Transversity and Transverse-Momentum-Dependent Partonic Functions
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Transcript of Transversity and Transverse-Momentum-Dependent Partonic Functions
Transversity andTransverse-Momentum-
Dependent Partonic Functions
Alessandro Bacchetta
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Outline
• Theoretical framework
• Transversity
• Sivers function
• Boer-Mulders function
Theoretical framework
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Semi-inclusive Deep Inelastic Scattering
proton
pionlepton
l p l X2 2 virtuality ( ') of photonk k Q
k
'k
transverse momentum of pionhP
hP
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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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Factorization proofsFactorization proofs for 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)
QCD hP Q
See talk by A. Metz
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Nonperturbative elements
2
=
4
4 [0, ]( , , ) , (0) ( ) ,(2 )
i kij j i
dk P S e P S P S
U
,P S
k
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Origin of the gauge link
P
k ......
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Different processes
SIDIS
q
q
Drell-Yan
q
q
q
q
pp to hadrons
Bacchetta, Bomhof, Mulders, Pijlman, hep-ph/0505268
See talk by C. Bomhof
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[0, ] , ]
2
[
( ) ( , , )
, (0) ( ) ,2
ij T ij k x P
i Pj a
xi
x dk d k k P S
de P S a P S
U U
The correlation functionintegrated over kT
P
( ,0 ,0 )a
k
axis 0 a
SIDIS
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[0, ] [ , ]
2( ) ( , , )
, (0) ( ) ,2
ij T ij k x P
ia
x Pj i
x dk d k k P S
de P S a P S
U U
The correlation functionintegrated over kT
P
( ,0 ,0 )a
k
axis0 a
Drell-Yan
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[0,
2
3 ] [ , ]
( , ) ( , , )
(0) ( )(2 )
ij T ij k x P
i kTj a i
x k dk k P S
d de P a P
U U
The correlation functionUNintegrated over kT
( ,0 , )Ta
( )
[0, ] [0, ] [ ,
2
3 ]
( , ) ( , , )
(0) ( )(2 )
ij T ij k x P
i kT Taj ia
x k dk k P S
d de P a P
U U U
P
kaxis
0a
Belitsky, Ji, Yuan, NPB656 (03)
SIDIS
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The correlation functionUNintegrated over kT
( ,0 , )Ta
( )
[0, ] [0, ] [ , ]
2
3
( , ) ( , , )
(0) ( )(2 )
ij T ij k x P
i Ta a
kTj i
x k dk k P S
d de P a P
U U U
P
k
0a
axis
Ji, Yuan, PLB 543 (02); Belitsky, Ji, Yuan, NPB656 (03)
Drell-Yan
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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
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Decomposition of the correlation function integrated over kT
1 5151( ) ( )1
( )2
() Lqq q
Thx f x g x S x S
Leading twist only
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 )Q
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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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Parton distribution functions with transverse spin
-Sivers
-Boer-Mulders
-Transversity
Photon moves into the screen/ proton moves out of the screen
Transversity
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Definition of transversity distribution function
Probability to find a quark with momentum xP+ and transverse spin sq in a proton with transverse spin S
11
1( ) ( )
2( )q q
qq pf x f x h S sx
-
NOTE: transverse momentum is integrated over!
25(1
1( ) T ( , )
4)r ,Tq p
kq
x P
f x dk d k k Ps S
Notation of
Anselmino et al.
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Transversity: birth and growth
Citations
050100150200250300350400
1980-1985
1986-1990
1991-1995
1996-2000
2001-2005
Ralston, Soper, NPB 152 (79)
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Helicity and transversity
1qh q
1qg q
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Transversity vs helicity• Different evolution
h1@ 0.079 GeV2
g1h1@ 0.75 GeV2
g1
S. Scopetta, V. Vento,PLB 424 (1997)
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Transversity vs helicity• Different evolution• Different integrals (axial and tensor charge of the
nucleon). E.g. 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)
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Transversity vs helicity• Different evolution• Different integrals (axial and tensor charge of the
nucleon). E.g. from lattice QCD:
• Different sum-rules Bakker, Leader, Trueman, PRD 70 (04)
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)
,1 1
2 2q g
q
q G L ,1 1
2 2q gT
q
q L
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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
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First data: Collins asymmetry
1
1 1
1
sin( )sin( ) h S
h S UT
H
f D
h
hadron plane
lepton plane
S
h
Airapetian et al. [HERMES], PRL 94 (05)
Alexakhin et al. [COMPASS], PRL 94 (05)
See talk by G. Schnell, A. Martin
hP
'kk
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First attempt to combine BELLE and HERMES dataEfremov, Goeke, Schweitzer, PRD 73 (06)
1. Get Collins function from BELLE
Talk by R. Seidl
Talks by P. Schweitzer and M. Boglione
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First attempt to combine BELLE and HERMES data
2. Check size of transversity at HERMES
See talk by P. Schweitzer
