Accessing transversity via single spin (azimuthal) asymmetries

Accessing transversity via single spin (azimuthal) asymmetries

P.J. MuldersVrije Universiteit

Amsterdam

[email protected]

COMPASS workshopParis, March 2004

Universality of T-odd effects in single spin and azimuthal asymmetries, D. Boer, PM and F. Pijlman, NP B667 (2003) 201-241; hep-ph/0303034

1-3-2004 COMPASS p j mulders 2

Content

Soft parts in hard processes twist expansion gauge link Illustrated in DIS

Two or more (separated) hadrons transverse momentum

dependence T-odd phenomena Illustrated in SIDIS and DY

Universality Items relevant for other processes Illustrated in high pT hadroproduction


Soft physics in hard processes (e.g. inclusive deep inelastic leptoproduction)

(calculation of) cross sectionDIS

“Full” calculation

+ …

+ +

+PARTONMODEL

Lightcone dominance in DIS


Leading order DIS In limit of large Q2 the result

of ‘handbag diagram’ survives … + contributions from A+ gluons

A+

A+ gluons gauge link

Ellis, Furmanski, PetronzioEfremov, Radyushkin

Color gauge link in correlator Matrix elements

A+ produce the gauge link U(0,) in leading quark lightcone correlator

A+

Distribution functions

Parametrization consistent with:Hermiticity, Parity & Time-reversal

SoperJaffe & Ji NP B 375 (1992) 527


M/P+ parts appear as M/Q terms in T-odd part vanishes for distributions but is important for fragmentation Jaffe & Ji NP B 375 (1992)

527Jaffe & Ji PRL 71 (1993) 2547

leading part


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

= ½

Sum over lightcone wf

squared

Production matrix:

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 representationfor M = [(x)+]T

Related to thehelicity formalism

Anselmino et al.

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


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 Perturbative QCD evolution


Hard processes with two or more hadrons

SIDIS cross section variables hadron tensor

(calculation of) cross sectionSIDIS

“Full” calculation

+

+ …

+

+PARTONMODEL

Lightfront dominance in SIDIS

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 result

of ‘handbag diagram’ survives

Isolating parts encoding soft physics

? ?

Lightfront correlator(distribution)

Lightfront correlator (fragmentation)+

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

Distribution

A+

A+including the gauge link (in SIDIS or

DY)SIDIS

SIDIS [-]

DYDY [+]

Distribution

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

T = U

this does affect (x,pT) appearance of T-odd functions in (x,pT)

including the gauge link (in SIDIS or DY)

Parameterizations including pT

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

2) Without T: h1

and f1T

nonzero! 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 with pTRalston & 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


Remains valid for (x,pT)… and also after inclusion of links for (x,pT)

Sum over lightcone wf

squared

Brodsky, Hoyer, Marchal, Peigne, Sannino PR D 65 (2002) 114025

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

Difference between [+] and [-]

Integrateover pT

Integrated distributions

T-odd functions only for fragmentation

Weighted distributions

Appear in azimuthal asymmetries in SIDIS or DYThese are process-dependent (through gauge link) and thus need in fact [±] superscript!

Collinear structure of the nucleon!

Matrix representationfor M = [(x)+]T

reminder

pT-dependent functions

T-odd: g1T g1T – i f1T and h1L

h1L + i

h1

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


Matrix representationfor M = [±](z,kT) ]T


FF’s: f D g G h H

No T-inv constraints H1

and

D1T

nonzero!

Matrix representationfor M = [±](z,kT) ]T


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


Summarizing SIDIS Beyond just extending DIS by tagging

quarks … Transverse momenta of partons become

relevant, appearing in azimuthal asymmetries DF’s and FF’s depend on two variables, (x,pT) and (z,kT) Gauge link structure is process dependent (

pT-dependent distribution functions and (in general) fragmentation functions are not constrained by time-reversal invariance

This allows T-odd functions h1 and f1T

(H1 and

D1T) appearing in single spin asymmetries


T-odd effects in single spin asymmetries

T-odd single spin asymmetry

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

Conclusion:

with time reversal constraint only even-spin asymmetriesBut time reversal constraint cannot be applied

in DY or in 1-particle inclusive DIS or e+e

Example of a single spin asymmetry

example of a leading azimuthal asymmetry T-odd fragmentation function (Collins function) 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

example:OTO inep epX

Single spin asymmetriesOTO

T-odd fragmentation function (Collins function) or T-odd distribution function (Sivers function) Both of the above also appear in SSA in pp X Different asymmetries in leptoproduction! But be aware now of [±] dependence

Boer & MuldersPR D 57 (1998) 5780

Boglione & MuldersPR D 60 (1999) 054007

CollinsNP B 396 (1993) 161

SiversPRD 1990/91


Process dependence and universality


integrated quarkdistributions

transverse moments

measured in azimuthal asymmetries

±


gluonic pole m.e.


Time reversal constraints for distribution functions

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

G

T-even(real)

T-odd(imaginary)

Consequences for distribution functions

(x,pT) = (x,pT) ± G

Time reversal

SIDIS[+]

DY [-]


(x,pT) = (x,pT) ± G

Sivers effect in SIDISand DY opposite in sign

Collins hep-ph/0204004


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)

Fragmentation functions

(x,pT) = (x,pT) ± G

Time reversal does not lead to constraints

Collins effect in SIDISand e+e unrelated!

If G = 0

But at present this seems (to me) unlikely


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

T-invariance does not constrain (x,pT) T-odd DF’s and thus SSA in SIDIS (in combination with

azimuthal asymmetries) are identified with gluonic poles that also appear elsewhere (Qiu-Sterman, Schaefer-Teryaev)

Sign of gluonic pole contribution process dependent In fragmentation soft T-odd and (T-odd and T-even) gluonic pole

effects arise No direct comparison of Collins asymmetries in SIDIS and e+e

(unless G = 0)


What about hadroproduction?


Issues in hadroproduction

Weighted functions will appear in L-R asymmetries (pT now hard scale!)But which one?

There are (moreover) various possibilities with gluons G(x,pT) – unpolarized gluons in unpolarized nucleon G(x,pT) – transversely polarized gluons in a longitudinally polarized nucleon GT(x,pT) – unpolarized gluons in a transversely polarized nucleon (T-odd) H(x,pT) – longitudinally polarized gluons in an unpolarized nucleon …


Issues in hadroproduction Contributions of (x,pT) and G not necessarily in one

combinationAN ~ … G(xa) f1T

(1)[-](xb) D1(zc) + … f1(xa) f1T (1)[+](xb) D1(zc)

+ … f1(xa) h1(xb) H1[-] (zc) + … f1(xa) h1(xb) H1

[+] (zc)

+ … f1(xa) GT(xb) D1(zc)

Many issues to be sorted out


Thank you for your attention

Documents

Accessing transversity via single spin (azimuthal) asymmetries