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)