single spin asymmetries in hard reactions

single spin asymmetries inhard reactions

P.J. MuldersVrije Universiteit

Amsterdam

mulders@nat.vu.nl

Los Alamos5 Aug. 2002

05/08/2002 lightcone2002 p j mulders

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Content Observables in (SI)DIS in field theory language

lightcone/lightfront correlations Relation to lightcone wave functions Single-spin asymmetries in hard reactions

T-odd correlations T-odd in final (fragmentation) and initial state

(distribution) correlations Conclusions and further work

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Soft physics in inclusive deep inelastic leptoproduction

(calculation of) cross sectionDIS

Full calculation

+ …

+ +

+PARTONMODEL

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Leading order DIS In limit of large Q2 only result

of ‘handbag diagram’ survives

Isolate part encoding soft physics

? ?

Lightcone dominance in DIS

Distribution functions

Parametrization consistent with:Hermiticity, Parity & Time-reversal

SoperJaffe Ji NP B 375 (1992) 527

Distribution functions

Jaffe JiNP B 375 (1992) 527

Selection via specific probing operators(e.g. appearing in leading order DIS, SIDIS or DY)

Lightcone correlator

momentum density

Bacchetta Boglione Henneman MuldersPRL 85 (2000) 712

= ½

Sum over lightcone wf

squared

Lightfront quantization

Kogut & Soper

• Good fields are independent, satisfying CCR’s• Suitable to define the partons in QCD

Basis for partons

‘Good part’ of Dirac space is 2-dimensional

Interpretation of DF’s

unpolarized quarkdistribution

helicity or chiralitydistribution

transverse spin distr.or transversity

Off-diagonal elements (RL or LR) are chiral-odd functions Chiral-odd soft parts must appear with partner in e.g. SIDIS, DY

Matrix representation

Related to thehelicity formalism

Anselmino et al.

chiral-odd functions

diagonalin transversespin basis

Color gauge link in correlator

Diagrams containing correlator A+ produce the gauge link U(0,) in quark-quark lightcone correlator

(x) subleading

include M/P+ parts gives M/Q terms in T-odd only for FF in e.g. e+e

Jaffe Ji NP B 375 (1992) 527Jaffe Ji PRL 71 (1993) 2547

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Summarizing DIS Structure functions (observables) are identified with

distribution functions (lightcone quark-quark correlators)

DF’s are quark densities that are directly linked to lightcone wave functions squared

There are three DF’s f1

q(x) = q(x), g1q(x) =q(x), h1

q(x) =q(x) Longitudinal gluons (A+, not seen in LC gauge) are

absorbed in DF’s Transverse gluons appear at 1/Q and are contained in

(higher twist) qqG-correlators

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Soft physics in semi-inclusive (1-particle incl) leptoproduction

SIDIS cross section

variables hadron tensor

(calculation of) cross sectionSIDIS

Full calculation

+

+ …

+

+PARTONMODEL

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Leading order SIDIS In limit of large Q2 only result

of ‘handbag diagram’ survives

Isolating parts encoding soft physics

? ?

Lightfront dominance in SIDIS

Lightfront dominance in SIDIS

Three external momentaP Ph q

transverse directions relevantqT = q + xB P – Ph/zh

orqT = -Ph/zh

Lightfront correlator(distribution)

Lightfront correlator (fragmentation)+

no T-constraintT|Ph,X>out = |Ph,X>in

Collins SoperNP B 194 (1982) 445

distribution functions in SIDIS

Constraints from Hermiticity & Parity Dependence on …(x, pT

2) T-invariance: h1

= f1T = 0?

T-odd functions

Ralston SoperNP B 152 (1979) 109

Tangerman MuldersPR D 51 (1995) 3357

Fragmentation f D g G h H No T-constraint: H1

and D1T

nonzero!

Distribution functions in SIDISRalston SoperNP B 152 (1979) 109

Tangerman MuldersPR D 51 (1995) 3357

Selection via specific probing operators(e.g. appearing in leading order SIDIS or DY)

Lightcone correlator

momentum density

Bacchetta Boglione Henneman MuldersPRL 85 (2000) 712

Remains valid for (x,pT)

= ½

Sum over lightcone wf

squared

Interpretation

unpolarized quarkdistribution

helicity or chiralitydistribution

transverse spin distr.or transversity

need pT

need pT

need pT

need pT

need pT

T-odd

T-odd

Collinear structure of the nucleon!

