SPIN 2006, Kyoto, October 6, 2006 Aram Kotzinian 1
Double spin azimuthal asymmetries ALT and ALL in semi-inclusive DIS
Aram Kotzinian YerPhI, Armenia and JINR, Russia and INFN, Torino
In collaboration withB. Parsamyan, & A. Prokudin, hep-ph/0603194; Phys.Rev. D73 (2006)
M. Anselmino, A. Efremov & B. Parsamyan, hep-ph/0608048, submitted to PRD
IntroductionPositivity bounds for TMD DFs
Status of Lorentz Invariance relations
Predictions for azimuthal and phT- dependence of ALT
Predictions for azimuthal and phT- dependence of ALL
Conclusions
SPIN 2006, Kyoto, October 6, 2006 Aram Kotzinian 2
TMD quark DFs of Nucleon
Twist-two:
6 T-even & 2 T-
odd
At twist-three:
8 T-even & 8 T-
odd
N
@ twist 2
2
1 ( , )q
T Th x k
2
1 ( , )q
L Th x k2
1 ( , )q
Th x k
2
1 ( , )q
T Th x k
2
1 ( , )q
T Tf x k
2
1 ( , )q
Tf x k
2
1 ( , )q
L Tg x k
2
1 ( , )q
T Tg x k
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Generalized positivity bounds
Sivers function
For d-quark this bound is reached at
for the best fit parameters. But error bars of extracted parameters
are large enough and allow to fulfill the general positivity bound.
Naïve bound was used during fitting of Sivers DF:
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Status of Lorentz invariance (LI) relations
2001: explicit model calculations of DFs by Kundu & Metz show that these relations are violated2003: Goeke, Metz, Pobylitsa & Polyakov demonstrated that the same Wilson link in the definition of DFs which makes possible the existence of nonzero Sivers effect leads to violation of LI relations among DFs.
Without Wilson link Sivers effect = 0. If observed by HERMES asymmetry is due to Sivers effect then LI relations can be strongly violated
is g1T smaller or larger than proposed by LI relation?
12 1 ...( )( ) ( ),q q
T
dg x g x
dx
Relations between kT-integrated twist-3 and twist-2 PDFs
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SIDIS and TMD DF21 ( , )T Tg x k
2 21
2 2 21 1
( , ) ( , ),
( , ) ( , ) ( , )
q qN T T
Nq q q q T T
N T L L T N T T
x k f x k
k Sx k s g x k g x k
M
P
P
cos( ) cos( ) cos( ) term in the SIDIS x-sectionN hT T q s h s sk S
1 1 1
22
2 2
21 1
...)
l
( ) ( ) cos( )
(
L T
N lhXN
f l N g T g h sh
dy D y
dxdydzd P Q y
SSA Sivers Collins
SH H H
1
1
1
2 2 2 21
2 2 2 21
2 2 2 21
T
L
q hf q T T q hT T
q
q hT hTg q T T T q hT T
q hT
q hg q T L T q hT T
q
e d k f x D z z
e d k g x D z zM
e d k g x D z z
k P k
k Pk P k
P
k P k
( , ) ( ,( ) )
( , ) ( ,( ) )
( , ) ( ,( ) )
H
H
H
TS
N
qLs
Tk
qs
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Double Spin Asymmetry
1
1
1( , , , , ) ( ) cos( ) ...Tgh h h
LT hT s sNfl T
d dA x y z P D y
d d
S
H
H
2 (1)21
221
( ) ( )cos( )( )| |cos( ) ( , , ) 2 2 ( )
( ) ( )( )
q hhhTq T qT s
qh hhTs LT q h
q qTq
e g x D zd P d dMA x y z zD y
M e f x D zd P d d
PP
2(1) 2 2
1 12( ) ( , )
2q qTT T T Tg x d k g x k
M
kWeighted asymmetry is related to the first kT-momentum of g1T:
There exists a relation between first momentum of g1T and g2
(follows from Lorentz invariance, Tangerman & Mulders) :1 1
(1) 11 2 2
0
( )( ) ( ) ( ) ( )
x qq q qT
x x
g yg x g y dy g y dy WW appr x dy
y
All ingredients (DFs & FFs) are known!
First estimations by A.K. & P.Mulders, PRD 54, 1229 (1996)
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CUTS
Low
x, y, z & pT
High
x, y, z & pT+
Low
x, y, z & pT
High
x, y, z & pT
+
ATMD=
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PT-weighted asymmetries
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Factorized Gaussian parameterization
From analysis
of unpolarized
PhT dependence
and Cahn effect
All μ are constant
independent
of flavor, x and z
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Transverse momentum dependence (Proton)
Proton
Deuteron
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Orbital momentum and g1L
2 222 2
1 diquark 2 2 2spin-1( , ) [ ( ) ](1 ) (1 )
T TL T
k k mg x k M
x x x x
Model by Brodsky, Hwang, Ma & Schmidt, NPB 593 (2001)
311
222 2
1 diquas rkin-0 2p( , ) [( ) ]TL T
kmg x k M
x x
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PT-dependence of ALL
With factorized Gaussian parameterization of DFs and FFs we have simple expressions
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Predicted PT-dependence of ALL
Proton
Deuteron
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Azimuthal dependence of ALL
0S
l’l
γ*
SST
x
z
In O(1/Q) approximation we have kinematical (Cahn) corrections
and contribution from transverse component of target polarization
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Cahn effect in ALL
1/Q kinematical correction:
azimuthal modulation of lepton-
quark non-coplanar hard scattering
cross-section in presence of nonzero
transverse momentum of quark in
nucleon
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cosφh-weighted ALL
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Results PT-dependence
Influence of
ΔσLT
is negligibleProton
Deuteron
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Results x,y,z-dependence at COMPASS
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ConclusionsTo check the self-consistency of LO QCD picture of polarized SIDIS with TMD DFs & FFs it is important
to measure at different energies the azimuthal and PhT dependence of
unpolarized SIDIS ALL & ALT
These and also Sivers asymmetry measurements will allow
to check the Lorentz invariance relationsto perform a “global” analysis and extract the parameters describing g1T, g1L and Sivers function taking into account the positivity bounds for TMF DFs.
Data are already available at COMPASS, HERMES & JLab !