# Presentation: 700736

## Single spin asymmetries in hard scattering processes

- Vrije Universiteit Amsterdam

- Fetze Pijlman

- D. Boer
- C.J. Bomhof
- P.J. Mulders

- in collaboration with

- Partonic structure of hadrons
- 9 / 14 May 2005, Trento

## Contents

- Factorization and universality
- Wilson lines in transverse momentum dependent parton distribution functions
- Experimental input
- Interpretation of the Sivers effect

## Factorized approach

- Cross-sections are usually
-      expressed as:
- Questions: Which observables allow for a factorized description?
- 	                  Are the parton distributions universal?
- Recent claims on transverse momentum dependent factorization:

| (DY / SIDIS / e+ e- ) | Factorization & universality |
| --- | --- |
| Ji, Ma, Yuan (2004) | yes |
| Bomhof, Mulders, Pijlman (2004) | unclear due to gauge link |
| Pijlman (2004) | looks problematic for some processes |
| Collins, Metz (2004) | yes |
| Ji, Ma, Yuan (2005) | probably yes |

## Diagrammatic approach

- A field-theory-based approach to hard scattering processes in which hadrons are observed in initial and final state
- Extension of the parton model
- Valid in perturbation theory
- The diagrammatic approach does not
-      make any assumption on factorization
-      or universality

## Diagrammatic approach  in semi-inclusive DIS

- Resummation of a class of interactions leads to a gauge link in parton distributions
- fragmentation functions also contain a gauge link (via minus infinity)

- Efremov Radyushkin, Theor.Math.Phys. 44 (1981) 774,     Belitsky Ji, Yuan, Nucl.Phys.B656 (2003) 165,                   Boer, Mulders, F.P., Nucl.Phys. B667 (2003) 201

## Leading order cross-section of SIDIS

- The other interactions are believed to be       suppressed; they need to be worked out and verified.
- At this order the cross-section appears to be factorized:

## The gauge link in Drell-Yan

- Resumming the same class of interactions leads also
- to gauge links in the distribution functions
- The result can also be written in a factorized form, but it contains different functions. What about universality??

## Universality relations

**Transverse momentum dependent distribution functions**
- T-even/T-odd:
- Matrix element T-odd:
**Transverse momentum dependent fragmentation functions**
- in semi-inclusive DIS via (-) infinity
- in electron-positron annihilation via (+) infinity
- there is no relation between the two (unless some higher symmetry)
- problem for extracting transversity via Collins effect

- Boer, Mulders, F.P., Nucl.Phys. B667 (2003) 201

- Metz, Phys.Lett.B549 (2002) 139

## Higher order in SIDIS

- Bomhof, Mulders, F.P., Phys.Lett.B596 (2004) 277

## Higher order in Drell-Yan

- Bomhof, Mulders, F.P., Phys.Lett.B596 (2004) 277

## Evolution of parton distributions

**Semi-inclusive DIS**
- Real:
- Virtual:

**Drell-Yan**
- Real:
- Virtual:

- Is the evolution of (transverse momentum dependent)
- distribution functions is process-dependent?

- not a peculiarity
-  of QCD
- also appears in QED

## Factorization

- For general tree-level diagrams the gauge link can be calculated with a straight-forward algorithm. The gauge link turns out to be diagram-dependent
- At first order the gauge link is not modified by virtual corrections
- Gauge links in distribution functions get modified                under gluon radiation if there are two incoming                 hadrons (such as pp scattering)
- Gauge links in fragmentation functions get modified                      if there is an incoming hadron (like semi-inclusive DIS, but not in electron-positron annihilation)

- F.P.,  hep-ph/0409332 & hep-ph/0411307

## How to tackle the problems experimentally?

- Measure the Sivers function        , and Boer-Mulders function
-     ( if zero ):   no effect from gauge link, factorization & universality
-      if nonzero: factorization & universality needs to be studied
- Compare T-odd distribution functions in different processes,         is there a sign change or more?
- Compare transverse momentum dependent fragmentation functions
-      T-even          and T-odd
- Measure                  in jet-production in electron-proton scattering

##           in jet-production

- Gauge link in the correlator introduces an extra vector, violating
-      the *Lorentz invariance relations*
- This extra vector leads at twist 3 to new functions for transversely polarized hadrons and unpolarized hadrons,
- Beam-spin asymmetry for the jet

- Goeke, Metz, Pobylitsa, Polyakov, Phys.Lett.B567:27,2003
- Bacchetta, Mulders, F.P., Phys.Lett.B595:309 (2004)

- Metz, Schlegel, Eur.Phys.J.A. 22:489 (2004)
- Goeke, Metz, Schlegel, hep-ph0504130

## Interpretation of the Sivers effect

**Aharanov-Bohm effect**
- L-R asymmetry
- magnetic fields only    nonzero in solenoid, vector fields provide the effect
- topological effect (1, 2, etc)

- The first QCD analogue of the Aharanov-Bohm effect?

## Summary

- Transverse momentum dependent distribution and fragmentation functions contain rich information on the structure of the nucleon and theoretical progress is being made
- Access to non-perturbative physics with interesting interpretations
- Gauge links are essential for understanding SSA’s
-      (such as SIDIS, Drell-Yan, and also p+p ->  + X)
- In general gauge loops can appear in processes and will probably be an essential ingredient in considerations on factorization

## Slide 17