1. Single spin asymmetries in semi-inclusive DIS
Semi-Inclusive Reactions Workshop 2005
May 18 – 20, Jefferson Lab, Newport News
Vrije Universiteit Amsterdam
Single spin asymmetries in semi-inclusive DIS
Fetze Pijlman
in collaboration with
D. Boer
C.J. Bomhof
P.J. Mulders

2. Outline Factorization, universality and Wilson lines
Experimental input
Interpretation of the Sivers effect

3. 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 in SIDIS, Drell-Yan (and e+ e- annihilation):
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)

4. Separation of longitudinally polarized gluons and Wilson lines
Diagrammatic expansion
Take all possible interactions into account
Separation of longitudinally polarized gluons
Ward identities
gauge link
F.P. (2004) Brodsky, Hwang, Schmidt (2002)

5. Wilson lines at tree-level in SIDIS and Drell-Yan
matrix elements
integrated:
unintegrated:
gauge links in SIDIS
integrated:
unintegrated:
gauge links in Drell-Yan
integrated:
unintegrated:
outgoing particle-lines yield gauge link via (+) infinity
incoming particle-lines yield gauge link via (-) infinity
T-even/T-odd:
T-odd:
Efremov, Radyushkin (1981) Belitsky, Ji, Yuan (2003) Boer, Mulders, Pijlman (2003)

6. Wilson lines in fragmentation functions
SIDIS
incoming quark: link via - infinity
e+ e- annihilation
outgoing quark: link via + infinity
Time reversal
We have no relation between the fragmentation functions
of the two processes, but of course there could be one.
Boer, Mulders, Pijlman (2003)
Metz (2002) Gamberg, Goldstein (2003) Collins, Metz (2004)

7. Wilson lines at higher orders
results:
results:
Is evolution process-dependent? Universality & factorization?
Bomhof, Mulders, F.P. (2004) F.P. (2004)

8. Factorization & universality
general result: gauge links in a hadron change if there are other
incoming and outgoing QCD (bound) states
possible appearance of loops:
fragmentation in SIDIS
distributions in pp scattering
no loops:
distribution functions in SIDIS
e+ e- annihilation
probably safe observables
integrated SIDIS and Drell-Yan
“two hadron production” in DIS, PP scattering, e+ e- annihilation
if fragmentation is link independent, then unintegrated SIDIS
parts of the azimuthal asymmetries in Drell-Yan in which T-odd effects do not contribute

9. 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 (Drell-Yan / SIDIS), 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

10. Lorentz invariance relations
Vector dependence also leads to new functions
Beam-spin-asymmetry for the jet (neglecting quark masses)
Goeke et al. (2003) Bacchetta, Mulders, F.P. (2004)

11. Speculations on 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)
Sivers effect
L-R asymmetry
gluon-magnetic field only nonzero in hadron, vector fields provide the effect
topological effect (1, 5/4,etc)
The first QCD analogue of the Aharanov-Bohm effect?

12. 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