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A sideways look into the proton
Transverse momentum and transverse spin in QCD
Alessandro Bacchetta
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Outline
• Introduction• Transverse spin• Transverse momentum• Transverse spin and transverse
momentum• Factorization and hadron collisions• Conclusions
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Introduction
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Deep Inelastic Scattering - DIS
target proton
electron
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Inclusive DIS
proton
lepton l
'l
tran
sver
se
longitudinal
( ) ( ) ( ) l p P l X
2 2 virtuality ( ' of pho o) t n l l Q2
2 ( ')
Q
xP l l
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Semi-inclusive DIS (SIDIS)
proton
pion
transverse momentum of pionhP
hP lepton l
'l
( ) ( ) ( ) ( ) hl p P l h P X
2 2 virtuality ( ' of pho o) t n l l Q2
2 ( ')
Q
xP l l
( ')
hP P
zP l l
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Vectors and angles involved
DIS
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Vectors and angles involved
SIDIS
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Transverse momentum effects
hP
SIDIS
Tq
Drell-Yan
TK
e-e+ to pions
TR
3-Dp-p to pions
Whenever we measure transverse-momentum effects, we need kT-factorization and we need transverse momentum dependent (or unintegrated) parton distributions
Collins, Soper, NPB 193 (81)
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Feynman diagrams & Factorization
proton
lepton lepton
pion
SIDIS
Partonic scattering amplitude
Fragmentation amplitude
Distribution amplitude
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Feynman diagrams & Factorization
Partonic scattering amplitude
Fragmentation amplitude
Distribution amplitude
proton
lepton lepton
pion
SIDIS
electron
positron
pion
pion
e–e+ to pions
proton
proton lepton
antilepton
Drell-Yan
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Feynman diagrams & Factorization
proton
lepton lepton
pion
SIDIS
proton
proton lepton
antilepton
Drell-Yan
proton
proton
pion
pion
p-p to pions
electron
positron
pion
pion
e–e+ to pions
??
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Nonperturbative elements
2
,
0
[0 ]( ; ) , (0) ( ) ,2
i x Pij j i
dx S e P S P SU
,P S
k
=
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Decomposition of the correlation function
1 1 5 1 5( ) ( )1
)2
(( ); Lq
Tq qf x g x h xx S S S
1
1
1
( )
(
(
)
( ) (
)
) )
(
(
)
q
q
qT
q x
q x
q x
f x
g x
q xh x
Unpolarized distr. func.
Helicity distr. func.
Transversity distr. func.
0(1 ), leading twistO Q
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Longitudinal vs transverse momentum
target
parton
Tk
P
partonk
longk xP
photon
Long. view
long. spinphoton
Trans. view Photon moves into the screen/ proton moves out of the screen
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Longitudinal vs transverse momentum
Tkpartonk
longk xP
photon
Long. view
trans. spinphoton
Trans. view Photon moves into the screen/ proton moves out of the screen
target
partonP
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Probabilistic interpretation
( ) ( )qTh x q x 1
_
( ) ( )qf x q x 1
( ) ( )qg x q x 1_
Photon moves into the screen/ proton moves out of the screen
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Decomposition of the correlation function UNintegrated over kT
12
1
21
2 ( , )1
( , ( , ))2
..( , ) .
qT T
qT
qT
T T TT
T
f x kS k
x kM
ki
Mh
k
k
f x
x
Boer-Mulders
Sivers
Sivers, PRD 43 (91)
Boer, Mulders, PRD 57 (98)
Mulders, Tangerman, NPB 461 (96)Goeke, Metz, Schlegel, PLB 618 (05)
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Probabilistic interpretation
-Sivers
-Boer-Mulders
Photon moves into the screen/ proton moves out of the screen
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Unpolarized distribution functions q(x)
ZEUS Coll, EPJ C42 (05)
7 groups are working on the extraction of these PDFs (see www-spires.dur.ac.uk/hepdata/pdf.html)
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Helicity distribution function q(x)
AAC, Hirai et al. PRD69 (04)
6 groups are working on the extraction of these PDFs (see www-spires.dur.ac.uk/hepdata/pdf.html)
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• What about the transverse spin distribution h1(x) ?
• What about the transverse momentum dependence, e.g. f1(x,kT
2)?
