Transversity and
Transverse-Momentum-Dependent
Partonic Functions
Alessandro Bacchetta
Outline
•
•
•
•
Theoretical framework
Transversity
Sivers function
Boer-Mulders function
Alessandro Bacchetta - Transverse-momentumdependent functions
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Theoretical framework
Semi-inclusive
Deep Inelastic Scattering
l p  l X
(k  k ')2  Q 2  virtuality of photon
k'
proton
lepton
k
Ph 
pion
Ph   transverse momentum of pion
Alessandro Bacchetta - Transverse-momentumdependent functions
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Feynman diagrams & Factorization
lepton
lepton
proton
pion
SIDIS
Partonic scattering
amplitude
Fragmentation amplitude
Distribution amplitude
Alessandro Bacchetta - Transverse-momentumdependent functions
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Feynman diagrams & Factorization
lepton
lepton
proton
lepton
proton
pion
proton
antilepton
SIDIS
electron
Drell-Yan
pion
Partonic scattering
amplitude
Fragmentation amplitude
positron
pion
e–e+
to pions
Alessandro Bacchetta - Transverse-momentumdependent functions
Distribution amplitude
6/58
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
p-p to pions
Alessandro Bacchetta - Transverse-momentumdependent functions
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Factorization proofs
Factorization proofs for  QCD Ph Q
involving TMD functions
LEADING TWIST (1/Q0), NOT for pp to pions
–
–
–
–
Collins, Soper, NPB 193 (81)
Ji, Ma, Yuan, PRD 71 (04)
Ji, Ma, Yuan, PLB 597 (04)
Collins, Metz, PRL 93 (04)
Alessandro Bacchetta - Transverse-momentumdependent functions
See talk by
A. Metz
8/58
Nonperturbative elements
k
2
=
P, S
d 4 i k 
 ij (k, P, S )  
e
P, S  j (0) U[0, ]  i ( ) P, S
4
(2 )
Alessandro Bacchetta - Transverse-momentumdependent functions
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Origin of the gauge link
...
k

P
Alessandro Bacchetta - Transverse-momentumdependent functions
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Different processes
q
q
q
q
SIDIS
Drell-Yan
q
pp to hadrons
See talk by
C. Bomhof
q
Bacchetta, Bomhof, Mulders, Pijlman, hep-ph/0505268
Alessandro Bacchetta - Transverse-momentumdependent functions
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The correlation function
integrated over kT
 ij ( x )   dk  d 2kT  ij (k , P, S )
k  x P
d  i x P  



e
P, S  j (0) U[0,
U
 ] [ ,a ]  i (a ) P , S
2
a  (  ,0 ,0 )
0
a
 axis
k
SIDIS

P
Alessandro Bacchetta - Transverse-momentumdependent functions
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The correlation function
integrated over kT
 ij ( x )   dk  d 2kT  ij (k , P, S )
k  x P
d  i x P  



e
P, S  j (0) U[0,
U
 ] [ ,a ]  i (a ) P , S
2
a  (  ,0 ,0 )
0
a  axis
Drell-Yan
k

P
Alessandro Bacchetta - Transverse-momentumdependent functions
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The correlation function
UNintegrated over kT
 ijij ( x, kTT )   dk   ijij ( k , P, S )
kk xx P
P
d d 22TT ii kk

T(  )


e
P

(0)
U
U

U
(a
Pi (a ) P
j
[0, ]
[0,
[ a
,a]]
[i 
,a ]) 
33
(2 )
a  (  ,0 ,T )
T
a
0
 axis
k
SIDIS

P
Belitsky, Ji, Yuan, NPB656 (03)
Alessandro Bacchetta - Transverse-momentumdependent functions
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The correlation function
UNintegrated over kT
 ij ( x, kT )   dk   ij (k , P, S )
k  x P
d d 2T i k 

T (  )


e
P

(0)
U
U
U
j
[0, ]
[0,a ]
[ ,a ]  i (a ) P
3
(2 )
a  (  ,0 ,T )
a
0
Drell-Yan
T
 axis
k

