1. Theoretical developments
Transversity workshop
Trento, 17/06/2004
Theoretical developments
piet mulders

2. Content Spin structure & transversity
Transverse momenta & azimuthal asymmetries
T-odd phenomena & single spin asymmetries

3. Collinear hard processes, e.g. DIS
Structure functions (observables) are identified with distribution functions (matrix elements of quark-quark correlators along lightcone)
Longitudinal gluons (A+, not seen in LC gauge) are part of DF’s, rendering them gauge inv.
There are three ‘leading’ DF’s
f1q(x) = q(x), g1q(x) = Dq(x), h1q(x) = dq(x)
Transverse gluons appear at 1/Q and are contained in (higher twist) qqG-correlators
DF’s are quark densities that are directly linked to lightcone wave functions squared
Perturbative QCD  evolution (reflecting the large pT-behavior of the correlator proportional to as/pT2 in the pT-integration)

4. Leading order DIS A+ In limit of large Q2 the result
of ‘handbag diagram’ survives
… + contributions from A+ gluons
ensuring color gauge invariance
A+ gluons
 gauge link
Ellis, Furmanski, Petronzio
Efremov, Radyushkin A+

5. Collinear hard processes, e.g. DIS
Structure functions (observables) are identified with distribution functions (matrix elements of quark-quark correlators along lightcone)
Longitudinal gluons (A+, not seen in LC gauge) are part of DF’s, rendering them gauge inv.
There are three ‘leading’ DF’s
f1q(x) = q(x), g1q(x) = Dq(x), h1q(x) = dq(x)
Transverse gluons appear at 1/Q and are contained in (higher twist) qqG-correlators
DF’s are quark densities that are directly linked to lightcone wave functions squared
Perturbative QCD  evolution (reflecting the large pT-behavior of the correlator proportional to as/pT2 in the pT-integration)

6. 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
leading part

7. Collinear hard processes, e.g. DIS
Structure functions (observables) are identified with distribution functions (matrix elements of quark-quark correlators along lightcone)
Longitudinal gluons (A+, not seen in LC gauge) are part of DF’s, rendering them gauge inv.
There are three ‘leading’ DF’s
f1q(x) = q(x), g1q(x) = Dq(x), h1q(x) = dq(x)
Transverse gluons appear at 1/Q and are contained in (higher twist) qqG-correlators
DF’s are quark densities that are directly linked to lightcone wave functions squared
Perturbative QCD  evolution (reflecting the large pT-behavior of the correlator proportional to as/pT2 in the pT-integration)

8. Matrix representation for M = [F(x)g+]T
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

9. Collinear hard processes, e.g. DIS
Structure functions (observables) are identified with distribution functions (matrix elements of quark-quark correlators along lightcone)
Longitudinal gluons (A+, not seen in LC gauge) are part of DF’s, rendering them gauge inv.
There are three ‘leading’ DF’s
f1q(x) = q(x), g1q(x) = Dq(x), h1q(x) = dq(x)
Transverse gluons appear at 1/Q and are contained in (higher twist) qqG-correlators
DF’s are quark densities that are directly linked to lightcone wave functions squared
Perturbative QCD  evolution (reflecting the large pT-behavior of the correlator proportional to as/pT2 in the pT-integration)

10. Non-collinear processes, e.g. SIDIS
Relevant in electroweak processes with two hadrons (SIDIS, DY)
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,
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

11. ? ? Leading order SIDIS In limit of large Q2 only result
of ‘handbag diagram’ survives
Isolating parts encoding soft physics ? ?

12. Lightfront correlators
Collins & Soper
NP B 194 (1982) 445
no T-constraint
T|Ph,X>out = |Ph,X>in
Jaffe & Ji,
PRL 71 (1993) 2547;
PRD 57 (1998) 3057

13. Non-collinear processes, e.g. SIDIS
Relevant in electroweak processes with two hadrons (SIDIS, DY)
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,
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

14. including the gauge link (in SIDIS)
Distribution
including the gauge link (in SIDIS) A+
One needs also AT
G+a =  +ATa
ATa(x)= ATa(∞) +  dh G+a
Belitsky, Ji, Yuan, hep-ph/
Boer, M, Pijlman, hep-ph/
From <y(0)AT()y(x)> m.e.

