Transcript Template

‘Lepton scattering and the
structure of nucleons and nuclei’
September 16-24, 2004
Polarized structure functions
Piet Mulders
[email protected]
Content
• Spin structure & transversity
• Transverse momenta & azimuthal asymmetries
• T-odd phenomena & single spin asymmetries
DIS
• Known leptonic part
• Completeness allows
reduction in hadronic
tensor to commutator
[Jm(x),Jn(0)]
• Known structure of
current in terms of quarks
• OPE
• ….
Deep inelastic scattering (DIS)
Lepton tensor
• Lepton tensor can
also be expanded
using the spacelike
and timelike vectors
• Tensor encompasses
many ‘polarization
options’
Polarized DIS
Semi-inclusive deep inelastic scattering
• Known lepton part
with much flexibility
(unused in DIS)
• Involves two hadrons
and hence a much
more complex
hadronic tensor
SIDIS
(calculation of) cross section in DIS
Full calculation
+
PARTON
MODEL
+
+
+…
Lightcone dominance in DIS
Leading order DIS
• In limit of large Q2 the result
of ‘handbag diagram’ survives
• … + contributions from A+ gluons
ensuring color gauge invariance
A+
Ellis, Furmanski, Petronzio
Efremov, Radyushkin
A+ gluons
 gauge link
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
leading part
Jaffe & Ji
Jaffe & Ji
NP B 375 (1992) 527
PRL 71 (1993) 2547
Basis of
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
Matrix representation
for M = [F(x)g+]T
Bacchetta, Boglione, Henneman & Mulders
PRL 85 (2000) 712
Quark production
matrix, directly
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
Results for DIS
• Structure functions in (sub)leading order in 1/Q
F2 ( x , Q )  2 xF1 ( x , Q ) 
2
2

2
q
e q xf 1 ( x )
q
2 g1 ( x, Q ) 
2

2
q
e q xg 1 ( x )
q
2 gT ( x, Q ) 
2

2
q
e q xg T ( x )
q
• Two of three (Polarized) quark densities for each flavor:
f1 ( x )  q ( x )
q
g1 ( x )   q ( x )
q
h1 ( x )   q ( x )
q
Not accessible in DIS
(calculation of) cross section in SIDIS
“Full” calculation
+
PARTON
MODEL
+
+
+…
Lightfront dominance in SIDIS
Three external momenta
P Ph q
transverse directions relevant
qT = q + xB P – Ph/zh
or
qT = -Ph^/zh
Leading order SIDIS
• In limit of large Q2 only result
of ‘handbag diagram’ survives
• Isolating parts encoding soft physics
?
?
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
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/0208038
Boer, M, Pijlman, hep-ph/0303034
From <y(0)AT()y(x)> m.e.
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^
Interpretation
unpolarized quark
distribution
need pT
helicity or chirality
distribution
need pT
transverse spin distr.
or transversity
need pT
need pT
Matrix representation
for M = [F[±](x,pT)g+]T
 pT-dependent
functions
T-odd: g1T  g1T – i f1T^ and h1L^  h1L^ + i h1^ (imaginary parts)
Bacchetta, Boglione, Henneman & Mulders
PRL 85 (2000) 712
T-odd  single spin asymmetry
 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)
___ _
_
*
* (q;P,S;P ,S ) = W (q;P,S;P ,S )
 Wmn
h h
mn
h h
_
_
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 e+e• In those cases single spin asymmetries can be used to select T-odd
quantities
Leptoproduction of pions
H1^ is T-odd
and chiral-odd
COLLINS ASYMMETRY
RESULTS OF COMPASS
Acoll depends on phT, zh, xBj
with more statistics, the full analysis is foreseen
from 2002 data:
Acoll vs xBj
COLLINS ASYMMETRY
RESULTS OF COMPASS
from 2002 data:
AColl vs zh
all the tests made are consistent with the fact that systematic
effects, if present, are smaller than statistical errors
Distribution
including the gauge link (in SIDIS or DY)
A+
SIDIS
A+
DY
SIDIS  F[-]
DY  F[+]
Difference between F[+] and F[-] upon integration
Back to the lightcone (theoretically clean)
integrated quark
distributions

twist 2
transverse
moments
measured in
azimuthal
asymmetries
twist 2 & 3
±
Difference between F[+] and F[-] upon integration
In momentum space:
gluonic pole
m.e. (T-odd)
Conclusion: T-odd parts
are gluon-driven (QCD
interactions)
Time reversal constraints for
distribution functions
T-odd
(imaginary)
pFG
Time reversal: F[+](x,pT)  F[-](x,pT)
F[+]
F
F[-]
T-even
(real)
Conclusion:
T-odd effects in SIDIS
and DY have opposite
signs
Time reversal constraints for
fragmentation functions
T-odd
(imaginary)
pG
Time reversal: [+]out(z,pT)  [-]in(z,pT)
[+]

[-]
T-even
(real)
Time reversal constraints for
fragmentation functions
T-odd
(imaginary)
pG out
Time reversal: [+]out(z,pT)  [-]in(z,pT)
[+]out
 out
[-]out
T-even
(real)
Conclusion:
T-odd effects in SIDIS
and e+e- are not
related
C. Bomhof, P.J. Mulders and F. Pijlman
PLB 596 (2004) 277
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
e.g.
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 (T-odd) gluonic
pole contribution involving
also specific color factors
• Finding the right observables,
however is crucial
Conclusions
• Hard processes  quark and gluon structure of hadrons (quark
distributions, their chirality and transverse polarization)
• Many new observables accessible when going beyond collinearity,
often in combination with (transverse) polarization (among others
the simplest access to transverse quark polarization)
• Going beyond collinearity gives access to gluon dynamics in
hadrons, which can be done in a controlled way via weighted
asymmetries (twist limited, t  3), use of chirality, and the specific
time-reversal behavior of single spin asymmetries.