# Presentation: 103211

## Understanding Single Spin Asymmetry in QCD

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

- RBRC Seminar

- Jan. 13, 2005

## Outline

**Introduction: what is ****SSA?**
**Why naïve parton model fails**
**Quark-gluon correlation and Qiu-Sterman mechanism**
**Sivers function and ****SSA**** in SIDIS**
**Sivers function at RHIC**
**What we learn from ****SSA?**
**Summary**

- RBRC Seminar

- Jan. 13, 2005

## 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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- Jan. 13, 2005

## 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).

- RBRC Seminar

- Jan. 13, 2005

## 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
-     Therefore a generic pQCD prediction goes like
-          AN ~ αS mq/Q  less than 0.1 per cent
***Every factor suppresses the SSA*****!**

## Parton Orbital Angular Momentum and Gluon Spin

- The hadron helicity flip can be generated by other mechanism in QCD
- Quark orbital angular momentum (OAM): Therefore, the hadron helicity flip can occur without requiring the quark helicity flip.

- Beyond the naïve parton model in which quarks are collinear

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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!

- -1

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## What are the consequences?

- The correlations effect replaces the current quark mass by a typical QCD scale ΛQCD
-            **A****N   ****~ ****α****S**** ****Λ****QCD****/Q**
-     which now has a size at percent level.
- How to generate phase?
- Perturbative QCD loops
- A new possibility: *through the tree diagrams*. Scattering of initial and final state partons in the Coulomb field of the polarized nucleon.
-      **A****N   ****~  ****Λ****QCD****/Q**:   tens of percent level!

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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**

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## In Deep Inelastic Scattering, HERMES, A. Airapetian et al., 

**hep-ex/0408013**,

**<Q****2****>**** ****=2.41GeV****2****, <P****?****> =0.41GeV**
**0.023<x<0.4,   0.2<z<0.7**

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## Semi-inclusive DIS

**Semi-inclusive DIS:**
**Probe additional information including the transverse distribution **

** ****The ****Sivers**** function leads to SSA in order of 1 in the **
**   ****scaling (Bjorken) limit, i.e., twist-2 effect**
** ****Transverse momentum could be small on the order**
**   ****of ********QCD****, require new factorization theorem****.**

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## What is Sivers function

**Sivers function is the asymmetric part of k****?**** distribution in a transversely polarized hadron**
**It generates novel ****SSA**** in SIDIS and hadron-hadron collisions**

**K****?**

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## Physics of Sivers function

**Hadron helicity flip**
**	****quarks can be in S- and P-waves**
**Final state interactions (FSI)**
**The hadron structure has no FSI phase, therefore Sivers function vanishes by time-reversal (Collins, 1993)**
**There is final state interaction leading to single spin asymmetry at leading twist in SIDIS (Brodsky et al. 2002) **
***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.**

**Non-perturbative hadron structure involving**

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## Physical origin of the gauge link

**Factorizable Final State Interactions**

**Parton distributions with full gauge link**

- n

- n

- n

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## TMD Parton Distributions

**The gauge invariant definition**

**Ji, Yuan (02)**
**Belitsky, Ji, Yuan (03)**

**Light-cone gauge: **
**Additional Gauge link **
**at ********-****=****§1**

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## These definitions are also consistent with QCD factorization

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## SIDIS Cross Section: QCD Factorization 

**At leading power of 1/Q**

**Sivers Asym**.

**Collins Asym.**

- Ji, Ma, Yuan, hep-ph/0404183,hep-ph/0405085

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## How does factorization work?

- Four possible regions for the gluon momentum k:
-   1) k is collinear to p (parton distribution)
-   2) k is collinear to p′ (fragmentation)
-   3) k is soft (Soft factor)
-   4) k is hard (pQCD correction)

- p

- p′

- k

**Vertex corrections at one-loop (single quark target)**

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## One-Loop Factorization (cont.)

- Three reagions for the gluon momentum k:
-   1) k is collinear to p (parton distribution)
-   2) k is collinear to p′ (fragmentation)
-   3) k is soft  (Soft factor)

- p

- p′

- k

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## Model calculations

- Bag model calculation of
- The Sivers function, F. Yuan
- Phys. Lett. B575, 45 (2003):
- In the bag model, quarks are
- In S- and P-wave states, and
- The quark orbital angular
- Momentum contributes to 40%
- Of the proton spin
- Exp. Data: HERMES,
- Hep-ph/0408013

- RBRC Seminar

- Jan. 13, 2005

## SSA at Large P? in SIDIS(Sivers function and twist-3 correl.)

- Sivers function from Qiu-Sterman mechanism
- It is not suppressed by 1/Q, but by 1/P?
- eRHIC and/or EIC can measure this asymmetry,  plus many others, and provide information on quark-gluon correlations in nucleon

**Ji, Ma, Yuan, in progress**

**Quark-gluon correlations**

## Transition from Perturbative region  to Nonperturbative region?

- Compare different region of P?

**Nonperturbative TMD**

**Perturbative region**

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## Sivers function at RHIC

- RBRC Seminar

- Jan. 13, 2005

## Opportunities at RHIC

- Di-jet correlation  (Boer & Vogelsang, 2003)
- Heavy quark/quarkonium
- Di-photon, di-hadron, hadron+photon, ...

**Quark Sivers function**

**Gluon and quark Sivers function**

- SSA for Drell-Yan: Sivers function has opposite sign, qTDY=**-**qTDIS, because of the gauge link changing direction.

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

- Boer, Vogelsang, PRD69:094025,2004

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## What do we learn from SSA?

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## Nonzero Sivers function probes 

**Quark Orbital Angular Momentum**
-     e.g, Siver’s function ~ the wave function amplitude with orbital angular momentum! Vanishes if quarks only in s-state!
-     Friends:
- Pauli Form Factor F2(t)
- Spin-dependent structure function g2(x)
- Generalized Parton Distribution E(x, ξ, t)

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- Jan. 13, 2005

## Lz≠0 Amplitude and Sivers Function 

- All distributions can be calculated using the wave function. The amplitudes are not real because of FSI.  Siver’s function:
- Similar expressions for F2(Q), g2(x) and E(x,t)

- Ji, Ma, Yuan, Nucl. Phys. B (2003)

- Lz=0

- Lz=1

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## A perturbative QCD calculation of F2(Q)

- Belitsky, Ji & Yuan,
- PRL, 2003

- It predicts that F2 goes like
- (ln2Q2)/Q6 and hence
- F2/F1 ~ (ln2Q2)/Q2

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- Jan. 13, 2005

## Spin Structure of the Proton

- Pin down the Lz≠0 component of the proton wave function is very important. It can help us to determine how much of the nucleon spin is carried by orbital angular momentum.
- It also help to determine to what extent, the proton is not spherically symmetric (or deformed).
- It is wonderful to see that all seemingly unrelated physics are tied together after all.

- RBRC Seminar

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## Summary

- SSA in hadronic reactions provides much more information about nucleon structure: especially the quark orbital angular momentum, and the quark-gluon quark correlations.
- A QCD factorization has been proved for Semi-inclusive DIS
- Two mechanisms generating SSA in QCD merge at large P? SSA in SIDIS or Drell-Yan

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## TMD Gluon Distributions

- The definition

**Gauge Link**

- Fa=  Aa - Aa-gfabc Ab Ac

- RBRC Seminar

- Jan. 13, 2005