Aram Kotzinian 1
Single transversely polarized DY, observables, TMDs
• Introduction– TMD DFs– SIDIS
• DY– General expression and LO parton model
• Intrinsic transverse parton momenta – P. Schweitzer, T. Teckentrup and A. Metz article review– Application to Sivers effect in DY
• Discussion
Aram KotzinianTorino Uni&INFN
On leave in absence from YerPhI, Armenia
4/26/2010, CERN
24/26/2010, CERN Aram Kotzinian
TMD quark DFs of Nucleon
At twist-two we have 6 T-even & 2 T-odd
21 ( , )q
Th x k
21 ( , )qT Th x k
21 ( , )qT Tf x k
21 ( , )q
Tf x k
21 ( , )qL Tg x k
21 ( , )qT Tg x k
21 ( , )qT Th x k
21 ( , )qL Th x k
3
SIDIS
4/26/2010, CERN Aram Kotzinian
Using current conservation + parity conservation + hermiticity one can show that 18 independent Structure Functions describe one particle SIDIS cross section in one photon exchange approximation.Moreover, the dependences on azimuthal angle of produced hadron and of the target nucleon polarization are calculated explicitly and factorized
Aram Kotzinian
SIDIS, polarized lepton and target
4/26/2010, CERN 4
cos, ,
2 2 2
2 2
2
cos2 sin sin sin 2
cos
1 2 (1 )cos2(1 ) 2
cos(2 ) 2 (1 )sin 2 (1 )sin sin(2 )
1 2 (1 )cos
( ){
[ ][ ]
h
h h h h
h
UU T UU L UU
UU LU U
hh h
h h h hL
L
L UL
L T
e
e L L
l
Lh
d ydxdy dzd dP xyQ x
F F F
F F F F
FS F S
S
sin( ) sin( ), ,
sin( ) sin(3 ) sin
sin(2 ) co )2 s(
sin( )
sin( ) sin(3 ) 2 (1 )sin
2 (1 )sin(2 ) 1 cos( )
2 (1 )cos
[ ( )
] [
h S h S
h S h S S
h S h S
h S
h S h S S
h S h S
UT T UT L
UT UT UT
UT LT
L
e
S
T
F F
F F F
F S F
F
cos cos(2 )2 (1 )cos(2 ) ]}S h ShT LTS F
( , ) ( , ) ( ) ( )hl N P S l h P X
( , )18 structure functions h SfABF
2 2 2 2 2 21 1 11 1 , 4 2 4
Mxy y y y yQ
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SIDIS, unpolarized lepton
4/26/2010, CERN
cos cos2, ,
sin sin 2
sin( ) sin( )
2 2 2
,
2 2
,
12(1 ) 2
2 (1 )cos cos(2 )
2 (1 )sin sin(2 )
sin( )
sin( )
( ){
[ ][ ( )
h h
h h
h S h S
h h
h hUU T UU L UU UU
ULh h
h S
UL
UT T
h S
UT L
l
L
T
d ydxdy dzd dP xyQ x
F F F F
F F
F
F
S
FS
sin( ) sin(3 )
sin sin(2 )
sin(3 )
2 (1 )sin 2 (1 )sin(2 ) ]}h S h S
S h S
UT UT
UT
h S
S Uh S T
F
F F
( ) ( , ) ( ) ( )hl N P S l h P X
( , )12 structure functions h SfABF
Aram Kotzinian 6
SIDIS, unpolarized lepton, LO parton model
4/26/2010, CERN
cos2,
sin 2 sin( )
sin( )
2 2 2
2 2
sin(3 )
1 cos(2 )2(1 ) 2
sin(2 ) sin( )
sin( ) sin(3 )
( ){
[]}
h
h h S
h S h S
U
L
U T UU
UL UT
U
hh h
h h S
h S h ST
l
UT
T
F F
F F
d ydxdy d
S
zd dP xyQ
S
F F
x
( , )6 structure functions h SfABF
cos(2 ) sin(2 ),
sin( ) sin( ) si
1 1 1 1 1 1
1 1 1 1 1(3 )
1n
, ,
, ,
h h
h s h s h s
q h q h q hq q L q
q h q h q hT q
UU T UU UL
UT T q T qU UT
f D h H h H
f D h H h
F F F
F F F H
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Crossing
4/26/2010, CERN
( , ) ( )
( , )
( )
(
( )
( () ) )
h
h
h P
h
N P S l X
N PP S l X
l
l
8
DY processes
4/26/2010, CERN Aram Kotzinian
2 22 2, Q , , ,
2 2a b F a ba b
q qq l l q x x x x xP q P q
( ) ( , )a bP p P S X
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Angular variables
4/26/2010, CERN
Target rest frame (TF)
Collins-Soper frame (CS)
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General expression for DY cross section(only target is polarized)
