A sideways look into the proton
Transverse momentum and transverse spin in QCD
Alessandro Bacchetta
Outline
•
•
•
•
Introduction
Transverse spin
Transverse momentum
Transverse spin and transverse
momentum
• Factorization and hadron collisions
• Conclusions
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Introduction
Deep Inelastic Scattering - DIS
electron
target proton
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Inclusive DIS
(l )  p (P )  (l )  X
(l  l ')2  Q 2  virtuality of photon
l'
Q2
x
2P  (l  l ')
lepton
transverse
proton
l
longitudinal
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Semi-inclusive DIS (SIDIS)
(l )  p (P )  (l )  h(Ph )  X

(l  l ')2  Q 2  virtuality of photon
l'
Q2
x
2P  (l  l ')
proton
lepton
Ph
l
P  Ph
z
P  (l  l ')
Alessandro Bacchetta
pion
Ph  transverse momentum of pion
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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
Drell-Yan
SIDIS
Ph
KT
qT
p-p to pions
3-D
e-e+ 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)
RT
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Feynman diagrams & Factorization
lepton
lepton
proton
pion
SIDIS
Partonic scattering
amplitude
Fragmentation amplitude
Distribution amplitude
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Feynman diagrams & Factorization
lepton
lepton
proton
lepton
proton
pion
proton
antilepton
SIDIS
Drell-Yan
electron
pion
Partonic scattering
amplitude
Fragmentation amplitude
positron
e–e+
pion
to pions
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Distribution amplitude
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Feynman diagrams & Factorization
lepton
lepton
proton
lepton
proton
pion
proton
antilepton
SIDIS
Drell-Yan
electron
pion
proton
positron
pion
proton
e–e+ to pions
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pion
?
pion
p-p to pions
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Nonperturbative elements
k
2
=

P,S
d  i x P  
 ij ( x; S )  
e
P, S  j (0) U[0, ]  i ( ) P, S
2
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    0
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Decomposition of the
correlation function
O(1 Q0 ), leading twist
( x; S )


1 q
f1 ( x )  g1q ( x )  5 SL  h1q ( x )  5S T  
2
f1q ( x )  q( x )
Unpolarized distr. func.
g1q ( x )  q( x )
Helicity distr. func.
h1q ( x )   q( x )  T q( x )
Transversity distr. func.
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Longitudinal vs transverse momentum
Long. view
Trans. view
Photon moves into the screen/
proton moves out of the screen
kparton
kT
photon
parton
P
klong  xP
photon
target
long. spin
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Longitudinal vs transverse momentum
Long. view
Trans. view
Photon moves into the screen/
proton moves out of the screen
kparton
kT
photon
parton
P
klong  xP
photon
trans. spin
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target
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Probabilistic interpretation
Photon moves into the screen/
proton moves out of the screen
f ( x )  q( x ) 
q
1
g ( x )  q( x ) 
q
1
h ( x )  T q( x ) 
q
1
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
_


_
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Decomposition of the correlation
function UNintegrated over kT
Mulders, Tangerman, NPB 461 (96)
Goeke, Metz, Schlegel, PLB 618 (05)

