1. Single-Transverse Spin Asymmetries in Hadronic Scattering
Werner Vogelsang (& Feng Yuan)
BNL Nuclear Theory
ECT, 06/13/2007

2. Mostly based on: X. Ji, J.W. Qiu, WV, F. Yuan,
Phys. Rev. Lett. 97, (2006)
Phys. Rev. D73, (2006)
Phys. Lett. B638, 178 (2006)
C. Kouvaris, J.W. Qiu, WV, F. Yuan,
Phys. Rev. D74, (2006)
( C. Bomhof, P. Mulders, WV, F. Yuan,
Phys. Rev. D75, (2007) )
J.W. Qiu, WV, F. Yuan,
arXiv: [hep-ph] (Phys. Lett. B, to appear)
arXiv: [hep-ph]

3. Outline: • Introduction • Single-spin asymmetries in pp  hX
• How are mechanisms for Single-spin asymmetries
related ?
• Conclusions

4. I. Introduction

5. • SSA for single-inclusive process
 example: pp  X L R 
 a single large scale (pT)
 power-suppressed
 collinear factorization (Efremov,Teryaev / Qiu,Sterman TF)
• SSA with small & measured qT , large scale Q
 examples: typical AN measured in lepton-scattering,
“back-to-back” jets in pp
 need not be suppressed with 1/Q
 may have TMD factorization (Sivers & other fcts.)

6. II. Asymmetry in pphX

7. L R 
E704
STAR

8. collinear factorization
Brahms
y=2.95
STAR

9. STAR

10. • typically, hard-scattering calculations based on
LO/NLO fail badly in describing the cross section
√s=23.3GeV
Apanasevich et al.
Bourrely and Soffer
 Resummation of important higher-order corrections beyond NLO de Florian, WV

11. higher-order corrections beyond NLO ?
de Florian, WV
higher-order corrections beyond NLO ?
“threshold” logarithms
Real emission inhibited
Only soft/collinear gluons allowed

12. expect large enhancement !
Mellin moment in
Leading logarithms
expect large enhancement !
de Florian, WV

13. de Florian, WV
E706

14. WA70
Effects start to become visible at S=62 GeV…
Rapidity dependence ? Spin dependence ?

15. ~ Im • Kane, Pumplin, Repko ‘78 In helicity basis: _ + transversity +
• lesson from this: AN in pph X is power-suppressed !

16. • power-suppressed effects in QCD much richer than
just mass terms (Efremov,Teryaev; Qiu,Sterman;
Kanazawa, Koike) _ x1 x2
x2-x1

17. • ingredients:
Collinear factorization.
quark-gluon correlation
function TF(x1, x2)
provides helicity flip
unpol. pdf x1 x2
x2-x1
Phase from imaginary part of propagator
~ i (x1-x2) (soft-gluon-pole contributions)

18. • full structure:
Qiu,Sterman
Transversity
Kanazawa,Koike

21. Position of pole may
depend on
k of initial partons FS IS

22. “derivative terms”
• plus, non-derivative terms !
Qiu & Sterman argue:
At forward xF , collisions are asymmetric:
large-x parton hits
“small-x” parton
 TF (x, x) mostly probed at relatively large x

23. xF=0.15
xF=0.4

24. • derivative terms only
Assumptions in Qiu & Sterman :
• derivative terms only
• valence TF only,
• neglect gluonpion fragmentation
In view of new data, would like to relax some of these.
Kouvaris, Qiu, Yuan, WV

25. Remarkably simple answer:
Recently: proof by Koike & Tanaka

26.  for RHIC, use data with pT>1 GeV
 Ansatz:
usual pdf
 Fit to E704, STAR, BRAHMS
 for RHIC, use data with pT>1 GeV
for E704, choose pT=1.2 GeV
allow normalization of theory to float (~0.5)

27. Fit I: “two-flavor / valence”
Fit II: allow sea as well

28. solid: Fit I, dashed: Fit II

30. Our TF functions:

31. pT dependence

32. Dependence on RHIC c.m.s. energy:

33. III. How are the mechanisms for
single-spin asymmetries related ?

34. Q: In what way are mechanisms connected ?
• have two “mechanisms”
• tied to factorization theorem that applies
Q: In what way are mechanisms connected ?
• Boer, Mulders, Pijlman
• see interplay of mechanisms in a physical process ?

35. TF “Unification” / Consistency of formalisms
• consider Drell-Yan process at measured qT and Q qT
d/dqT
QCD
qT~Q coll. fact.
Sivers TF
qT<<Q kT fact.
QCD << qT << Q same physics ?
“Unification” / Consistency of formalisms
• verify at 1-loop
X. Ji, J.W. Qiu, WV, F. Yuan

36. Step 1: calculate SSA for DY at qT ~ Q
use Qiu/Sterman formalism
Because of Q2 ≠ 0, there are also “hard poles”:
Propagator (H) has pole at xg0
No derivative terms in hard-pole contributions.

37. soft-pole
hard-pole

38. • result for qq process is (completely general!) _
soft-pole
hard-pole
derivative
non-deriv.
(recently also: Koike, Tanaka)

39. Step 2: expand this for qT << Q
Unpol.
Pol.

40. Step 3: calculate various factors in TMD factorized formula
Collins, Soper, Sterman
Ji, Ma, Yuan
At QCD << qT can calculate each factor from one-gluon emission

41. Unpolarized pdf:

42. Sivers function:
soft-pole
hard-pole
w/ correct direction of gauge link

43. Precisely what’s needed to make factorization work
soft-pole, deriv.
hard-pole
hard-pole
soft-pole, non-deriv.
Precisely what’s needed to make factorization work
and match on to the Qiu/Sterman result at small q!
So:
Step 4: compare both results and find agreement !

44. Take a closer look: if one works directly in small q limit 
Here for soft-pole, but happens separately for:
derivative / non-derivative / hard-pole  +  ( )

45. The interesting question now:
What happens in more general QCD hard-scattering ?
Consider ppjet jet X
= jet pair transv. mom.
Underlying this: all QCD 22 scattering processes

46. Example: qq’  qq’ • for Qiu/Sterman calculation: subset of diagramsIS
FS1
FS2
(these are soft-pole)

47. Simplify: • assume q << P from the beginning
• more precisely, assume k’ nearly parallel to hadron A or B
and pick up leading behavior in q / P
• reproduces above Drell-Yan results

48. k’ parallel to pol. hadron:
(partly even on individual diagram level, as in Drell-Yan)
Likewise for hard-pole contributions

49. What this means: When k’ nearly parallel to pol. hadron,
structure at this order can be organized as

50. Some remarks: • highly non-trivial. Relies on a number of “miracles”:
color structure
no derivative terms when k’ parallel to hadron B …
Calculation seems to “know” how to organize itself
• happens for all partonic channels:
individual
diagrams

51. Some further remarks:
• the obtained Sivers partonic hard parts are identical
to the ones obtained by Amsterdam group
• the obtained unpolarized partonic hard parts are identical
to the standard 22 ones
• complete calculation can be redone in context of
Brodsky-Hwang-Schmidt model:
identical results as from collinear-factorization approach

54. IV. Conclusions

55. • Single-inclusive case: use Qiu/Sterman formalism
Non-derivative terms have simple form
Not all aspects of data understood
• Connection between mechanisms for single-spin asym.
Drell-Yan as case study:
qT ~ Q Qiu/Sterman, matches TMD formalism for qT<<Q
Important input for phenomenology (Note: Sudakov logs)
• The same happens for pp  jet jet X
1-loop results for qT<<Q consistent with TMD factorization