P
Ji, Yuan, PLB 543 (02); Belitsky, Ji, Yuan, NPB656 (03)
Alessandro Bacchetta - Transverse-momentumdependent functions
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Practical consequences
• Some TMD functions will not be sensitive to the
differences in the structure of the gauge link (i.e. they
will be connected to standard partonic cross-sections)
• Some TMD functions will be sensitive to it and will be
multiplied by prefactors to be computed for each
partonic subprocess (i.e. they will be connected to
gluonic-pole cross-sections)
See talk by
C. Bomhof
Alessandro Bacchetta - Transverse-momentumdependent functions
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Decomposition of the correlation
function integrated over kT
Leading twist only (1 Q 0 )
( x )


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.
Alessandro Bacchetta - Transverse-momentumdependent functions
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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)
Alessandro Bacchetta - Transverse-momentumdependent functions
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Parton distribution functions
with transverse spin
Photon moves into the screen/
proton moves out of the screen
Transversity
-
Sivers
-
Boer-Mulders
-
Alessandro Bacchetta - Transverse-momentumdependent functions
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Transversity
Definition of transversity
distribution function
Probability to find a quark with momentum xP+ and transverse
spin sq in a proton with transverse spin S
1
fq p ( x )   dk d 2kT Tr   (1   5 sq ) (k, P, S )
4
k  x P
Notation of
Anselmino et al.
NOTE: transverse momentum is
integrated over!

1 q
fq p ( x )  f1 ( x )  h1q ( x ) S  sq
2

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Transversity: birth and growth
Ralston, Soper, NPB 152 (79)
Citations
400
350
300
250
200
150
100
50
0
19801985
19861990
19911995
19962000
Alessandro Bacchetta - Transverse-momentumdependent functions
20012005
22/58
Helicity and transversity
g  q 
q
1


 



h  q 
q
1
Alessandro Bacchetta - Transverse-momentumdependent functions
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Transversity vs helicity
• Different evolution
g1
h1@ 0.75 GeV2
h1@ 0.079 GeV2
g1
S. Scopetta, V. Vento,PLB 424 (1997)
Alessandro Bacchetta - Transverse-momentumdependent functions
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Transversity vs helicity
• Different evolution
• Different integrals (axial and tensor charge of the
nucleon). E.g. from lattice QCD:
 u   h1u ( x)dx 0.84,
 d   h1d ( x)dx 0.23,
u   g1u ( x)dx
d   g1d ( x)dx
0.64,
0.35
S. Aoki et al., PRD 56 (1997)
M. Göckeler et al. [QCDSF/UKQCD], PLB (05)
Alessandro Bacchetta - Transverse-momentumdependent functions
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Transversity vs helicity
• Different evolution
• Different integrals (axial and tensor charge of the
nucleon). E.g. from lattice QCD:
 u   h1u ( x)dx 0.84,
u   g1u ( x)dx
 d   h1d ( x)dx 0.23,
d   g1d ( x)dx
0.64,
0.35
S. Aoki et al., PRD 56 (1997)
M. Göckeler et al. [QCDSF/UKQCD], PLB (05)
• Different sum-rules
1 1

2 2
Bakker, Leader, Trueman, PRD 70 (04)

q  G  Lq , g
q
1 1

2 2

 q  LqT, g
q
Alessandro Bacchetta - Transverse-momentumdependent functions
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Comparison of tensor charges
c quark
spectator
lattice
non-relativistic
bag
c quark soliton 2
c quark soliton 1
Barone, Drago, Ratcliffe, PR 359 (2002)
Alessandro Bacchetta - Transverse-momentumdependent functions
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First data: Collins asymmetry
k
Airapetian et al. [HERMES], PRL 94 (05)
k' 
S
Ph 
h
Alexakhin et al. [COMPASS], PRL 94 (05)
See talk by
G. Schnell,
A. Martin
sin(h  S )
UT
sin(h  S ) h1  H1
f1  D1
Alessandro Bacchetta - Transverse-momentumdependent functions
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First attempt to combine BELLE and
HERMES data
Efremov, Goeke, Schweitzer, PRD 73 (06)
1. Get Collins function from BELLE
Talk by R.
Seidl
Talks by P. Schweitzer and M. Boglione
Alessandro Bacchetta - Transverse-momentumdependent functions
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First attempt to combine BELLE and
HERMES data
2. Check size of transversity at HERMES
See talk by P.
Schweitzer
Alessandro Bacchetta - Transverse-momentumdependent functions
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Where to observe it
Process
l p  l  X
l p  l   X
l p  l  X


