Single Transverse-Spin Asymmetries and Single Transverse-Spin Asymmetries and Twist-3 FactorizationTwist-3 Factorization

J.P. Ma, ITP, BeijingJ.P. Ma, ITP, Beijing

Weihai, 2011.08.08

Content:

1. Introduction2. Partonic states and SSA3. Collinear Factorization of Partonic Results4. Summary

1. Introduction Single transverse-Spin Asymmetries(SSA) are asymmetries in the case where one initial hadron or one produced hadron is transversely polarized. Taking Drell-Yan processes as an example:

The initial hadron is transversely polarized.

SSA can only be generated if there exist scattering absorptive parts in scattering amplitudes AND helicity-flip interactions. Studies of SSA provide ways to explore the structure of hadrons. Theoretical concept: Factorization, if Q2 is large. Collinear factorization with twist-3 operators TMD factorization, if the transverse momentum of the lepton pair is small. So far, all factorizations are derived with the diagram approach at tree-level.

The diagramatic approach at hadron level:

Quark density matrix of B

Quark-Gluon density matrix of A

Hard scattering

Expanding momenta of incoming partons collinearly, one derives the factorized form in the case:

All A’s are perturbative functions. TF is the Qiu-Sterman matrix elements:

Hard pole contribution

Soft-gluon-pole contribution

It is a factorization involving twist-3 operators. Three partons enter the hard scattering, and the gluon can be soft with zero momentum. This gives the so called soft-gluon pole contribtutions. Note: The derivation is not a standard calculation of standard scattering amplitudes. Let’s look at the familiar factorization of DIS:

We consider here only the quark sector. To determine the hard part, one can replace the initial hadron with a quark. Then everything can be calculated perturbatively.

At tree –level:

At one-loop level:

The collinear divergence in F2 is the same as that in the first term, so that H does not contain it. This is the sense of factorization.

Important: The collinear divergence at one-loop in F2 is “determined” by the tree-level H…..

Can we do the same for SSA?? Yes or No……??

If we replace the hadron A with a transversely polarized quark, one can not have a nonzero SSA, because helicity conservation of QCD. One needs to consider multi-parton states for the replacement. The talk presents a study by using partonic states to der

ive the factorization of SSA.

2. Partonic states and SSATransverse spin corresponds to the non-diagonal part of spin density matrix in helicity space. Define a spin ½ state as:

Using this state to replace the transversely polarized hadron A, one will get nonzero non-diagonal part of spin density matrix because of the interference between the single quark- and the quark-gluon state. E.g., at tree-level:

It is nonzero!

One can construct more multi-parton states to study the factorization. One can follow the procedure, discussed for DIS, to derive the factorization by calculating the spin-dependent cross-section and those pdf’s and twist-3 matrix elements.

We will replace the hadron A with the multi-parton state, the hadron B with an antiquark for our purpose. We consider the kinematic region:

3. Collinear Factorization of Partonic ResultsAt tree-level only one diagram:

The absorptive part is generated by cutting the quark propagator.

With this and the tree-level result of twist-3 matrix element one finds:

With tree-level results one can only find the hard pole contribution. The soft-gluon pole contributions can not be derived at tree-level, because one can not define a state of gluon with zero-momentum.

One has to go beyond tree-level…..

Adding one-gluon to the tree-level diagram:

The gluon can be soft (Glauber gluon) If this gluon is collinear to the incoming gluon

Therefore, the contribution with the soft gluon is collinearly divergent.

And: This divergence can not be factorized with the tree-level H of the hard pole contribution, because the diagram is not a simple correction of the tree-level diagram……

This type of contributions should be factorized in another way as:

= H

The soft gluon

The collinear gluon

Calculating all those diagrams with the collinear gluons , one finds everything:

Observation: Although soft-pole contributions to the structure function are at one-loop, but their hard parts are at the same order of the hard part of the hard pole contribution at tree-level.

Part of discrepancies of the evolution is solved with our result of the twist-3 matrix element.

Detailed results and discussions can be found: J.P. Ma and H.Z. Sang, JHEP 1104:062,2011. e-Print: arXiv:1102.2679 [hep-ph] H.G. Cao, J.P. Ma and H.Z. Sang, Commun.Theor.Phys.53:313-324,2010. e-Print: arXiv:0901.2966 [hep-ph]

In the full kinematic region:

Hard-pole contribution Soft-gluon pole contributions Soft-quark pole contributions Soft-gluon pole contributions of three-gluon correlator Two quark-quark-gluon twist-3 matrix elements Two gluon-gluon-gluon twist-3 matrix elements So far no completed results…..

4. Summary ● Using multi-parton states, one can study the factorization of SSA by calculating parton scattering amplitudes. ● There is a nontrivial order-mixing between hard-pole- and soft-pw can sole contributions. ● One can systematically derive the factorization for other processes.