Structure functions are parton densities P.J. Mulders Vrije Universiteit Amsterdam UIUC March 2003 Universality of T-odd effects in single spin and azimuthal asymmetries, D. Boer, PJM and F. Pijlman, hep-ph/

27/03/2003UIUC p j mulders2 Content Observables in (SI)DIS in field theory language  lightcone/lightfront correlations Single-spin asymmetries in hard reactions  T-odd correlations T-odd observables in final (fragmentation) and initial state (distribution) correlations Structure functions remain parton densities Universality of T-odd phenomena

27/03/2003UIUC p j mulders3 Soft physics in inclusive deep inelastic leptoproduction

(calculation of) cross section DIS Full calculation + … + + +PARTON MODEL

Lightcone dominance in DIS

27/03/2003UIUC p j mulders6 Leading order DIS In limit of large Q 2 the result of ‘handbag diagram’ survives … + contributions from A + gluons A+A+ A + gluons  gauge link Ellis, Furmanski, Petronzio Efremov, Radyushkin

Color gauge link in correlator Matrix elements  A +  produce the gauge link U(0,  ) in leading quark lightcone correlator A+A+

Distribution functions Parametrization consistent with: Hermiticity, Parity & Time-reversal Soper Jaffe & Ji NP B 375 (1992) 527

Distribution functions  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) 527 Jaffe & Ji PRL 71 (1993) 2547 leading part

Distribution functions Jaffe & Ji NP 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

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

 Off-diagonal elements (RL or LR) are chiral-odd functions  Chiral-odd soft parts must appear with partner in e.g. SIDIS, DY Matrix representation Related to the helicity formalism Anselmino et al. Bacchetta, Boglione, Henneman & Mulders PRL 85 (2000) 712

27/03/2003UIUC p j mulders14 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 f 1 q (x) = q(x), g 1 q (x) =  q(x), h 1 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

27/03/2003UIUC p j mulders15 Soft physics in semi-inclusive (1-particle incl) leptoproduction

SIDIS cross section  variables  hadron tensor

(calculation of) cross section SIDIS Full calculation + + … + + PARTON MODEL

Lightfront dominance in SIDIS

Three external momenta P P h q transverse directions relevant q T = q + x B P – P h /z h or q T = -P h  /z h

27/03/2003UIUC p j mulders20 Leading order SIDIS In limit of large Q 2 only result of ‘handbag diagram’ survives Isolating parts encoding soft physics ? ?

Lightfront correlator (distribution) Lightfront correlator (fragmentation) + no T-constraint T|P h,X> out = |P h,X> in Collins & Soper NP B 194 (1982) 445 Jaffe & Ji, PRL 71 (1993) 2547; PRD 57 (1998) 3057

Distribution From  A T ( J )  m.e. including the gauge link (in SIDIS) A+A+ One needs also A T G +  =  + A T   A T  (  )= A T  ( J ) +  d  G +  Ji, Yuan, PLB 543 (2002) 66 Belitsky, Ji, Yuan, hep-ph/

Distribution A+A+ A+A+ including the gauge link (in SIDIS or DY) SIDIS SIDIS   [-] DY DY   [+] hep-ph/

Distribution  for plane waves T|P> = |P>  But... T U  J   T = U  J    this does affect   (x,p T )  it does not affect  (x)   appearance of T-odd functions in   (x,p T ) including the gauge link (in SIDIS or DY)

Parameterizations including p T Constraints from Hermiticity & Parity  Dependence on …(x, p T 2 )  Without T: h 1  and f 1T  nonzero! 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: H 1  and D 1T  nonzero!

