Accessing transversity via single spin (azimuthal) asymmetries

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Accessing transversity via single spin
(azimuthal) asymmetries
COMPASS workshop Paris, 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

• P.J. Mulders
• Vrije Universiteit
• Amsterdam
• pjg.mulders_at_few.vu.nl

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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

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Soft physics in hard processes (e.g. inclusive
deep inelastic leptoproduction)
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(calculation of) cross sectionDIS
Full calculation




PARTON MODEL
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Lightcone dominance in DIS
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Leading order DIS

• In limit of large Q2 the result
• of handbag diagram survives
• contributions from A gluons

A
Ellis, Furmanski, Petronzio Efremov, Radyushkin
A gluons ? gauge link
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Color gauge link in correlator
Matrix elements ltyAygt produce the gauge link
U(0,x) in leading quark lightcone correlator
A
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Distribution functions
Soper Jaffe Ji NP B 375 (1992) 527
Parametrization consistent with Hermiticity,
Parity Time-reversal
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Distribution functions

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

Jaffe Ji NP B 375 (1992) 527 Jaffe Ji
PRL 71 (1993) 2547
leading part
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Distribution functions
Selection via specific probing operators (e.g.
appearing in leading order DIS, SIDIS or DY)
Jaffe Ji NP B 375 (1992) 527
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Lightcone correlatormomentum density
Production matrix
y ½ g-g y
Sum over lightcone wf squared
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Basis for partons

• Good part of Dirac
• space is 2-dimensional
• Interpretation of DFs

unpolarized quark distribution
helicity or chirality distribution
transverse spin distr. or transversity
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Matrix representationfor M F(x)gT
Bacchetta, Boglione, Henneman Mulders PRL 85
(2000) 712
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

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Summarizing DIS

• Structure functions (observables) are identified
  with distribution functions (lightcone
  quark-quark correlators)
• DFs are quark densities that are directly linked
  to lightcone wave functions squared
• There are three DFs
• f1q(x) q(x), g1q(x) Dq(x), h1q(x) dq(x)
• Longitudinal gluons (A, not seen in LC gauge)
  are absorbed in DFs
• Transverse gluons appear at 1/Q and are contained
  in (higher twist) qqG-correlators
• Perturbative QCD ? evolution

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Hard processes with two or more hadrons
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SIDIS cross section

• variables
• hadron tensor

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(calculation of) cross sectionSIDIS
Full calculation


PARTON MODEL


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Lightfront dominance in SIDIS
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Lightfront dominance in SIDIS
Three external momenta P Ph q transverse
directions relevant qT q xB P Ph/zh or qT
-Ph/zh
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Leading order SIDIS

• In limit of large Q2 only result
• of handbag diagram survives
• Isolating parts encoding soft physics

?
?
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Lightfront correlator(distribution)

Lightfront correlator (fragmentation)
Collins Soper NP B 194 (1982) 445
no T-constraint TPh,Xgtout Ph,Xgtin
Jaffe Ji, PRL 71 (1993) 2547 PRD 57 (1998)
3057
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Distribution

A
including the gauge link (in SIDIS)
One needs also AT Ga ? ATa ATa(x) ATa(8)
?dh Ga
Belitsky, Ji, Yuan, hep-ph/0208038 Boer, M,
Pijlman, hep-ph/0303034
From lty(0)AT(?)y(x)gt m.e.
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Distribution

A
including the gauge link (in SIDIS or DY)
SIDIS
A
DY
SIDIS ? F-
DY ? F
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Distribution

• for plane waves
• TPgt Pgt
• But...
• T U0, ? T U0,- ?
• this does affect
• F?(x,pT)
• ? appearance of
• T-odd functions
• in F?(x,pT)

including the gauge link (in SIDIS or DY)
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Parameterizations including pT
Ralston Soper NP B 152 (1979) 109
Tangerman Mulders PR D 51 (1995) 3357
Constraints from Hermiticity Parity

• Dependence
• on (x, pT2)
• Without T
• h1 and f1T
• nonzero!
• T-odd functions

• Fragmentation
• f ? D
• g ? G
• h ? H
• No T-constraint
• H1 and D1T
• nonzero!

