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Los Alamos 5 Aug. 2002. single spin asymmetries in hard reactions. P.J. Mulders Vrije Universiteit Amsterdam mulders@nat.vu.nl. Content. Observables in (SI)DIS in field theory language  l ightcone/lightfront correlations Relation to lightcone wave functions

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single spin asymmetries in hard reactions

Los Alamos

5 Aug. 2002

single spin asymmetries inhard reactions

P.J. Mulders

Vrije Universiteit

Amsterdam

mulders@nat.vu.nl

content
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

lightcone2002 p j mulders

calculation of cross section dis
(calculation of) cross sectionDIS

Full calculation

+

+

+ …

PARTON

MODEL

+

leading order dis
Leadingorder DIS
  • In limit of large Q2 only result

of ‘handbag diagram’ survives

  • Isolate part encoding soft physics

?

?

lightcone2002 p j mulders

distribution functions
Distribution functions

Soper

Jaffe Ji NP B 375 (1992) 527

Parametrization

consistent with:

Hermiticity, Parity

& Time-reversal

distribution functions1
Distribution functions

Selection via specific probing operators

(e.g. appearing in leading order DIS, SIDIS or DY)

Jaffe Ji

NP B 375 (1992) 527

lightcone correlator momentum density

Bacchetta Boglione Henneman Mulders

PRL 85 (2000) 712

Lightcone correlatormomentum density

y+ = ½ g-g+ y

Sum over

lightcone wf

squared

lightfront quantization

Kogut & Soper

Lightfront quantization
  • Good fields are
  • independent,
  • satisfying CCR’s
  • Suitable to define
  • the partons in QCD
basis for partons
Basis for partons
  • ‘Good part’ of Dirac

space is 2-dimensional

  • Interpretation of DF’s

unpolarized quark

distribution

helicity or chirality

distribution

transverse spin distr.

or transversity

matrix representation
Matrix representation

Related to the

helicity formalism

Anselmino et al.

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

diagonal

in transverse

spin basis

color gauge link in correlator
Color gauge link in correlator
  • Diagrams containing correlator
  • <yA+y> produce the gauge
  • link U(0,x) in quark-quark
  • lightcone correlator
f x subleading
F(x) subleading
  • include M/P+ parts
  • gives M/Q terms in s
  • T-odd only for FF in e.g. e+e-

Jaffe Ji NP B 375 (1992) 527

Jaffe Ji PRL 71 (1993) 2547

summarizing dis
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

f1q(x) = q(x), g1q(x) =Dq(x), h1q(x) =dq(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

lightcone2002 p j mulders

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

lightcone2002 p j mulders

sidis cross section
SIDIS cross section
  • variables
  • hadron tensor
calculation of cross section sidis
(calculation of) cross sectionSIDIS

Full calculation

+

+

PARTON

MODEL

+ …

+

leading order sidis
Leading order SIDIS
  • In limit of large Q2 only result

of ‘handbag diagram’ survives

  • Isolating parts encoding soft physics

?

?

lightcone2002 p j mulders

lightfront dominance in sidis1
Lightfront dominance in SIDIS

Three external momenta

P Ph q

transverse directions relevant

qT = q + xB P – Ph/zh

or

qT = -Ph^/zh

lightfront correlator distribution
Lightfront correlator(distribution)

+

Lightfront correlator (fragmentation)

no T-constraint

T|Ph,X>out =|Ph,X>in

Collins Soper

NP B 194 (1982) 445

distribution functions in sidis

Ralston Soper

NP B 152 (1979) 109

distribution functions in SIDIS

Tangerman Mulders

PR D 51 (1995) 3357

Constraints from Hermiticity & Parity

  • Dependence

on …(x, pT2)

  • T-invariance:
  • h1^ = f1T^ = 0?
  • T-odd functions
  • Fragmentation

f  D

g  G

h  H

  • No T-constraint:

H1^ and D1T^

nonzero!

distribution functions in sidis1

Ralston Soper

NP B 152 (1979) 109

Distribution functions in SIDIS

Tangerman Mulders

PR D 51 (1995) 3357

Selection via specific probing operators

(e.g. appearing in leading order SIDIS or DY)

lightcone correlator momentum density1

Bacchetta Boglione Henneman Mulders

PRL 85 (2000) 712

Lightcone correlatormomentum density

Remains valid for F(x,pT)

y+ = ½ g-g+ y

Sum over

lightcone wf

squared

interpretation
Interpretation

unpolarized quark

distribution

need pT

T-odd

helicity or chirality

distribution

need pT

T-odd

need pT

transverse spin distr.

