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Universality of T-odd effects in single spin azimuthal asymmetries
P.J. MuldersVrije Universiteit
Amsterdam
BNLDecember 2003
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
4-12-2003 BNL 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
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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
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) 527Jaffe & Ji PRL 71 (1993) 2547
leading part
Distribution functions
Jaffe & JiNP B 375 (1992) 527
Selection via specific probing operators(e.g. appearing in leading order DIS, SIDIS or DY)
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 representation
Related to thehelicity formalism
Anselmino et al.
Bacchetta, Boglione, Henneman & MuldersPRL 85 (2000) 712
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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 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
Lightfront dominance in SIDIS
Three external momentaP Ph q
transverse directions relevantqT = q + xB P – Ph/zh
orqT = -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
? ?
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+
Ji, Yuan, PLB 543 (2002) 66Belitsky, Ji, Yuan, hep-ph/0208038
Distribution
for plane waves T|P> = |P> But... T U
T = U
this does affect (x,pT) it does not affect (x) 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
Bacchetta, Boglione, Henneman & MuldersPRL 85 (2000) 712
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
Weighted distributions
Appear in azimuthal asymmetries in SIDIS or DYThese are process-dependent (through gauge link)
pT-dependent functions
T-odd: g1T g1T – i f1T and h1L
h1L + i
h1
Matrix representationfor M = [(x,pT)+]T
Bacchetta, Boglione, Henneman & MuldersPRL 85 (2000) 712
Matrix representationfor M = [(z,kT) ]T
pT-dependent functions
FF’s: f D g G h H
No T-inv constraints H1
and
D1T
nonzero!
Matrix representationfor 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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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 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)
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
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!
Boer & MuldersPR D 57 (1998) 5780
Boglione & MuldersPR D 60 (1999) 054007
CollinsNP B 396 (1993) 161
SiversPRD 1990/91
Difference between [+] and [-]
integrated quarkdistributions
transverse moments
measured in azimuthal asymmetries
±
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Time reversal constraints for distribution functions
Time reversal(x,pT) (x,pT)
G
T-even(real)
T-odd(imaginary)
Distribution functions
(x,pT)
= (x,pT) ± G
Sivers effect in SIDISand DY opposite in sign
Collins hep-ph/0204004
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Time reversal constraints for fragmentation functions
Time reversalout(z,pT)
in(z,pT)
G
T-even(real)
T-odd(imaginary)
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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
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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
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)
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Issues in hadroproduction
Weighted functions will appear in L-R asymmetries (pT now hard scale!) There are 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 …
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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
Relations among distribution functions
1. Equations of motion2. Define interaction dependent functions3. Use Lorentz invariance
Distribution functions
(x,pT)
= (x,pT) ± G
(omitting mass terms)
Sivers effect in SIDISand DY opposite in sign
Collins hep-ph/0204004
Fragmentation functions
(x,pT)
= (x,pT) ± G
Collins effect in SIDISand e+e unrelated!
including relations