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 & non-perturbative,pp to pp at low energy: non-perturbativeWhat 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 spin-averaged cross sections

SSA & processesDIS & Drell-Yanpp -> X & friendsHard scaleQ2PTSmall PT~QCDQCD factorizationIn TMDsNon-perturbativeQ2,s PT QCDQCD factorizationIn TMDsTwist-3 effectsQCD factorizationIn TMDs ?Twist-3 effects

SIDIS at low pT Single-jet 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 twist-2 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 one-loopVertex 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 light-cone gauge link is

the usual parton distribution may be regarded as

ClassificationThe leading-twist ones are classified by Boer, Mulders, and Tangerman (1996,1998)There are 8 of them, corresponding to the number of quark-quark scattering amplitudes without T-constraint

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 transverse-momentum-dependent 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 fromnon-perturbative structure physics.

Physics of a Sivers FunctionHadron helicity flipThis can be accomplished through non-perturbative 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 time-reversal (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.

Light-Cone Gauge PitfallsIt seems that if one choose the light-cone 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 gauge-independent TMD PD contains a additional FSI gauge link at = which does not vanish in the light-cone 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 light-cone 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 light-cone gauge (Ji & Yuan)

Factorization theoremFor semi-inclusive DIS with small pT

~ 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 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 one-gluon exchange

The soft matrix element is the twist-3 matrix elements TD

Putting all togetherOne should obtain a SSA calculated in Qiu-Sterman approach (H. Eguchi & Y. Koike)Therefore, SSA becomes twist-3, JI, Ma, Yuan (to be published)

Relation between TMDs & Twist-3?The TMD approach for DIS/DY works for both small and perturbative, but moderate PT. At small PT, it is a twist-two effectAt moderate PT, it is a twist-three effect.The TMD approach is more general, but not necessary at moderate PTThe twist-3 approach works only at large PT, but is the most economical there!

SSA & processesDIS & Drell-Yanpp -> X & friendsHard scaleQ2PTSmall PT~QCDQCD factorizationIn TMDsNon-perturbativeQ2,s PT QCDQCD factorizationIn TMDsTwist-3 effectsQCD factorizationIn TMDs ?Twist-3 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 twist-3 approach is sufficient.Phases are generated perturbatively.

Perturbative Way to Generate PhaseSome propagators in the tree diagrams go on-shellNo 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 transverse-momentum 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 twist-3 mechanism (ETQS).For pp->X at perturbative P, twist-3 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.