T ST  kT 
1 q
2
q
2
( x, kT )
f1 ( x, kT )  f1T ( x, kT )
2
M
T
 
q
2 k
 i h1 ( x, kT )  ...  
M


Sivers, PRD 43 (91)
Boer, Mulders, PRD 57 (98)
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Probabilistic interpretation
Photon moves into the screen/
proton moves out of the screen
Sivers
-
Boer-Mulders
-
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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,kT2)?
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Transverse spin
Transversity in SIDIS
d
...  ST sin(h  S ) h1 ( x, pT2 )  H1 ( z, kT2 )  ...
see e.g. A.B., Diehl, Goeke, Metz,
Mulders, Schlegel, JHEP 0702,093
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Data from Hermes and Compass
DEUTERON
PROTON
HERMES, hep-ex/0507013
COMPASS, NPB 765 (07)
HERMES, PRL 94 (05)
COMPASS, PRL 94 (05)
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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)
[4] Wakamatsu, PLB 509 (01)
[2] Korotkov et al. EPJC 18 (01)
[5] Pasquini et al., PRD 72 (05)
[3] Schweitzer et al., PRD 64 (01)
[6] Anselmino et al., PRD 75 (07)
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Tensor charge from Lattice QCD
Compared to
axial charge
 u   h1u ( x)dx 0.84,
 d   h1d ( x)dx 0.23,
u   g1u ( x)dx 0.64,
d   g1d ( x)dx
0.35
S. Aoki et al., PRD 56 (1997)
M. Göckeler et al. [QCDSF/UKQCD], PLB (05)
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Comparison of tensor charges
c quark
spectator
2.4 GeV2
non-relativistic
lattice
0.3 GeV2
bag
c quark soliton 2
c quark soliton 1
Wakamatsu, 0705.2917[hep-ph]
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Barone, Drago, Ratcliffe, PR 359 (2002)
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Evolution equations
Barone, Drago, Ratcliffe, PR 359 (2002)
Hayashigaki, Kanazawa, Koike, PRD56 (97)
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Other observables
d
...  ST sin(h  S ) h1 ( x, pT2 )  H1 ( z, kT2 )  ...
Dihadron fragmentation
Efremov, Mankiewicz, Tornquist, PLB 284 (92)
Collins, Heppelmann, Ladinsky, NPB 420 (94)
Jaffe, Jin, Tang, PRL 80 (98)
d
...  ST sin(h  R ) h1 ( x ) H1 ( z, , M )  ...
2
Doubly polarized Drell-Yan
d
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Ralston, Soper, NPB 152 (79)
...  S1T S2T h1 ( x1 ) h1( x2 )  ...
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Transverse momentum
Semi-inclusive DIS
l p  l  X
(l  l ')2  Q 2  virtuality of photon
l'
proton
lepton
Ph
l
pion
Ph  transverse momentum of pion
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Example of model calculation
quark
diquark
proton
Jakob, Mulders, Rodrigues, NPA 626 (1997)
f1q ( x, kT2 )
f1q ( x, kT2 )
x  0.5
ky
x  0.01
ky
k x (GeV)
k x (GeV)
kT2  0.13 GeV 2
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kT2  0.48 GeV 2
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Example of a measurement
H1 Coll, NPB485(97)
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Transverse momentum distribution
h
fragmentation
q
q
pQCD
intrinsic
Koike, Nagashima, Vogelsang, NPB744 (06)
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Differences between flavors?
u
proton
d
ud diquark
uu diquark
proton
Assuming Mud  Muu
f1d (kT2 )x 2
f1u (kT2 )
ky
ky
k x (GeV)
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k x (GeV)
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Transverse momentum and
transverse spin
Semi-inclusive
Deep Inelastic Scattering
l p  l  X

left
l'

lepton
proton
l
N  N
NR  NL
R
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L
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pion
right
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Measured asymmetries in DIS
HERMES Coll, hep-ex/0507013
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Parton distribution functions
with transverse spin
Photon moves into the screen/
proton moves out of the screen
Sivers
-
Boer-Mulders
-
Transversity
-
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Chromodynamic lensing
Burkardt, PRD 66 (02)
Long. view
Trans. view
left
quarks
proton
right
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Chromodynamic lensing
Long. view
Trans. view
left
photon
right
NOTE: QCD tells us that the FSI has to be attractive, since
quark and remnants form a color antisymmetric state
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Chromodynamic lensing
Long. view
Trans. view
left
photon
right
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Distortions in transverse space
Long. view
Trans. view
left
quarks
proton
right
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Distortions in transverse space
Long. view
Trans. view
left
quarks

proton
right
The presence of spin can distort the distribution of quarks in
transverse space (orbital angular momentum of quarks is required)
A distortion in the distribution of quarks in transverse space
can give rise to a nonzero Sivers function
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Results from Lattice QCD
Sivers function expected to be:
•NEGATIVE for up quarks
•POSITIVE for down quarks
QCDSF/UKQCD Collab. (see e.g. hep-ph/05110032)
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Data from Hermes and Compass
PROTON
HERMES, hep-ex/0507013
DEUTERON
HERMES, PRL 94 (05)
COMPASS, PRL 94 (05)
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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
Feynman diagrams & Factorization
lepton
lepton
proton
lepton
proton
pion
proton
antilepton
SIDIS
Drell-Yan
electron
pion
proton
pion
positron
pion
proton
pion
e–e+ to pions
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p-p to pions
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Sivers effect in SIDIS
Long. view
Trans. view
left
quarks

proton
right
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Sivers effect in Drell-Yan
Long. view
Trans. view
left
photon
antiquark

proton
right
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
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Collins, PLB 536 (02)
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Final/initial state interactions
q
q
q
q
Drell-Yan
SIDIS
q
pp to hadrons
q
A.B., Bomhof, Mulders, Pijlman, hep-ph/0505268
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kT k-factorization
is broken ?
Is
T -factorization
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
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An example
A.B., D’Alesio, Bomhof, Mulders, Murgia, hep-ph/0703153
p p   jet X atRHIC
“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
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