Integrated distributions T-odd functions only for fragmentation

Weighted distributions Appear in azimuthal asymmetries in SIDIS or DY

T-odd  single spin asymmetry  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 h 1 q (x) Tangerman & Mulders PL B 352 (1995) 129 Collins NP B 396 (1993) 161 example:  OTO in ep   epX

Single spin asymmetries  OTO  T-odd fragmentation function (Collins function) or  T-odd distribution function (Sivers function)  Both of the above can explain SSA in pp    X  Different asymmetries in leptoproduction! Boer & Mulders PR D 57 (1998) 5780 Boglione & Mulders PR D 60 (1999) Collins NP B 396 (1993) 161 Sivers PRD 1990/91

27/03/2003UIUC p j mulders30 Summarizing SIDIS Beyond just extending DIS by tagging quarks … Transverse momenta of partons become relevant, effects appearing in azimuthal asymmetries DF’s and FF’s depend on two variables,   (x,p T ) and   (z,k T ) Gauge link structure is process dependent (   p T -dependent distribution functions and (in general) fragmentation functions are not constrained by time- reversal invariance This allows T-odd functions h 1  and f 1T  (H 1  and D 1T  ) appearing in single spin asymmetries

27/03/2003UIUC p j mulders31 Structure functions are parton densities

Distribution functions with p T 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)

Lightcone correlator momentum density Bacchetta, Boglione, Henneman & Mulders PRL 85 (2000) 712 Remains valid for  (x,p T ) … and also after inclusion of links for   (x,p T ) Sum over lightcone wf squared Brodsky, Hoyer, Marchal, Peigne, Sannino PR D 65 (2002)

Interpretation unpolarized quark distribution helicity or chirality distribution transverse spin distr. or transversity need p T T-odd

Collinear structure of the nucleon! Matrix representation for M = [  (x)  + ] T

 p T -dependent functions T-odd: g 1T  g 1T – i f 1T  and h 1L   h 1L  + i h 1  Matrix representation for M = [  (x,p T )  + ] T

Positivity and bounds

Matrix representation for M = [  (z,k T )   ] T  p T -dependent functions  FF’s: f  D g  G h  H  No T-inv constraints H 1  and D 1T  nonzero!

Matrix representation for M = [  (z,k T )   ] T  p T -dependent functions  FF’s after k T -integration leaves just the ordinary D 1 (z)  R/L basis for spin 0  Also for spin 0 a T-odd function exist, H 1  (Collins function) e.g. pion

27/03/2003UIUC p j mulders41 Process dependence and universality

Difference between  [+] and  [-] Integrate over p T

Difference between  [+] and  [-]  integrated quark distributions transverse moments measured in azimuthal asymmetries ±

Difference between  [+] and  [-] gluonic pole m.e.

27/03/2003UIUC p j mulders45 Time reversal constraints for distribution functions Time reversal   (x,p T )    (x,p T )  G        T-even (real) T-odd (imaginary)

Consequences for distribution functions    (x,p T ) =   (x,p T ) ±  G Time reversal  SIDIS  [+] DY  [-]

Distribution functions    (x,p T ) =   (x,p T ) ±  G Sivers effect in SIDIS and DY opposite in sign Collins hep-ph/

Relations among distribution functions 1. Equations of motion 2. Define interaction dependent functions 3. Use Lorentz invariance

Distribution functions    (x,p T ) =   (x,p T ) ±  G (omitting mass terms) Sivers effect in SIDIS and DY opposite in sign Collins hep-ph/

27/03/2003UIUC p j mulders50 Time reversal constraints for fragmentation functions Time reversal   out (z,p T )    in (z,p T )  G        T-even (real) T-odd (imaginary)

27/03/2003UIUC p j mulders51 Time reversal constraints for fragmentation functions  G out    out   out    out T-even (real) T-odd (imaginary) Time reversal   out (z,p T )    in (z,p T )

Fragmentation functions    (x,p T ) =   (x,p T ) ±  G Time reversal does not lead to constraints Collins effect in SIDIS and e + e  unrelated!

Fragmentation functions    (x,p T ) =   (x,p T ) ±  G Collins effect in SIDIS and e + e  unrelated! including relations

27/03/2003UIUC p j mulders54 T-odd phenomena T-invariance does not constrain fragmentation T-odd FF’s (e.g. Collins function H 1  ) T-invariance does constrain  (x) No T-odd DF’s and thus no SSA in DIS T-invariance does not constrain  (x,p T ) 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 