Polarized structure functions

Piet Mulders

‘Lepton scattering and the structure of nucleons and nuclei’September 16-24, 2004

pjg.mulders@few.vu.nl

Content

• Spin structure & transversity

• Transverse momenta & azimuthal asymmetries

• T-odd phenomena & single spin asymmetries

DIS

• Known leptonic part• Completeness allows

reduction in hadronic tensor to commutator [J(x),J(0)]

• Known structure of current in terms of quarks

• OPE• ….

Deep inelastic scattering (DIS)

Lepton tensor

• Lepton tensor can also be expanded using the spacelike and timelike vectors

• Tensor encompasses many ‘polarization options’

Polarized DIS

Semi-inclusive deep inelastic scattering

• Known lepton part with much flexibility (unused in DIS)

• Involves two hadrons and hence a much more complex hadronic tensor

SIDIS

(calculation of) cross section in DIS

Full calculation

+ …

+ +

+PARTONMODEL

Lightcone dominance in DIS

Leading order DIS

• In limit of large Q2 the resultof ‘handbag diagram’ survives

• … + contributions from A+ gluonsensuring color gauge invariance

A+ gluons gauge link

Ellis, Furmanski, PetronzioEfremov, Radyushkin

A+

Parametrization of lightcone correlator

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

leading part

• M/P+ parts appear as M/Q terms in • T-odd part vanishes for distributions but is important for fragmentation

Basis of 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 representationfor M = [(x)+]T

Quark production matrix, directly related to thehelicity formalism

Anselmino et al.

Bacchetta, Boglione, Henneman & MuldersPRL 85 (2000) 712

Results for DIS

• Structure functions in (sub)leading order in 1/Q

• Two of three (Polarized) quark densities for each flavor:

1

1

1

( ) ( )

( ) ( )

( ) ( )

q

q

q

f x q x

g x q x

h x q x

Not accessible in DIS

2 2 22 1 1

2 21 1

2 2

( , ) 2 ( , ) ( )

2 ( , ) ( )

2 ( , ) ( )

qq

q

qq

q

qT q T

q

F x Q xF x Q e xf x

g x Q e xg x

g x Q e xg x

(calculation of) cross section in SIDIS

“Full” calculation

+

+ …

+

+PARTONMODEL

Lightfront dominance in SIDIS

Three external momentaP Ph q

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

orqT = -Ph/zh

Leading order SIDIS

• In limit of large Q2 only resultof ‘handbag diagram’ survives

• Isolating parts encoding soft physics

? ?

Lightfront correlators

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

Collins & SoperNP B 194 (1982) 445

Jaffe & Ji, PRL 71 (1993) 2547;PRD 57 (1998) 3057

Distribution

From AT() m.e.

including the gauge link (in SIDIS)

A+

One needs also AT

G+ = +AT

AT()= AT

(∞) + d G+

Belitsky, Ji, Yuan, hep-ph/0208038Boer, M, Pijlman, hep-ph/0303034

Parametrization of (x,pT)

• Link dependence allows also T-odd distribution functions since T U[0,] T = U[0,-]

• Functions h1 and f1T

(Sivers) nonzero!

• These functions (of course) exist as fragmentation functions (no T-symmetry) H1

(Collins) and D1T

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

pT-dependent functions

T-odd: g1T g1T – i f1T and h1L

h1L + i h1

(imaginary parts)

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

Bacchetta, Boglione, Henneman & MuldersPRL 85 (2000) 712

T-odd single spin asymmetry

• with time reversal constraint only even-spin asymmetries• the time reversal constraint cannot be applied in DY or in 1-

particle inclusive DIS or ee

• In those cases single spin asymmetries can be used to select T-odd quantities

*

*

W(q;P,S;Ph,Sh) = W(q;P,S;Ph,Sh)

W(q;P,S;Ph,Sh) = W(q;P,S;Ph,Sh)

W(q;P,S;Ph,Sh) = W(q;P, S;Ph, Sh)

W(q;P,S;Ph,Sh) = W(q;P,S;Ph,Sh)

_

___

_ ____

__ _time

reversal

symmetrystructure

parity

hermiticity

*

*

Leptoproduction of pions

H1 is T-odd

and chiral-odd

COLLINS ASYMMETRYRESULTS OF COMPASS

Acoll depends on phT, zh, xBj

with more statistics, the full analysis is foreseen

from 2002 data:

Acoll vs xBj

Sign!Sign!

COLLINS ASYMMETRYRESULTS OF COMPASS

from 2002 data:

AColl vs zh

all the tests made are consistent with the fact that systematic effects, if present, are smaller than statistical errors

Sign!Sign!

Distribution

A+

A+

including the gauge link (in SIDIS or DY)

SIDIS

SIDIS [-]DY DY [+]

Difference between [+] and [-] upon integration

integrated quarkdistributions

transverse moments

measured in azimuthal asymmetries

±

Back to the lightcone (theoretically clean)

twist 2

twist 2 & 3

Difference between [+] and [-] upon integration

gluonic pole m.e. (T-odd)

In momentum space:

Conclusion: T-odd parts are gluon-driven (QCD interactions)

Time reversal constraints for distribution functions

Time reversal(x,pT) (x,pT)

G

T-even(real)

T-odd(imaginary)

Conclusion:T-odd effects in SIDIS and DY have opposite signs

Time reversal constraints for fragmentation functions

Time reversalout(z,pT)

in(z,pT)

G

T-even(real)

T-odd(imaginary)

Time reversal constraints for fragmentation functions

G out

out

out

out

T-even(real)

T-odd(imaginary)

Time reversalout(z,pT)

in(z,pT)

Conclusion:T-odd effects in SIDIS and ee are not related

other hard processes

• qq-scattering as hard subprocess

• insertions of gluons collinear with parton 1 are possible at many places

• this leads for ‘external’ parton fields to gauge link to lightcone infinity

e.g.

C. Bomhof, P.J. Mulders and F. PijlmanPLB 596 (2004) 277

other hard processes

• qq-scattering as hard subprocess

• insertions of gluons collinear with parton 1 are possible at many places

• this leads for ‘external’ parton fields to gauge link to lightcone infinity

• The correlator (x,pT) enters for each contributing term in squared amplitude with specific link

• The link may enhance the effect of the (T-odd) gluonic pole contribution involving also specific color factors

• Finding the right observables, however is crucial

Conclusions

• Hard processes quark and gluon structure of hadrons (quark distributions, their chirality and transverse polarization)

• Many new observables accessible when going beyond collinearity, often in combination with (transverse) polarization (among others the simplest access to transverse quark polarization)

• Going beyond collinearity gives access to gluon dynamics in hadrons, which can be done in a controlled way via weighted asymmetries (twist limited, t 3), use of chirality, and the specific time-reversal behavior of single spin asymmetries.