p p ll X
p p  l l X
 p  l l X
pp  X

p p   X

Experiment
Hermes, Compass,
Clas, EIC
Hermes, Compass,
EIC
Compass, EIC
Rhic, Pax, JPARC
Panda
Compass
Rhic
Observable Grade
h1  H1
h1 H
h1 H1
Talks by
Schnell,Martin,
Avakian,
Elschenbroich
Martin, Radici,
Giordano
h1 H1
h1 h1
h1  h1
(1)
1
h1 h
f1  h1  H1
f1  h1  H1
Alessandro Bacchetta - Transverse-momentumdependent functions
Heppelman,
Dalpiaz
Koch, Fischer
Heppelman,
Aidala
31/58
The Sivers function
Introducing the Sivers function
Probability to find a quark with momentum xP+ and kT in a proton
with transverse spin S
1
fq p ( x, kT )   dk  Tr   (k, P, S )
2
k  x P
fq
p
q
1T
( x, kT )  f ( x, k )  f
q
1
2
T
(Pˆ  kT )  S
( x, k )
M
T-odd
2
T
see e.g. Bacchetta, D’Alesio, Diehl, Miller, PRD 70 (04)

kT 
2 M 

-
Alessandro Bacchetta - Transverse-momentumdependent functions





33/58
Sivers function: birth and growth
D. Sivers, PRD41 (90)
Citations
200
150
100
50
0
1990-1993
1994-1997
1998-2001
2001-2005
Alessandro Bacchetta - Transverse-momentumdependent functions
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Two ingredients
• Final-state interactions (included in the gauge link)
Ji, Yuan, PLB 543 (02); Belitsky, Ji, Yuan, NPB656 (03)
• Transverse-spin dependent distribution of quarks
in transverse space
Burkardt, PRD 66 (02); Diehl, Hägler, EPJ C44 (05)
Alessandro Bacchetta - Transverse-momentumdependent functions
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Final-state interactions
Side view
Front view
up
quarks
proton
down
Alessandro Bacchetta - Transverse-momentumdependent functions
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Final-state interactions
Side view
Front view
up
photon
down
NOTE: QCD tells us that the FSI has to be attractive, since
quark and remnants form a color antisymmetric state
Alessandro Bacchetta - Transverse-momentumdependent functions
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Final-state interactions
Side view
Front view
up
photon
down
Chromodynamic lensing
Burkardt, PRD 66 (02)
Alessandro Bacchetta - Transverse-momentumdependent functions
See talk by
M. Burkardt
38/58
Change of sign in Drell-Yan
Side view
Front view
up
quarks
antiquark
proton
photon
down
Clear-cut prediction of QCD
Collins, PLB 536 (02)
Alessandro Bacchetta - Transverse-momentumdependent functions
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Distortions in transverse space
Side view
Front view
up
quarks
proton
down
Alessandro Bacchetta - Transverse-momentumdependent functions
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Distortions in transverse space
Side view
Front view
up
quarks

proton
down
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
Alessandro Bacchetta - Transverse-momentumdependent functions
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Relation to GPDs
• The transverse-space deformation of unpolarized
quarks in a transversely polarized nucleon is
described by the generalized parton distribution
function E
2
d
T
1

