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 hadrons9 / 14 May 2005, Trento

2Contents

• Factorization and universality

• Wilson lines in transverse momentum dependent parton distribution functions

• Experimental input

• Interpretation of the Sivers effect

3Factorized 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

4Diagrammatic 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

5Diagrammatic 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)

semi-inclusive DIS at leading order in M/Q

+ + + + ... =

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

6Leading 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:

7The gauge link in Drell-Yan

Resumming the same class of interactions leads alsoto gauge links in the distribution functions

The result can also be written in a factorized form, but it contains different functions. What about universality??

8Universality 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

9Higher order in SIDIS

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

10Higher order in Drell-Yan

What is happening here?

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

11Evolution 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

12Factorization

• 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

13How 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

14 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.B567:27,2003Bacchetta, Mulders, F.P., Phys.Lett.B595:309 (2004)

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

15Interpretation 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?

Sivers effect

• L-R asymmetry• gluon-magnetic field only

nonzero in hadron, vector fields provide the effect

• topological effect (1, 5/4,etc)

16Summary

• 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

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