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Where to observe itProcess Experiment Observable Grade
Hermes, Compass, Clas, EIC
Hermes, Compass, EIC
Compass, EIC
Rhic, Pax, JPARC
Panda Compass
Rhic
l p l X
p p l l X
p l l X
p p l l X
l p l X
l p l X
1 1
1
h H
h H
1 1h H
1 1h H
1 1h h
1 1
(1)1 1
h h
h h
p p X
p p X
1 1 1
1 1 1
f h H
f h H
Schnell,Martin, Avakian,
Elschenbroich
Martin, Radici, Giordano
Heppelman, Dalpiaz
Talks by
Koch, Fischer
Heppelman, Aidala
The Sivers function
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Introducing the Sivers function
1( , ) Tr ( , , )
2Tq pk x P
f x k dk k P S
Probability to find a quark with momentum xP+ and kT in a proton
with transverse spin S
21
21
ˆ( )( , ) ( , )( , ) q
Tq T
T Tq Tp
P k Sf x k f x kf x k
M
see e.g. Bacchetta, D’Alesio, Diehl, Miller, PRD 70 (04)
2Tk
M
-
T-odd
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Sivers function: birth and growth
Citations
0
50
100
150
200
1990-1993 1994-1997 1998-2001 2001-2005
D. Sivers, PRD41 (90)
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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 spaceBurkardt, PRD 66 (02); Diehl, Hägler, EPJ C44 (05)
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up
Final-state interactions
proton
quarks
Side view Front view
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up
Final-state interactions
photon
Side view Front 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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up
Final-state interactions
photon
Burkardt, PRD 66 (02)
Chromodynamic lensing
Side view Front view
See talk by M. Burkardt
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down
up
Change of sign in Drell-Yan
proton
photon
quarks
Side view Front view
antiquark
Clear-cut prediction of QCD
Collins, PLB 536 (02)
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up
Distortions in transverse space
proton
quarks
Side view Front view
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up
Distortions in transverse space
proton
quarks
Side view Front 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)
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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
22
1 22(
1( , ) ( , )
(2 ),0, ) T T
q Tibq T
T Tq py
df x b f x b e
ME x
b
anomalous( ,0 mag,0 netic moment) qqE kxdx Work of Burkardt
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Lattice-QCD studies
Preliminary results by QCDSF Collab. (see e.g. hep-ph/05110032)
up quarks
Sivers function for up quarks expected to be NEGATIVE
See talk by P. Hägler
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Model calculations
spectator
proton
1 2 4 5F Se e C
1e
2e
x
(1 2)1Txf
up
down
Bacchetta, Schäfer, Yang, PLB 578 (04)
Brodsky, Hwang, Schmidt PLB530 (02)
Gamberg, Goldstein, Oganessyan, PRD 67 (03)
See talk by L. Gamberg
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Fits to HERMES and COMPASS data
Airapetian et al. [HERMES], PRL 94 (05) Anselmino et al., PRD72 (05)
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Comparison with COMPASS data
Alexakhin et al. [COMPASS], PRL94 (05), Anselmino et al., PRD72 (05)
deut
eron
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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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New preliminary kaon data
See talk by U. Elschenbroich
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Future experimental data• 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)
See talk by U. D’Alesio
The Boer-Mulders function
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Definition of the Boer-Mulders function
5,
1( , ) Tr ( , , 0)
4(1 )
T
q pk x P
T qk
k sf x dk k P S
Probability to find a quark with momentum xP+ and kT and spin sq
12
12 ( ,
ˆ( )1( , ) ( , ) )
2T qq
T Tq
q
p Th x kP k s
f x k f x kM
T-odd
-Tk
M
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Boer-Mulders function: birth and growth
Citations
01020304050607080
1998-1999 2000-2001 2002-2003 2004-2005
Boer, Mulders, PRD57 (98)
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Relation to transverse space
proton
quarks
Side view Front view
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Relation to transverse space
proton
quarks
Side view Front view
A distortion in the distribution of quark spin in transverse space can give rise to a Boer-Mulders function Burkardt, hep-ph/0510408
Quark spin can be unevenly distributed in transverse space (orbital angular momentum of quarks is required)
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Lattice-QCD studies
Preliminary results by QCDSF Collab. (see e.g. hep-ph/05110032)
up quarks
Boer-Mulders function for up quarks expected to be NEGATIVE
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Model calculations
x
(1 2)1xh
up
down
Bacchetta, Schäfer, Yang, PLB 578 (04)
See talk by L. Gamberg, G. Goldstein
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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)
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Conclusions
Many progresses from the theoretical, experimental, phenomenological sides…
...Pay attention to the rest of the workshop!
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down
up
Semi-inclusive Deep Inelastic Scattering
proton
pion
lepton
l p l X
up down
up down
N N
N N