Matrix representationfor M = [(x)+]T

Matrix representationfor M = [(x,pT) +]T

pT-dependent functions

T-odd: g1T g1T – i f1T and h1L

h1L + i

h1

Matrix representation for M = [(z,kT) ]T

pT-dependent functions

FF’s: f D g G h H

No T-inv constraints H1

and

D1T

nonzero!

Matrix representation for M = [(z,kT) ]T

pT-dependent functions

FF’s after kT-

integration leaves just the ordinary D1(z)

R/L basis for spin 0 Also for spin 0 a T-odd function exist, H1

(Collins function)

e.g. pion

Distribution and fragmentation functions

pT-integrated

pT-dependent

no T-odd

T-odd ?

Mulders TangermanNP B 461 (1996) 197

Sample azimutal asymmetryLTO

example of a leading azimuthal asymmetry without appropriate weights this would be a convolution instead of a factorized expression

Tangerman MuldersPL B 352 (1995) 129

KotzinianNP B 441 (1995) 234

Sample single spin asymmetryOTO

example of a leading azimuthal asymmetry T-odd fragmentation function (Collins function) T-odd single spin asymmetry involves two chiral-odd functions Best way to get transverse spin polarization h1

q(x)

Tangerman MuldersPL B 352 (1995) 129

CollinsNP B 396 (1993) 161

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Summarizing SIDIS Transverse momenta of partons become

relevant, effects appearing in azimuthal asymmetries

DF’s and FF’s depend on two variables, (x,pT) and (z,kT) Fragmentation functions are not

constrained by time-reversal invariance This allows T-odd functions H1

and D1T,

appearing in single spin asymmetries

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T-odd phenomena T-invariance does not constrain fragmentation

T-odd FF’s (e.g. Collins function H1)

T-invariance does constrain (x) No T-odd DF’s and thus no SSA in DIS

What about T-invariance and (x,pT)?

Color gauge link in correlator

?

Diagrams containing correlator A+ produce the gauge links U(0,-) and U(-,) in quark-quark matrix element

Boer MuldersNP B 569 (2000) 505hep-ph/9906223

Distribution

for plane waves T|P> = |P> But...T U

T = Uthis does not affect (x) it does affect (x,pT) appearance of T-odd functions in (x,pT) color gauge inv requires leading transverse gluons

?

including the gauge link

From AT(-)y m.e.?

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T-odd phenomena T-invariance does not constrain (x,pT) Effects are connected to gauge link

They require ‘leading’ pT-effects

They change of sign from SIDIS to DY

They spoil density interpretation of functions and relation to lc wf

Phenomenology of T-odd functions exist

Similar T-odd effects exist at subleading (twist-3) level Experiment can give answers via specific experiments

Brodsky Hwang Schmidt hep-ph/0201296

Ji Yuan hep-ph/0206057

Collins hep-ph/0204004

Brodsky Hoyer PR D 65 (2002) 114025

Boer Mulders TeryaevPR D 57 (1998) 3057hep-ph/9710223

Qiu Sterman

Schaefer Teryaev

Related work:

Mulders Tangerman NP B 461 (1996) 197

Single spin asymmetriesOTO

T-odd fragmentation function (Collins function) or T-odd distribution function (Sivers function) Both of the above can explain pp X SSA Different asymmetries in leptoproduction!

Boer MuldersPR D 57 (1998) 5780

Boglione MuldersPR D 60 (1999) 054007

CollinsNP B 396 (1993) 161

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Conclusions Hard reactions with two hadrons (DY, SIDIS)

offer possibilities to access parton transverse momenta pT

Experimental access via azimuthal asymmetries, often also requiring polarization

T-odd phenomena occur in single-spin asymmetries

These are natural for fragmentation functions and seem possible for pT-dependent distribution functions involving necessary AT pieces in gauge link