SingleTransverse Spin Asymmetries in Hadronic Scattering Werner Vogelsang

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. D 73, 094017 (2006) Phys. Lett. B 638, 178 (2006) C. Kouvaris, J. W. Qiu, WV, F. Yuan, Phys. Rev. D 74, 114013 (2006) ( C. Bomhof, P. Mulders, WV, F. Yuan, Phys. Rev. D 75, 074019 (2007) ) J. W. Qiu, WV, F. Yuan, ar. Xiv: 0704. 1153 [hep-ph] (Phys. Lett. B, to appear) ar. Xiv: 0706. 1196 [hep-ph]

Outline: • Introduction • Single-spin asymmetries in pp h. X • How are mechanisms for Single-spin asymmetries related ? • Conclusions

I. Introduction

• SSA for single-inclusive process example: pp X a single large scale (p. T) L R power-suppressed collinear factorization (Efremov, Teryaev / Qiu, Sterman TF) • SSA with small & measured q. T , 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 h. X

L R E 704 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. 3 Ge. V 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 “threshold” logarithms Real emission inhibited Only soft/collinear gluons allowed

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

de Florian, WV E 706

WA 70 Effects start to become visible at S=62 Ge. V… Rapidity dependence ? Spin dependence ?

• Kane, Pumplin, Repko ‘ 78 In helicity basis: ~ Im + + + _ _ 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) x 2 -x 1 _ x 1 x 2

• ingredients: Collinear factorization. x 1 x 2 -x 1 x 2 quark-gluon correlation function TF(x 1, x 2) provides helicity flip unpol. pdf Phase from imaginary part of propagator ~ i (x 1 -x 2) (soft-gluon-pole contributions)

• full structure: Qiu, Sterman Transversity Kanazawa, Koike

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

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

x. F=0. 15 x. F=0. 4

v 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: v Recently: proof by Koike & Tanaka

Ansatz: usual pdf Fit to E 704, STAR, BRAHMS for RHIC, use data with p. T>1 Ge. V § for E 704, choose p. T=1. 2 Ge. V 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:

p. T dependence

Dependence on RHIC c. m. s. energy:

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

• 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 ?

• consider Drell-Yan process at measured q. T and Q d /dq. T~Q coll. fact. TF q. T<<Q k. T fact. Sivers q. T QCD << q. T << 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 q. T ~ Q use Qiu/Sterman formalism Because of Q 2 ≠ 0, there 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 q. T << Q Unpol. Pol.

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

Unpolarized pdf:

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

soft-pole, deriv. soft-pole, non-deriv. hard-pole 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 X = jet pair transv. mom. Underlying this: all QCD 2 2 scattering processes

Example: qq’ • for Qiu/Sterman calculation: subset of diagrams IS FS 1 FS 2 (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: q. T ~ Q Qiu/Sterman, matches TMD formalism for q. T<<Q Important input for phenomenology (Note: Sudakov logs) • The same happens for pp jet X 1 -loop results for q. T<<Q consistent with TMD factorization