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Distribution functions with pT
Ralston Soper NP B 152 (1979) 109
Tangerman Mulders PR D 51 (1995) 3357
Selection via specific probing operators (e.g.
appearing in leading order SIDIS or DY)
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Lightcone correlatormomentum density
Bacchetta, Boglione, Henneman Mulders PRL 85
(2000) 712
Remains valid for F(x,pT)
and also after inclusion of links for
F?(x,pT)
Sum over lightcone wf squared
Brodsky, Hoyer, Marchal, Peigne, Sannino PR D
65 (2002) 114025
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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
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Difference between F and F-
Integrate over pT
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Integrated distributions
T-odd functions only for fragmentation
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Weighted distributions
Appear in azimuthal asymmetries in SIDIS or
DY These are process-dependent (through gauge
link) and thus need in fact superscript!
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Matrix representationfor M F(x)gT
reminder
Collinear structure of the nucleon!
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Matrix representationfor M F(x,pT)gT

• pT-dependent
• functions

T-odd g1T ? g1T i f1T and h1L ? h1L i h1
Bacchetta, Boglione, Henneman Mulders PRL 85
(2000) 712
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Matrix representationfor M D(z,kT) g-T

• pT-dependent
• functions

• FFs
• f ? D
• g ? G
• h ? H

• No T-inv
• constraints
• H1 and
• D1T
• nonzero!

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Matrix representationfor M D(z,kT) g-T

• pT-dependent
• functions

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

e.g. pion

• FFs after
• kT-integration
• leaves just the
• ordinary D1(z)

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Summarizing SIDIS

• Beyond just extending DIS by tagging quarks
• Transverse momenta of partons become relevant,
  appearing in azimuthal asymmetries
• DFs and FFs depend on two variables,
• F?(x,pT) and D?(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

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T-odd effects in single spin asymmetries
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T-odd ? single spin asymmetry

• Wmn(qP,SPh,Sh) -Wnm(-qP,SPh,Sh)
• Wmn(qP,SPh,Sh) Wnm(qP,SPh,Sh)
• Wmn(qP,SPh,Sh) Wmn(qP, -SPh, -Sh)
• Wmn(qP,SPh,Sh) Wmn(qP,SPh,Sh)

symmetry structure
hermiticity

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parity
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_
_
_
time reversal

Conclusion
with time reversal constraint only even-spin
asymmetries
But time reversal constraint cannot be applied
in DY or in ?1-particle inclusive DIS or ee-
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Example of a single spin asymmetry
examplesOTO in ep? ? epX

• example of a leading azimuthal asymmetry
• T-odd fragmentation function (Collins function)
• involves two chiral-odd functions
• Best way to get transverse spin polarization
  h1q(x)

Tangerman Mulders PL B 352 (1995) 129
Collins NP B 396 (1993) 161
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Single spin asymmetriessOTO

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

Boer Mulders PR D 57 (1998) 5780
Boglione Mulders PR D 60 (1999) 054007
Collins NP B 396 (1993) 161
Sivers PRD 1990/91
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Process dependence and universality
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Difference between F and F-
?
integrated quark distributions
transverse moments
measured in azimuthal asymmetries

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Difference between F and F-
gluonic pole m.e.
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Time reversal constraints for distribution
functions
T-odd (imaginary)
Time reversal F(x,pT) ? F-(x,pT)
pFG
F?
F?
T-even (real)
F?-
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Consequences for distribution functions
SIDIS F DY F-
F??(x,pT) F?(x,pT) pFG
Time reversal ?
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Distribution functions
F??(x,pT) F?(x,pT) pFG
Sivers effect in SIDIS and DY opposite in sign
Collins hep-ph/0204004
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Time reversal constraints for fragmentation
functions
T-odd (imaginary)
Time reversal Dout(z,pT) ? D-in(z,pT)
pDG
D?
D?
T-even (real)
D?-
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Time reversal constraints for fragmentation
functions
T-odd (imaginary)
Time reversal Dout(z,pT) ? D-in(z,pT)
D?out
pDG out
D? out
T-even (real)
D?-out
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Fragmentation functions
D??(x,pT) D?(x,pT) pDG
Collins effect in SIDIS and ee- unrelated!
Time reversal does not lead to constraints
If pDG 0
But at present this seems (to me) unlikely
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T-odd phenomena

• T-invariance does not constrain fragmentation
• T-odd FFs (e.g. Collins function H1)
• T-invariance does constrain F(x)
• No T-odd DFs and thus no SSA in DIS
• T-invariance does not constrain F(x,pT)
• T-odd DFs 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 ee- (unless pDG 0)

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What about hadroproduction?
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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
• DG(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

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Issues in hadroproduction

• Contributions of F?(x,pT) and pFG not necessarily
  in one combination
• AN 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
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Thank you for your attention