# Presentation: 202727

## (Single) Transverse Spin Physics-- from DIS to hadron collider

**Feng Yuan**
**RBRC** , **Brookhaven National Laboratory**

**W. Vogelsang, F.Y. , to be submitted**

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## Outline

**Introduction **
**More: RBRC Workshop on SSA (June 1-3) ****http://quark.phy.bnl.gov/~fyuan/workshop/summer05_program.htm**
**SSA in SIDIS, Physics of TMDs**
**SSA at hadron colliders (RHIC)**
**Summary**

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## What is Single Spin Asymmetry?

- Consider scattering of a transversely-polarized spin-1/2 hadron (S, p) with another hadron, observing a particle of momentum k

**S**

**k**

**p**

- The cross section can have a term depending on the azimuthal
- angle of **k**

- which produce an asymmetry AN when **S **flips: **SSA**

**p’**

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## Sample Exp. Data

- A. Bravar et al., E704, PRL77, 2626 (1996)

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## SSA at RHIC

**STAR Collaboration, ****Phys.Rev.Lett.92:171801,2004; hep-ex/0412035  **

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## PHENIX Collaboration, hep-ex/0410003

**PHENIX Collaboration, hep-ex/0410003**

**Central rapidity!!**

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## BRAHMS Collaboration, talk by Videbaek 

**<e>~ -0.035 => ****AN **= -0.08 +- 0.005
- +- [0.015]  in 0.17 < xF < 0.32

**<e>~ +0.022 => ****AN **= +0.05 +- 0.005
- +- [0.015]  in 0.17 < xF < 0.32

**BRAHMS Collaboration, talk by Videbaek **

- BRAHMS preliminary

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## Big SSA!

- Systematics
**A****N**** is significant in the fragmentation region of the polarized beam **
**A****N**** and its sign show a strong dependence on the type of polarized beam and produced particles**
- A related phenomenon: the transverse polarization of spin-1/2 particle in unpolarized hadron scattering.
- G. Bunce et. al, PRL36, 1113 (1976).

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## Why Does SSA Exist? 

**Single Spin Asymmetry is proportional to**
-  **Im (M****N**** * M****F****)**
**where ****M****N**** is the normal helicity amplitude **
**    ****and ****M****F**** is a spin flip amplitude**** **
**Helicity flip**: one must have a reaction mechanism for the hadron to change its helicity (in a cut diagram)
**Final State Interactions**** (FSI)****:** to generate a phase difference between two amplitudes
-    The phase difference is needed because the structure
-     S ·(p × k) formally violate time-reversal invariance

## Naïve Parton Model Fails

- If the underlying scattering mechanism is hard, the naïve parton model generates a very small SSA: (G. Kane et al, PRL41, 1978)
- The only way to generate the hadron helicity-flip is through quark helicity flip, which is proportional to current quark mass mq
- To generate a phase difference, one has to have pQCD loop diagrams, proportional to αS

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## Parton Orbital Angular Momentum and Gluon Spin

- The hadron helicity flip can be generated by other mechanism in QCD
- Quark orbital angular momentum (OAM):

- Beyond the naïve parton model in which quarks are collinear
**Transverse Momentum Dependent PDF!!**
**(TMDs)**

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## Parton OAM and Gluons (cont.)

- A collinear gluon carries one unit of angular momentum because of its spin. Therefore, one can have a coherent gluon interaction

- 1/2

- −1/2

- 1/2

- 1/2

- Quark-gluon quark correlation function!
**Qiu-Sterman Mechanism**

- -1

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## Novel Way to Generate Phase

- Some propagators in the tree diagrams go on-shell

- No loop is needed to generate the phase!

- Coulomb
- gluon

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

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## Comparison With Data

**Qiu & Sterman, PRD59, 014004 (1999)**

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## SSA In Semi-inclusive 

**Deep Inelastic Scattering**
**(TMDs)**

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## Inclusive and Semi-inclusive DIS

- Inclusive DIS:
**Partonic Distribution depending on **
**the longitudinal momentum fraction**

- Semi-inclusive DIS:
**Probe additional information for partons’**
**transverse distribution in nucleon **

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## Different P? Region

**Integrate out ****P****?**
**	****-- similar to inclusive DIS, probe int. PDF**
**Large ****P****? ****(****À********QCD****)**
**	****-- hard gluon radiation, can be calculated from perturbative QCD, **
**    ****Polarized ->q-g-q correlations**
**Low ****P****? ****(****»********QCD****)**
**	****-- nonperturbative information (****TMD****): new factorization formula**

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## TMD Physics

- A way to measure Transversity Distribution, the last **unkown** **leading twist** distribution
-     Collins 1993
- The Novel Single Spin Asymmetries
- Connections with GPDs, and Quantum Phase Space Wigner distributions
- Quark Orbital Angular Momentum and
- 	Many others ...