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Transverse spin
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Transversity in SIDIS
12
12(... sin( ) ( , ) ., .) .T h S TTh x pd S H z k
see e.g. A.B., Diehl, Goeke, Metz, Mulders, Schlegel, JHEP 0702,093
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Data from Hermes and Compass
HERMES, hep-ex/0507013
HERMES, PRL 94 (05)
COMPASS, NPB 765 (07)
COMPASS, PRL 94 (05)
PROTON DEUTERON
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Data from BELLE
BELLE, PRL 96 (06)
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First extraction of transversity
Anselmino et al., PRD 75, 054032 (07)
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Comparison with some models
[1] Soffer et al. PRD 65 (02)
[2] Korotkov et al. EPJC 18 (01)
[3] Schweitzer et al., PRD 64 (01)
[4] Wakamatsu, PLB 509 (01)
[5] Pasquini et al., PRD 72 (05)
[6] Anselmino et al., PRD 75 (07)
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Tensor charge from Lattice QCD
1 1
1 1
( ) 0.84, ( ) 0.64,
( ) 0.23, ( ) 0.35
u u
d d
u h x dx u g x dx
d h x dx d g x dx
S. Aoki et al., PRD 56 (1997)M. Göckeler et al. [QCDSF/UKQCD], PLB (05)
Compared to axial charge
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Comparison of tensor charges
Barone, Drago, Ratcliffe, PR 359 (2002)
lattice
quark soliton 1
spectator
bag
quark soliton 2
quark
non-relativistic
2.4 GeV2
0.3 GeV2
Wakamatsu, 0705.2917[hep-ph]
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Evolution equations
Barone, Drago, Ratcliffe, PR 359 (2002)
Hayashigaki, Kanazawa, Koike, PRD56 (97)
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Other observables
12
12(... sin( ) ( , ) ., .) .T h S TTh x pd S H z k
Efremov, Mankiewicz, Tornquist, PLB 284 (92)Collins, Heppelmann, Ladinsky, NPB 420 (94)Jaffe, Jin, Tang, PRL 80 (98)
12
1(... sin( ) ( , , ) .) ..T h R h xd S H z M
Dihadron fragmentation
11 1 212 ( )... ( ) ...T Td S h xS h x
Doubly polarized Drell-YanRalston, Soper, NPB 152 (79)
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Transverse momentum
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Semi-inclusive DIS
proton
pion
l p l X 2 2 virtuality of p( ') hotonl l Q
transverse momentum of pionhP
hP lepton l
'l
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Example of model calculation
diquarkproton
quark
(GeV)xk
yk
21 ( , )q
Tf x k
0.5x
(GeV)xk
yk
21 ( , )q
Tf x k
0.01x
2 20.13 GeVTk 2 20.48 GeVTk
Jakob, Mulders, Rodrigues, NPA 626 (1997)
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Example of a measurement
H1 Coll, NPB485(97)
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Transverse momentum distribution
q
q
h
intrinsic
pQCD
Koike, Nagashima, Vogelsang, NPB744 (06)
fragmentation
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Differences between flavors?
(GeV)xk (GeV)xk
yk yk
21 ( )u
Tf k 21 ( ) 2d
Tf k x
ud diquarkproton
u
uu diquarkproton
d
Assuming ud uuM M
Up and down quark can have different
transverse momentum distributions
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Transverse momentum and transverse spin
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right
left
lepton l
'l
Semi-inclusive Deep Inelastic Scattering
proton
pion
l p l X
R L
R L
N N
N N
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Measured asymmetries in DIS
HERMES Coll, hep-ex/0507013
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Parton distribution functions with transverse spin
-Sivers
-Boer-Mulders
-Transversity
Photon moves into the screen/ proton moves out of the screen
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right
left
Chromodynamic lensing
proton
quarks
Long. view Trans. view
Burkardt, PRD 66 (02)
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right
left
Chromodynamic lensing
photon
Long. view Trans. view
NOTE: QCD tells us that the FSI has to be attractive, since quark and remnants form a color antisymmetric state
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right
left
Chromodynamic lensing
photon
Long. view Trans. view
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right
left
Distortions in transverse space
proton
quarks
Long. view Trans. view
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right
left
Distortions in transverse space
proton
quarks
Long. view Trans. view
A distortion in the distribution of quarks in transverse space can give rise to a nonzero Sivers function
The presence of spin can distort the distribution of quarks in transverse space (orbital angular momentum of quarks is required)
Spin-orbit correlations
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Results from Lattice QCD
QCDSF/UKQCD Collab. (see e.g. hep-ph/05110032)
Sivers function expected to be:
•NEGATIVE for up quarks
•POSITIVE for down quarks
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Data from Hermes and Compass
HERMES, hep-ex/0507013
HERMES, PRL 94 (05)COMPASS, PRL 94 (05)
PROTON
DEUTERON
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Different fits
Anselmino et al., hep-ph/0511017
[20] Anselmino et al., PRD72 (05)
[21] Vogelsang, Yuan, PRD72 (05)
[23] Collins et al., hep-ph/0510342
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Factorization and hadron-hadron
collisions
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Feynman diagrams & Factorization
proton
lepton lepton
pion
SIDIS
proton
proton lepton
antilepton
Drell-Yan
proton
proton
pion
pion
p-p to pions
electron
positron
pion
pion
e–e+ to pions
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right
left
Sivers effect in SIDIS
proton
quarks
Long. view Trans. view
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right
left
Sivers effect in Drell-Yan
proton
Long. view Trans. view
photon
antiquark
The Sivers effect gets an extra minus sign due to the difference between initial and final state interactions. Impossible to predict
it in parton model.
Clear-cut prediction of QCD Collins, PLB 536 (02)
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Final/initial state interactions
SIDIS
q
q
q
q
pp to hadrons
A.B., Bomhof, Mulders, Pijlman, hep-ph/0505268
Drell-Yan
q
q
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kT -factorization is broken
My personal opinion: what is shown by Collins and Qiu is not sufficient to make such a statement. Maybe new concepts (i.e. “advanced” factorization and “generalized” universality) can work. Bomhof, Mulders, Pijlman, PLB596
A.B., Bomhof, Mulders, Pijlman, PRD72
Is kT -factorization broken ?
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An example
jet atRHICp p X
A.B., D’Alesio, Bomhof, Mulders, Murgia, hep-ph/0703153
“Standard” factorization & universality
“Generalized” factorization & universality
Broken factorization?
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Conclusions
• Transverse spin and transverse momentum observables uncover a new dimension of the structure of the nucleon
• Intense progress is taking place both from the theoretical and experimental point of view
• Impact also on hadron colliders and unpolarized physics