4/26/2010, CERN
22 2 2
4 2
2
1 2 cos cos2
sin sin 2
sin sin sin( ) sin(2 )
(1 cos ) (1 cos ) sin 2 cos sin cos 2
sin 2 sin sin sin 2
cos sin sin 2 sin( ) sin( )
sin
{( )
( )[( ) ( )S S S S
U U U U
L L
T T
e
T S
L
TS ST
mdd qd q
S
F F F F
F F
F FS F F
sin(2 ) s2 in(2 )sin(2 ) sin(2 )( )]}S ST STSF F
( , )2,
2 2 24 ( · ) - flux, ( , , ) Sa b a
fA a b Tb F x x qP P M M Q
( , )12 structure function s SfAF
as expected from crossing symmetry
S. Arnold, A. Metz and M. Schlegel, PRD 79, 034005 (2009)48 structure functions if both spin-1/2 hadrons are polarized
( ) ( , )a bP p P S X
Aram Kotzinian 11
DY-LO, only transversely polarized target
4/26/2010, CERN
2[ ( )] 2
1 ( )ˆ (1 cos ), 1 cos
LO LO
U fUfDF
2
2
cos2
sin sin(2 ) sin
2
4 2 [si
(2
n ]
[sin)
]
ˆ 1 cos 2
sin sin(2 ) sin(2 )
{
[ ( )]}S S S
LOem
U
T S S S
U
T T T
d Dd qd Fq
S
A
A AD A
( , )Only 5 structure fu nctions SfAF
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DY-LO asymmetries, transversely polarized target
4/26/2010, CERN
cos2 11 1
sin sin 11 1
sin(2 ) 2 2 11 1
sin(2 )
2 · · ·
·
2( · )[2( · )( · ) · ] (
· )
4
( )( ) )S S
S
S
LO
U aT bT aT bT a b U
LO LO
T T bT T b U
LO
T bT aT bT aT bT bT aT T a b U
T
A h h M M F
A A f f M F
A h h M M F
A
hk hk k k
hk
hk hk hk k k k hk
C
C
C1
1 1 · 2LO
aT a Uh h M F hkC
2
2
22
2 2 2
1 (2)( , ) ( ) ( , )1 2
( , ) ( , ) ( , ) ( , )1
1 2
2
aT bT q aT bT aT bT aT bTq
aT bT aT bT
w f f e wTNcq q q qf x f x f x f xb
d
a a
d
b
k k k k q k k k k
k k k k
C
1 LO
U a aF f f CT
Tqqh
Aram Kotzinian 13
Intrinsic transverse momentum in SIDIS
4/26/2010, CERN
P. Schweitzer, T. Teckentrup and A. Metz, arXiv:1003.2190v1 [hep-ph,]Comprehensive analysis of existing unpolarized SIDIS and DY data
SIDIS, factorized Gauss model:
Old extractions:Torino: EMC data
Bochum: HERMES
In many analyses, for example, extraction of transversity, BM function from SIDIS and pp and pd DY processes this values are applied
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New data from JLab
4/26/2010, CERN
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Test of Gauss model at HERMES
4/26/2010, CERN
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Parameters from HERMES and cross check with CLAS
4/26/2010, CERN
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Cahn effect at EMC
4/26/2010, CERN
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Intrinsic transverse momentum in DY: testing Gauss
4/26/2010, CERN
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Intrinsic transverse momentum in DY: pion vs nucleon
4/26/2010, CERN
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Energy dependence of intrinsic transverse momenta
4/26/2010, CERN
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STM conclusions
4/26/2010, CERN
• “Gauss model works very well in SIDIS and DY” – No evidence for flavor- or x- or z-dependence of
Gauss widths– Gauss width increase with energy in DY– This property holds in SIDIS too • “More precise data from SIDIS are needed…”
“The Gauss Anzatz remains to be tested, in particular, when polarization phenomena are included, and future data may demand to refine it, which will improve our understanding of intrinsic transverse parton momenta.”