or transversity

need pT

need pT

matrix representation for m f x g t
Matrix representationfor M = [F(x)g+]T

Collinear structure of the nucleon!

matrix representation for m f x p t g t

pT-dependent

functions

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

T-odd: g1T g1T – i f1T^ and h1L^  h1L^ + i h1^

matrix representation for m d z k t g t

pT-dependent

functions

Matrix representation for M = [D(z,kT) g-]T
  • FF’s:

f  D

g  G

h  H

  • No T-inv

constraints

H1^ and

D1T^

nonzero!

matrix representation for m d z k t g t1

pT-dependent

functions

Matrix representation for M = [D(z,kT) g-]T
  • R/L basis for spin 0
  • Also for spin 0

a T-odd function

exist, H1^

(Collins function)

e.g. pion

  • FF’s after kT-

integration

leaves just the

ordinary D1(z)

distribution and fragmentation functions

pT-dependent

Distribution and fragmentation functions

T-odd ?

  • pT-integrated

no T-odd

Mulders Tangerman

NP B 461 (1996) 197

sample azimutal asymmetry s lto
Sample azimutal asymmetrysLTO
  • example of a leading

azimuthal asymmetry

  • without appropriate

weights this would be

a convolution instead of

a factorized expression

Kotzinian

NP B 441 (1995) 234

Tangerman Mulders

PL B 352 (1995) 129

sample single spin asymmetry s oto
Sample single spin asymmetrysOTO
  • 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 h1q(x)

Collins

NP B 396 (1993) 161

Tangerman Mulders

PL B 352 (1995) 129

summarizing sidis
Summarizing SIDIS
  • Transverse momenta of partons become relevant, effects appearing in azimuthal asymmetries
  • DF’s and FF’s depend on two variables,

F(x,pT) and D(z,kT)

  • Fragmentation functions are not constrained by time-reversal invariance
  • This allows T-odd functions H1^ and D1T^, appearing in single spin asymmetries

lightcone2002 p j mulders

t odd phenomena
T-odd phenomena
  • T-invariance does not constrain fragmentation
    • T-odd FF’s (e.g. Collins function H1^)
  • T-invariance does constrain F(x)
    • No T-odd DF’s and thus no SSA in DIS
  • What about T-invariance and F(x,pT)?

lightcone2002 p j mulders

color gauge link in correlator1
Color gauge link in correlator

Diagrams containing correlator <yA+y>

produce the gauge links U(0,-J) and

U(-J,x) in quark-quark matrix element

Boer Mulders

NP B 569 (2000) 505

hep-ph/9906223

?

distribution
Distribution
  • for plane waves

T|P> = |P>

  • But...T U-T = U+
  • this does not

affect F(x)

  • it does affect

F(x,pT)

  • appearance of

T-odd functions

in F(x,pT)

  • color gauge inv

requires leading

transverse gluons

including the gauge link

?

From <y(0)AT(-J)y(x)> m.e.?

t odd phenomena1
T-odd phenomena

Related work:

  • T-invariance does not constrain F(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

Boer Mulders Teryaev

PR D 57 (1998) 3057

hep-ph/9710223

Brodsky Hwang Schmidt hep-ph/0201296

Qiu Sterman

Ji Yuan hep-ph/0206057

Schaefer Teryaev

Collins hep-ph/0204004

Brodsky Hoyer PR D 65 (2002) 114025

Mulders Tangerman NP B 461 (1996) 197

lightcone2002 p j mulders

single spin asymmetries s oto
Single spin asymmetriessOTO
  • T-odd fragmentation function (Collins function)

or

  • T-odd distribution function (Sivers function)
  • Both of the above can explain pp pX SSA
  • Different asymmetries in leptoproduction!

Collins

NP B 396 (1993) 161

Boer Mulders

PR D 57 (1998) 5780

Boglione Mulders

PR D 60 (1999) 054007

conclusions
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 <yATy> pieces in gauge link

lightcone2002 p j mulders