 ibT T
q
2
2
fq p ( x, bT )  f1 ( x, bT ) 
E
(
x
,0,

)
e
q
T
M by  (2 )2
 dx E ( x,0,0)  k
q
q
anomalous magnetic moment
Work of Burkardt
Alessandro Bacchetta - Transverse-momentumdependent functions
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Lattice-QCD studies
up quarks
Sivers function
for up quarks
expected to be
NEGATIVE
See talk by
P. Hägler
Preliminary results by QCDSF Collab. (see e.g. hep-ph/05110032)
Alessandro Bacchetta - Transverse-momentumdependent functions
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Model calculations
Brodsky, Hwang, Schmidt PLB530 (02)
See talk by
L. Gamberg
Gamberg, Goldstein, Oganessyan, PRD 67 (03)
xf1T(1 2)
e1
down
spectator
e2
up
proton
x
e1e2  4 CFS  5
Bacchetta, Schäfer, Yang, PLB 578 (04)
Alessandro Bacchetta - Transverse-momentumdependent functions
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Fits to HERMES and COMPASS data
Airapetian et al. [HERMES], PRL 94 (05) Anselmino et al., PRD72 (05)
Alessandro Bacchetta - Transverse-momentumdependent functions
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Comparison with COMPASS data
Alexakhin et al. [COMPASS], PRL94 (05), Anselmino et al., PRD72 (05)
Alessandro Bacchetta - Transverse-momentumdependent functions
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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
Alessandro Bacchetta - Transverse-momentumdependent functions
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New preliminary kaon data
See talk by U.
Elschenbroich
Alessandro Bacchetta - Transverse-momentumdependent functions
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Future experimental data
See talk by U.
D’Alesio
• HERMES@DESY: 0 and weighted
asymmetries (this year?)
• COMPASS@CERN: proton data (this year?)
• CLAS@Jlab
• STAR,PHENIX@RHIC: jet-production in pp
collisions
• EIC (DIS with polarized protons)
• FAIR@GSI (Drell-Yan with polarized protons)
Alessandro Bacchetta - Transverse-momentumdependent functions
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The Boer-Mulders function
Definition of the Boer-Mulders
function
Probability to find a quark with momentum xP+ and kT and spin sq
1
fq p ( x, kT )   dk  Tr   (1   5 sq ) (k , P, S  0)
4
k   x P , kT
ˆ k )s
(
P
1 q
T
q
fq p ( x, kT )   f1 ( x, kT2 )  h1q ( x, kT2 )
2 
M



kT 


M 



Alessandro Bacchetta - Transverse-momentumdependent functions
T-odd



51/58
Boer-Mulders function: birth and
growth
Boer, Mulders, PRD57 (98)
Citations
80
70
60
50
40
30
20
10
0
1998-1999
2000-2001 2002-2003 2004-2005
Alessandro Bacchetta - Transverse-momentumdependent functions
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Relation to transverse space
Side view
Front view
up
quarks
proton
down
Alessandro Bacchetta - Transverse-momentumdependent functions
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Relation to transverse space
Side view
Front view
up
quarks






proton
down
Quark spin can be unevenly distributed in transverse space (orbital
angular momentum of quarks is required)
A distortion in the distribution of quark spin in transverse space
can give rise to a Boer-Mulders function
Burkardt, hep-ph/0510408
Alessandro Bacchetta - Transverse-momentumdependent functions
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Lattice-QCD studies
up quarks
Boer-Mulders
function for up
quarks expected
to be NEGATIVE
Preliminary results by QCDSF Collab. (see e.g. hep-ph/05110032)
Alessandro Bacchetta - Transverse-momentumdependent functions
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Model calculations
xh1(1 2)
See talk by
L. Gamberg,
G. Goldstein
down
up
x
Bacchetta, Schäfer, Yang, PLB 578 (04)
Alessandro Bacchetta - Transverse-momentumdependent functions
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Where to look for it
• Unpolarized semi-inclusive DIS (HERMES,
COMPASS, H1, ZEUS, EIC, LHEC)
• Unpolarized Drell-Yan (GSI)
• Jet-production in unpolarized pp collisions (RHIC,
LHC)
Alessandro Bacchetta - Transverse-momentumdependent functions
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Conclusions
Many progresses from the theoretical, experimental,
phenomenological sides…
...Pay attention to the rest of the workshop!
Alessandro Bacchetta - Transverse-momentumdependent functions
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Semi-inclusive
Deep Inelastic Scattering
l p  l X

up

proton
lepton
N  N
Nup  Ndown
up
down
Alessandro Bacchetta - Transverse-momentumdependent functions
pion
down
59/58