Polarized structure functions

1
Polarized structure functions
Lepton scattering and the structure of nucleons
and nuclei September 16-24, 2004

• Piet Mulders

pjg.mulders_at_few.vu.nl
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Content

• Spin structure transversity
• Transverse momenta azimuthal asymmetries
• T-odd phenomena single spin asymmetries

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DIS

• Known leptonic part
• Completeness allows reduction in hadronic tensor
  to commutator Jm(x),Jn(0)
• Known structure of current in terms of quarks
• OPE
• .

4
Deep inelastic scattering (DIS)
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Lepton tensor

• Lepton tensor can also be expanded using the
  spacelike and timelike vectors
• Tensor encompasses many polarization options

6
Polarized DIS
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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

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SIDIS
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(calculation of) cross section in DIS
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
• ensuring color gauge invariance

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Parametrization of lightcone correlator

• 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
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Basis of 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
Quark production matrix, directly 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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Results for DIS

• Structure functions in (sub)leading order in 1/Q
• Two of three (Polarized) quark densities for each
  flavor

Not accessible in DIS
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(calculation of) cross section in SIDIS
Full calculation


PARTON MODEL


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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 correlators
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

including the gauge link (in SIDIS)
A
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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Parametrization of F(x,pT)

• Link dependence allows also T-odd distribution
  functions since T U0,? T U0,-?
• Functions h1 and f1T (Sivers) nonzero!
• These functions (of course) exist as
  fragmentation functions (no T-symmetry) H1
  (Collins) and D1T

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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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Matrix representationfor M F(x,pT)gT

• pT-dependent
• functions

T-odd g1T ? g1T i f1T and h1L ? h1L i h1
(imaginary parts)
Bacchetta, Boglione, Henneman Mulders PRL 85
(2000) 712
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T-odd ? single spin asymmetry
symmetry structure
hermiticity


parity
time reversal

• 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

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Leptoproduction of pions
H1? is T-odd and chiral-odd
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(No Transcript)
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COLLINS ASYMMETRYRESULTS OF COMPASS
Acoll depends on phT, zh, xBj with
more statistics, the full analysis is
foreseen from 2002 data
Sign!
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COLLINS ASYMMETRYRESULTS OF COMPASS
from 2002 data AColl vs zh
Sign!

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

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Distribution

including the gauge link (in SIDIS or DY)
A
SIDIS
A
SIDIS ? F-
DY
DY ? F
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Difference between F and F- upon
integration
Back to the lightcone (theoretically clean)
?
integrated quark distributions
twist 2
transverse moments
measured in azimuthal asymmetries
twist 2 3

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Difference between F and F- upon integration
In momentum space
gluonic pole m.e. (T-odd)
Conclusion T-odd parts are gluon-driven (QCD
interactions)
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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)
Conclusion T-odd effects in SIDIS and DY have
opposite signs
F?-
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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
Conclusion T-odd effects in SIDIS and ee- are
not related
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other hard processes
C. Bomhof, P.J. Mulders and F. Pijlman PLB 596
(2004) 277

• 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.
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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 F(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
  

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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.