Aram Kotzinian 22
Exercise for Sivers effect in DY@COMPASS
4/26/2010, CERN
2
2
cos2
sin sin(2 ) sin
2
4 2 [si
(2
n ]
[sin)
]
ˆ 1 cos 2
sin sin(2 ) sin(2 )
{
[ ( )]}S S S
LOem
U
T S S S
U
T T T
d Dd qd Fq
S
A
A AD A
sin sin 11 1 ·S S
LO LO
T T bT T b UA A f f M F hk C
1 LO
U a aF f f C2
2
22
2 2 2
1 (2)( , ) ( ) ( , )1 2
( , ) ( , ) ( , ) ( , )1
1 2
2
aT bT q aT bT aT bT aT bTq
aT bT aT bT
w f f e wTNcq q q qf x f x f x f xb
d
a a
d
b
k k k k q k k k k
k k k k
C
Unpolarized PDFs: GRV-P1 pion PDFs (Zeit.Phys.C53(1992)651) and GRV98 LO for proton
Aram Kotzinian 23
Parameterization of TMDs
4/26/2010, CERN
2 22
1 1 2
exp( , ) ( ) Sq q
T TS
f x f x
T TT
T
k kk
k
2 22
1 1 2
exp( , ) ( ) Uq q
U
f x f x
T T
TT
k kk
k
Torino parameterization for Sivers function (with changed sign for DY case)
2 21 1
2 21
1
2 22 1
1 12 2 21
2 22 1
2 21
( , ) ( ) ( ) ( , )
( ) 1 , ( ) 2 exp
exp( , ) 2 1 ( )
q q
q q
q q
q q
q qT q
q qq q
q q
Sq qq qT q
U b Sq q
U bS
U b
Mf x x h f x
x N x x h e MM
MMf x e N x x f xM
MM
T T TT
TT T
T TT
T T
TT
T
k k kk
kk k
k kk
k k
kkk
N
N
Extracted parameters (Melis talk) 21, , , , U b q q qM N Tk
Aram Kotzinian 24
Torino-STM
4/26/2010, CERN
2 2
2 2
2 21
2 2
0.25 (GeV/c)
0.25 (GeV/c)
0.34 (GeV/c)
Torino
0.144 (GeV/c)
U b
U a
S
M
T
T
T
k
k
k
2 2
2 2
2 21
2 2
(GeV/c)
(GeV/c)
0.34 (GeV/c)
STM
0.55
1.05
0.21 (GeV/c)
U b
U a
S
M
T
T
T
k
k
k
2 2
2 2
2 21
2 2
(GeV/c)
(
STM1
0.74
GeV/c)
(GeV/c
0.55
1.05
)
(GeV
8
)0 17 /c.3
U b
U a
S
M
T
T
T
k
k
k
Aram Kotzinian 25
Discussion
4/26/2010, CERN
• What is the origin of energy dependence of Gauss width? • SIDIS at COMPASS can provide at higher than HERMES and Jlab energy
• new acceptance corrected data on unpolarized TMD DFs• new data on Sivers asymmetry
• to check the energy dependence of the Gauss width of unpolarized DFs and FFs
• Energy dependence of parameters of Sivers DF• First DY measurements with transversely polarized target at COMPASS will allow to
study the universality of spin dependent TMD DFs• Size and shape of Sivers DY asymmetry dependence on xF and qT will are
sensitive to Gaussian width of Sivers function at high energy and Q2 and allow to understand if
• this width is energy dependent?• What is universal: the Sivers PDF or the ratio of Sivers PDF to unpolarized
one?