15. including the gauge link (in SIDIS or DY)
Distribution
including the gauge link (in SIDIS or DY) A+
SIDIS A+ DY
SIDIS  F[-]
DY  F[+]

16. Non-collinear processes, e.g. SIDIS
Relevant in electroweak processes with two hadrons (SIDIS, DY)
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,
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

17. Parametrization of F(x,pT)
Link dependence allows also T-odd distribution functions since T U[0,] T = U[0,-]
Functions h1^ and f1T^ (Sivers) nonzero!
These functions (of course) exist as fragmentation functions (no T-symmetry) H1^ (Collins) and D1T^

18. 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

19. Matrix representation for M = [F[±](x,pT)g+]T
pT-dependent
functions
T-odd: g1T  g1T – i f1T^ and h1L^  h1L^ + i h1^
Bacchetta, Boglione, Henneman & Mulders
PRL 85 (2000) 712

20. pT-dependent DF’s twist structure
For integrated correlator F(x) the (M/P+)t-2 expansion corresponds with definite twist assignments a la OPE
For unintegrated correlators F[](x,pT) the (M/P+)t-2 does not correspond with definite twist assignments; they contain operators of twist  t
Transverse moments Fa(x,pT)   d2pT pTa F(x,pT) project out the parts in F[](x,pT) proportional to pT . They correspond to matrix elements with operators of twist t and t+1 (quark-quark and quark-quark-gluon)
Transverse moments are measured in azimuthal asymmetries with in addition to angular averaging also require an explicit (transverse) momentum
The difference between F[+]a(x) and F[-]a(x) is a gluonic pole matrix element (soft gluon pole), which for distribution functions is T-odd
The socalled Lorentz invariance relations based on general structure of quark-quark correlators need to be augmented because of gauge link
Factorization of explicit pT-dependent functions requires ‘soft factors’
Universality of subleading functions (appearing at M/P+ level) in F[+]a(x) seems ok, but this seems not the case for subleading functions in F[+]a(x) [so f1T(1) ok but f(1) problematic: in cos fh asymmetry pTa f at order 1/Q gets pQCD correction as f1/|pT|, which shows up as as f1 at order 1]

21. Difference between F[+] and F[-] upon integration
Back to the lightcone 
integrated quark
distributions
transverse
moments
measured in azimuthal asymmetries

22. pT-dependent DF’s twist structure
For integrated correlator F(x) the (M/P+)t-2 expansion corresponds with definite twist assignments a la OPE
For unintegrated correlators F[](x,pT) the (M/P+)t-2 does not correspond with definite twist assignments; they contain operators of twist  t
Transverse moments Fa(x,pT)   d2pT pTa F(x,pT) project out the parts in F[](x,pT) proportional to pT . They correspond to matrix elements with operators of twist t and t+1 (quark-quark and quark-quark-gluon)
Transverse moments are measured in azimuthal asymmetries with in addition to angular averaging also require an explicit (transverse) momentum
The difference between F[+]a(x) and F[-]a(x) is a gluonic pole matrix element (soft gluon pole), which for distribution functions is T-odd
The socalled Lorentz invariance relations based on general structure of quark-quark correlators need to be augmented because of gauge link
Factorization of explicit pT-dependent functions requires ‘soft factors’
Universality of subleading functions (appearing at M/P+ level) in F[+]a(x) seems ok, but this seems not the case for subleading functions in F[+]a(x) [so f1T(1) ok but f(1) problematic: in cos fh asymmetry pTa f at order 1/Q gets pQCD correction as f1/|pT|, which shows up as as f1 at order 1]

