# Presentation: 202729

## single spin asymmetries inhard reactions

- P.J. Mulders
- Vrije Universiteit
- Amsterdam
- mulders@nat.vu.nl

- Los Alamos
- 5 Aug. 2002

## 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

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## Soft physics in inclusive deep inelastic leptoproduction

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## (calculation of) cross sectionDIS

- Full calculation

- + ...

- +

- +

- +

- PARTON
- MODEL

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## Leading order DIS

- In limit of large Q2 only result
- 	of ‘handbag diagram’ survives
- Isolate part encoding soft physics

- ?

- ?

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## Lightcone dominance in DIS

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## Distribution functions

- Parametrization
- consistent with:
- Hermiticity, Parity
- & Time-reversal

- Soper
- Jaffe Ji  NP B 375 (1992) 527

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## Distribution functions

- Jaffe Ji
- NP B 375 (1992) 527

- Selection via specific probing operators
- (e.g. appearing in leading order DIS, SIDIS or DY)

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## Lightcone correlatormomentum density

- Bacchetta Boglione Henneman Mulders
- PRL 85 (2000) 712

-  = 1⁄2 

- Sum over
- lightcone wf
- squared

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## Lightfront quantization

- Kogut & Soper

-  Good fields are
-   independent,
-   satisfying CCR’s
-  Suitable to define
-   the partons in QCD

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## 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

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## Matrix representation

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

- Related to the
- helicity formalism

- Anselmino et al.

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## chiral-odd functions

- diagonal
- in transverse
- spin basis

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## Color gauge link in correlator

-   Diagrams containing correlator
- A+  produce the gauge
-  link U(0,) in quark-quark
-  lightcone correlator

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## (x) subleading

-  include M/P+ parts
-  gives M/Q terms in 
- T-odd only for FF in e.g. e+e

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

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## 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) =q(x), h1q(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

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

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## SIDIS cross section

-  variables
-  hadron tensor

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## (calculation of) cross sectionSIDIS

- Full calculation

- +

- + ...

- +

- +

- PARTON
- MODEL

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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 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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## Lightfront correlator(distribution)

- Lightfront correlator (fragmentation)

**+**

**no T-constraint**
- T|Ph,X>out = |Ph,X>in

- Collins  Soper
- NP B 194 (1982) 445

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## distribution functions in SIDIS

- Constraints from Hermiticity & Parity

-  Dependence
-    on ...(x, pT2)
-  T-invariance:
-   h1= f1T = 0?
-   T-odd functions

- Ralston  Soper
- NP B 152 (1979) 109

- Tangerman  Mulders
- PR D 51 (1995) 3357

-  Fragmentation
-    f  D
-    g  G
-    h  H
-  No T-constraint:
-    H1 and D1T
-    nonzero!

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## Distribution functions in SIDIS

- 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 correlatormomentum density

- Bacchetta Boglione Henneman Mulders
- PRL 85 (2000) 712

- Remains valid for (x,pT)

-  = 1⁄2 

- Sum over
- lightcone wf
- squared

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## Interpretation

- unpolarized quark
- distribution

- helicity or chirality
- distribution

- transverse spin distr.
- or transversity

- need pT

- need pT

- need pT

- need pT

- need pT

- T-odd

- T-odd

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## Matrix representationfor M = [(x)+]T

- Collinear structure of the nucleon!

- Matrix representationfor M = [(x)+]T

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## Matrix representationfor M = [(x,pT) +]T

-  pT-dependent
-    functions

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

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## Matrix representation for M = [(z,kT) ]T

-  pT-dependent
-    functions

-  FF’s:
-    f  D
-    g  G
-    h  H

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

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## Matrix representation for M = [(z,kT) ]T

-  pT-dependent
-    functions

-  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

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## Distribution and fragmentation functions

-  pT-integrated

-  pT-dependent

- no T-odd

- T-odd ?

- Mulders  Tangerman
- NP B 461 (1996) 197

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## Sample azimutal asymmetryLTO

-  example of a leading
-    azimuthal asymmetry
-  without appropriate
-    weights this would be
-    a convolution instead of
-    a factorized expression

- Tangerman  Mulders
- PL B 352 (1995) 129

- Kotzinian
- NP B 441 (1995) 234

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## Sample single spin asymmetryOTO

-  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)

- Tangerman  Mulders
- PL B 352 (1995) 129

- Collins
- NP B 396 (1993) 161

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## Summarizing SIDIS

- Transverse momenta of partons become relevant, effects appearing in azimuthal asymmetries
- DF’s and FF’s depend on two variables,
- (x,pT) and (z,kT)
- Fragmentation functions are not constrained by time-reversal invariance
- This allows T-odd functions H1 and D1T, appearing in single spin asymmetries

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## 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
- What about T-invariance and (x,pT)?

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## Color gauge link in correlator

- ?

-  Diagrams containing correlator A+
-  produce the gauge links U(0,-) and
-  U(-,) in quark-quark matrix element

- Boer  Mulders
- NP B 569 (2000) 505
- hep-ph/9906223

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## Distribution

-  for plane waves
-   T|P> = |P>
-  But...T UT = U
- this does not
-   affect (x)
-  it does affect
-    (x,pT)
-   appearance of
-   T-odd functions
-   in (x,pT)
-  color gauge inv
-   requires leading
-   transverse gluons

- ?

- including the gauge link

- From AT(-)y m.e.?

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## T-odd phenomena

- T-invariance does not constrain (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

- Brodsky  Hwang  Schmidt      hep-ph/0201296

- Ji  Yuan      hep-ph/0206057

- Collins     hep-ph/0204004

- Brodsky  Hoyer     PR D 65 (2002) 114025

- Boer  Mulders  Teryaev
- PR D 57 (1998) 3057
- hep-ph/9710223

- Qiu  Sterman

- Schaefer   Teryaev

- Related work:

- Mulders  Tangerman       NP B 461 (1996) 197

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## Single spin asymmetriesOTO

-  T-odd fragmentation function (Collins function)
-    **or**
-  T-odd distribution function (Sivers function)
-  Both of the above can explain pp  X SSA
-  Different asymmetries in leptoproduction!

- Boer  Mulders
- PR D 57 (1998) 5780

- Boglione  Mulders
- PR D 60 (1999) 054007

- Collins
- NP B 396 (1993) 161

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## 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 AT pieces in gauge link

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