma, Yuan, PLB597, 299 (2004); PRD70:074021(2004)

Additional Structure FunctionsSivers effectCollins effect

As PT becomes largeIf PT become hard (PT QCD), so long as Q PT the above factorization formula still works! On the other hand, in this region one can calculate the PT dependence perturbatively, The pT dependence in the soft factor is easily to calculate..
Expanding in parton momentum, one leads to the following

As PT becomes largeThe pT dependence in the TMDs can also be calculated through onegluon exchange
The soft matrix element is the twist3 matrix elements TD

Putting all togetherOne should obtain a SSA calculated in QiuSterman approach (H. Eguchi & Y. Koike)Therefore, SSA becomes twist3, JI, Ma, Yuan (to be published)

Relation between TMDs & Twist3?The TMD approach for DIS/DY works for both small and perturbative, but moderate PT. At small PT, it is a twisttwo effectAt moderate PT, it is a twistthree effect.The TMD approach is more general, but not necessary at moderate PTThe twist3 approach works only at large PT, but is the most economical there!

SSA & processesDIS & DrellYanpp > X & friendsHard scaleQ2PTSmall PT~QCDQCD factorizationIn TMDsNonperturbativeQ2,s PT QCDQCD factorizationIn TMDsTwist3 effectsQCD factorizationIn TMDs ?Twist3 effects

pp X & friendsPT must be large so that perturbative QCD works.In this region, it is not need to use the TMD formalism. The twist3 approach is sufficient.Phases are generated perturbatively.

Perturbative Way to Generate PhaseSome propagators in the tree diagrams go onshellNo loop is needed to generate the phase!CoulombgluonEfremov & Teryaev: 1982 & 1984Qiu & Sterman: 1991 & 1999

A possible exception Is it possible that at moderate pT, the intrinsic transversemomentum effect is so large that it cannot be expanded? Soft function is still perturbative...One could include the Sudakov form factorsI dont know yet an argument to rule this out. However, I dont know an example where this is true.Difficulty: No proof of factorization (may be it will work!) The gauge links on the TMDs might be very complicated (both initial and final state interactions are present).

ConclusionFor SIDIS/DY with small and moderate transverse momentum, there is a QCD factorization theorems involving TMDs.At moderate P, one recovers the twist3 mechanism (ETQS).For pp>X at perturbative P, twist3 mechanism seems to be complete.One has yet to find a TMD type of factorization for pp>X at perturbative P; and the TMD distributions might not be related to those in SIDIS/DY.