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## TMD Distribution: the definition

- Gauge Invariance requires the Gauge Link

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

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## This definition is also consistent with the QCD factorization

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## Factorization

**Factorization**

**Soft Factor**

**Ji,Ma,Yuan, 04’**

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## The Factorization Applies to

- Semi-inclusive DIS (polarized and unpolarized)
- Drell-Yan at Low transverse momentum
- Di-hadron production in e+e- annihilation (extract the Collins function)
- Di-jet and/or di-hadron correlation at hadron collider (work in progress)
- Many others, ...

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## Phenomenology

- At current stage, it is difficult to implement the full factorization approach for the phenom. Studies
- As a first step, one may neglect all higher order effects, set S=H=1, forget the  and  in TMDs
- Back to Naïve Parton Model picture
- Mulders & Tangelmann 96

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## SIDIS Cross Sections

**Transversity**

**Collins Function**

**Sivers Function**

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## Asymmetries

- Integrate over the transverse momentum

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## Model for the Sivers functions

- Assume only valence quark Sivers functions
-               uT(1/2)/u=Su x(1-x)
-               dT(1/2)/u=Sd x(1-x)
- GRVLO for the unpolarized quark distribution Kretzer’s LO fragmentation function

**Valence feature**

**Power Suppressed at x->1**

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## Fit to HERMES Data

**S****u****=-0.81****§**** 0.07**
**S****d****=1.86****§**** 0.28**
******2****/d.o.f ****1⁄4**** 1.2**

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## Compare with COMPASS

**Assume the leading **
**hadrons are pions**

**HERMES large positive for ********+****, and almost zero for ********-****, **
**there is strong cancellation between u and d quark Sivers **
**function to explain the HERMES data!!**

## A Conjecture for Collins Function

- By summing up all hadrons, any quark Collins function vanishes,
- Due to quark-hadron duality, the fragmentation function to all hadrons equals to the fragment. to quark (quark+gluons) state, where the naïve T-odd effect is suppressed by quark mass
- Integrated over z with k? momentSchafer-Teryaev sumrule

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## Consequence on unfavor/favor

- Isospin/charge symmetry
- Under the above conjecture, neglecting Keons

**Support from string picture for fragmentation, Artzu**
**Also N. Makins’ talks**

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## Model for Collins Functions

- Two sets of parameterizations
-                     **Collins function vanishes at z********0**
**	****	****	****  ****(1-z) comes from Collins 93’**
- Transversity functions use the parameterization of Martin, Schafer, Stratmann, and Vogelsang, PRD 57(1998)

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## Set I Fit to HERMES

**C****f****=**** ****0.****29****§**** 0.0****4**
**C****d****=****-0.33****§**** 0.****04**
******2****/d.o.f ****1⁄4**** 0.8**

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## Compare to COMPASS

**Assume the leading **
**hadrons are pions**

**Discrepancy?**

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## Set II Fit

**C****f****=**** ****0.****29****§**** 0.0****2**
**C****d****=****-0.56****§**** 0.****07**
******2****/d.o.f ****1⁄4**** 0.7**

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## Compare to COMPASS

**Assume the leading **
**hadrons are pions**

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## Fitted Collins Functions

**Set I**

**Set II**

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## Transversity functions in the fit

**Unpolarized quark**
**distributions**

**If we get Collins functions from other processes, e.g., e^+e^-, **
**we can constrain the transversity functions !!**

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## TMDs at RHIC

- Drell-Yan
-    SSA for Drell-Yan: Sivers function has opposite sign, qTDY=**-**qTDIS, because of the gauge link changing direction.
- Di-Jet Correlation
- 	There is no factorization proof yet. It is likely factorizable in terms of TMDs. However, the universality of Sivers function for this case is not clear yet. We assume they are the same as DY.

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## SSA for Drell-Yan

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## Asym. Jet correlation probe Gluon Sivers function at RHIC

- Boer, Vogelsang, PRD69:094025,2004

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## Di-jet Correlation

******1**

******2**

**Jet1: P****1****?**

**Jet2: P****2****?**

**q****T****(1/2)****(x)**

**q****T****(1)****(x)**

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## cos Asymmetry

**Gluon Sivers**

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## Remarks on sin term

- It is inclusive Di-jet cross section, depending on the azimuthal angle between the di-jet and the polarization vector S?
- Remember that Qiu-Sterman twist-three also contributes to the Di-jet sin asymmetry
- The connection between the above two terms needs to be further investigated

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## Summary

- The simple parameterizations of the Sivers and Collins functions fit the HERMES data very well. Future data from both HERMES and COMPASS should solve the possible “discrepancy”
- The SSAs for Drell-Yan and Di-jet correlation were predicted.
- Further studies for Di-hadron correlation including Collins asymmetry should be carried out, and also the factorization

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