single-transverse spin asymmetries in hadronic scattering
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Single-Transverse Spin Asymmetries in Hadronic
Scattering
Werner Vogelsang (& Feng Yuan)BNL Nuclear Theory
ECT, 06/13/2007
Mostly based on:
X. Ji, J.W. Qiu, WV, F. Yuan,
Phys. Rev. Lett. 97, 082002 (2006)
Phys. Rev. D73, 094017 (2006)
Phys. Lett. B638, 178 (2006)
C. Kouvaris, J.W. Qiu, WV, F. Yuan,
Phys. Rev. D74, 114013 (2006)
( C. Bomhof, P. Mulders, WV, F. Yuan,
J.W. Qiu, WV, F. Yuan,
arXiv:0704.1153 [hep-ph] (Phys. Lett. B, to appear)
Phys. Rev. D75, 074019 (2007) )
arXiv:0706.1196 [hep-ph]
Outline:
• Single-spin asymmetries in pp hX
• How are mechanisms for Single-spin asymmetries related ?
• Conclusions
• Introduction
I. Introduction
• SSA for single-inclusive process
power-suppressed
a single large scale (pT)
example: pp X
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.)
L
R
II. Asymmetry in pphX
L
R
E704 STAR
STAR
collinear factorization
Brahmsy=2.95
STAR
Resummation of important higher-order corrections beyond NLO de Florian, WV
√s=23.3GeV
Bourrely and SofferApanasevich et al.
• typically, hard-scattering calculations based on LO/NLO fail badly in describing the cross section
higher-order corrections beyond NLO ?
Real emission inhibited Only soft/collinear gluons allowed
“threshold” logarithms
de Florian, WV
Leading logarithms
expect large enhancement ! de Florian, WV
Mellin moment in
de Florian, WV
E706
WA70
Effects start to become visible at S=62 GeV…
Rapidity dependence ? Spin dependence ?
transversity
−=1h
_+
+_
_
• lesson from this: AN in pph X is power-suppressed !
In helicity basis: ~ Im
+ _
• Kane, Pumplin, Repko ‘78
_
• power-suppressed effects in QCD much richer than just mass terms (Efremov,Teryaev; Qiu,Sterman; Kanazawa, Koike)
x1 x2x2-x1
• ingredients:
x1 x2
x2-x1
Collinear factorization.
Phase from imaginary part of propagator ~ i (x1-x2) (soft-gluon-pole contributions)
quark-gluon correlationfunction TF(x1, x2)provides helicity flip
unpol. pdf
• full structure:
Kanazawa,Koike
Qiu,Sterman
Transversity
IS
Position of pole may depend onk of initial partons
FS
“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
• plus, non-derivative terms !
xF=0.15 xF=0.4
Assumptions in Qiu & Sterman :
• derivative terms only
• valence TF only,
• neglect gluonpion fragmentation
In view of new data, would like to relax some of these. Kouvaris, Qiu, Yuan, WV
Remarkably simple answer:
Recently: proof by Koike & Tanaka
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)
Fit I: “two-flavor / valence”
Fit II: allow sea as well
solid: Fit I, dashed: Fit II
Our TF functions:
pT dependence
Dependence on RHIC c.m.s. energy:
III. How are the mechanisms for single-spin asymmetries related ?
• have two “mechanisms”
Q: In what way are mechanisms connected ?
• tied to factorization theorem that applies
• Boer, Mulders, Pijlman
• see interplay of mechanisms in a physical process ?
• consider Drell-Yan process at measured qT and Q
qT
d/dqT
QCD
qT<<Q kT fact.
qT~Q coll. fact.
QCD << qT << Q same physics ?
Sivers
TF
“Unification” / Consistency of formalisms
• verify at 1-loop X. Ji, J.W. Qiu, WV, F. Yuan
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 xg0 No derivative terms in hard-pole contributions.
soft-pole
hard-pole
• result for qq process is (completely general!)_
soft-pole
hard-pole
derivative
non-deriv.
(recently also: Koike, Tanaka)
Step 2: expand this for qT << Q
Unpol.
Pol.
Step 3: calculate various factors in TMD factorized formula
At QCD << qT can calculate each factor from one-gluon emission
Ji, Ma, Yuan
Collins, Soper, Sterman
Unpolarized pdf:
Sivers function:
soft-pole hard-pole
w/ correct direction of gauge link
Precisely what’s needed to make factorization workand match on to the Qiu/Sterman result at small q!
So:
Step 4: compare both results and find agreement !
soft-pole, deriv. hard-pole
soft-pole, non-deriv. hard-pole
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
+ +
( + + )
The interesting question now: What happens in more general QCD hard-scattering ?
Consider ppjet jet X
Underlying this: all QCD 22 scattering processes
= jet pair transv. mom.
Example: qq’ qq’
• for Qiu/Sterman calculation: subset of diagrams
IS
FS1
FS2
(these are soft-pole)
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
(partly even on individual diagram level, as in Drell-Yan)
Likewise for hard-pole contributions
k’ parallel to pol. hadron:
What this means:
When k’ nearly parallel to pol. hadron,structure at this order can be organized as
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
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 22 ones
• complete calculation can be redone in context of Brodsky-Hwang-Schmidt model: identical results as from collinear-factorization approach
IV. Conclusions
• 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
• The same happens for pp jet jet X
1-loop results for qT<<Q consistent with TMD factorization
Important input for phenomenology (Note: Sudakov logs)
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