1 SPIN 2006, Kyoto, October 6, 2006 Aram Kotzinian Double spin azimuthal asymmetries A LT and A LL in semi-inclusive DIS Aram Kotzinian YerPhI, Armenia

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


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


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:


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


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


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)


CUTS

Low

x, y, z & pT

High

x, y, z & pT+

Low

x, y, z & pT

High

x, y, z & pT

+

ATMD=


PT-weighted asymmetries


Factorized Gaussian parameterization

From analysis

of unpolarized

PhT dependence

and Cahn effect

All μ are constant

independent

of flavor, x and z


Transverse momentum dependence (Proton)

Proton

Deuteron


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


PT-dependence of ALL

With factorized Gaussian parameterization of DFs and FFs we have simple expressions


Predicted PT-dependence of ALL

Proton

Deuteron


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


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


cosφh-weighted ALL


Results PT-dependence

Influence of

ΔσLT

is negligibleProton

Deuteron


Results x,y,z-dependence at COMPASS


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 !

Documents

1 SPIN 2006, Kyoto, October 6, 2006 Aram Kotzinian Double spin azimuthal asymmetries A LT and A LL in semi-inclusive DIS Aram Kotzinian YerPhI, Armenia