Theoretical developments

1
Theoretical developments
Transversity workshop Trento, 17/06/2004

• piet mulders

pjg.mulders_at_few.vu.nl
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 DFs, rendering them gauge inv.
• There are three leading DFs
• 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
• DFs 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

• In limit of large Q2 the result
• of handbag diagram survives
• contributions from A gluons
• ensuring color gauge invariance

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 DFs, rendering them gauge inv.
• There are three leading DFs
• 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
• DFs 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
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 DFs, rendering them gauge inv.
• There are three leading DFs
• 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
• DFs 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 representationfor M F(x)gT
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 DFs, rendering them gauge inv.
• There are three leading DFs
• 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
• DFs 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
• DFs and FFs 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 TPh,Xgtout Ph,Xgtin
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
• DFs and FFs 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
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.
15
Distribution

including the gauge link (in SIDIS or DY)
A
SIDIS
A
SIDIS ? F-
DY
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
• DFs and FFs 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 U0,? T U0,-?
• 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 representationfor M F(x,pT)gT

• 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 DFstwist 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 t1
  (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 DFstwist 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 t1
  (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 DFstwist 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 t1
  (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
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

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 Dout(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 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
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