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, Phys. Rev. D75, 074019 (2007) ) J.W. Qiu, WV, F. Yuan, arXiv:0704.1153 [hep-ph] (Phys. Lett. B, to appear) arXiv:0706.1196 [hep-ph]

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

I. Introduction

• 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.)

II. Asymmetry in pphX

L R  E704 STAR

collinear factorization Brahms y=2.95 STAR

STAR

• 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

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

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

de Florian, WV E706

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

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

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

• 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)

• full structure: Qiu,Sterman Transversity Kanazawa,Koike

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

“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

xF=0.15 xF=0.4

• 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

Remarkably simple answer: Recently: proof by Koike & Tanaka

 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)

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 ?

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 ?

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

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.

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 Collins, Soper, Sterman Ji, Ma, Yuan At QCD << qT can calculate each factor from one-gluon emission

Unpolarized pdf:

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

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 !

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 ppjet jet X = jet pair transv. mom. Underlying this: all QCD 22 scattering processes

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

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

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 22 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 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