23. Difference between F[+] and F[-] upon integration
In momentum space:
gluonic pole m.e.
(T-odd)

24. pT-dependent DF’s twist structure
For integrated correlator F(x) the (M/P+)t-2 expansion corresponds with definite twist assignments a la OPE
For unintegrated correlators F[](x,pT) the (M/P+)t-2 does not correspond with definite twist assignments; they contain operators of twist  t
Transverse moments Fa(x,pT)   d2pT pTa F(x,pT) project out the parts in F[](x,pT) proportional to pT . They correspond to matrix elements with operators of twist t and t+1 (quark-quark and quark-quark-gluon)
Transverse moments are measured in azimuthal asymmetries with in addition to angular averaging also require an explicit (transverse) momentum
The difference between F[+]a(x) and F[-]a(x) is a gluonic pole matrix element (soft gluon pole), which for distribution functions is T-odd
The socalled Lorentz invariance relations based on general structure of quark-quark correlators need to be augmented because of gauge link
Factorization of explicit pT-dependent functions requires ‘soft factors’
Universality of subleading functions (appearing at M/P+ level) in F[+]a(x) seems ok, but this seems not the case for subleading functions in F[+]a(x) [so f1T(1) ok but f(1) problematic: in cos fh asymmetry pTa f at order 1/Q gets pQCD correction as f1/|pT|, which shows up as as f1 at order 1]

25. T-odd phenomena T-odd phenomena appear in single spin asymmetries
T-odd parts for distribution functions are in the gluonic pole part, hence in F[+]a(x) and F[-]a(x) they have opposite signs
T-odd parts for fragmentation functions in D[+]a(x) and D[-]a(x) are not related. This needs to be considered including QCD corrections, because of the interplay between T-behavior of hadronic states and gauge links
Contributions in other hard processes, such as pp  pX involving three hadrons require a careful analysis

26. T-odd  single spin asymmetry
structure
Wmn(q;P,S;Ph,Sh) = -Wnm(-q;P,S;Ph,Sh)
Wmn(q;P,S;Ph,Sh) = Wnm(q;P,S;Ph,Sh)
Wmn(q;P,S;Ph,Sh) = Wmn(q;P, -S;Ph, -Sh)
Wmn(q;P,S;Ph,Sh) = Wmn(q;P,S;Ph,Sh) _
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 e+e-
In those cases single spin asymmetries can be used to select T-odd quantities

27. T-odd phenomena T-odd phenomena appear in single spin asymmetries
T-odd parts for distribution functions are in the gluonic pole part, hence in F[+]a(x) and F[-]a(x) they have opposite signs
T-odd parts for fragmentation functions in D[+]a(x) and D[-]a(x) are not related. This needs to be considered including QCD corrections, because of the interplay between T-behavior of hadronic states and gauge links
Contributions in other hard processes, such as pp  pX involving three hadrons require a careful analysis

28. 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[-]

29. T-odd phenomena T-odd phenomena appear in single spin asymmetries
T-odd parts for distribution functions are in the gluonic pole part, hence in F[+]a(x) and F[-]a(x) they have opposite signs
T-odd parts for fragmentation functions in D[+]a(x) and D[-]a(x) are not related. This needs to be considered including QCD corrections, because of the interplay between T-behavior of hadronic states and gauge links
Contributions in other hard processes, such as pp  pX involving three hadrons require a careful analysis

30. Time reversal constraints for fragmentation functions
T-odd
(imaginary)
Time reversal: D[+]out(z,pT)  D[-]in(z,pT)
pDG
D[+] D
T-even
(real)
D[-]

31. Time reversal constraints for fragmentation functions
T-odd
(imaginary)
Time reversal: D[+]out(z,pT)  D[-]in(z,pT)
D[+]out
pDG out
D out
T-even
(real)
D[-]out
Conclusion:
T-odd effects in SIDIS and e+e- are not related

32. T-odd phenomena T-odd phenomena appear in single spin asymmetries
T-odd parts for distribution functions are in the gluonic pole part, hence in F[+]a(x) and F[-]a(x) they have opposite signs
T-odd parts for fragmentation functions in D[+]a(x) and D[-]a(x) are not related. This needs to be considered including QCD corrections, because of the interplay between T-behavior of hadronic states and gauge links
Contributions in other hard processes, such as pp  pX involving three hadrons require a careful analysis

33. 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

34. 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 gluonic pole contribution involving also specific color factors