1 Frascati, June 14, 2006 Aram Kotzinian Double spin azimuthal asymmetries A LT and A LL in semi-inclusive DIS Aram Kotzinian Torino University & INFN

Frascati, June 14, 2006 Aram Kotzinian 1

Double spin azimuthal asymmetries ALT and ALL in semi-inclusive DIS

Aram Kotzinian Torino University & INFN

On leave in absence from YerPhI, Armenia and JINR, Russia

In collaboration withB. Parsamyan, & A. Prokudin, hep-ph/0603194, PRD 2006;M. Anselmino, A. Efremov & B. Parsamyan, in preparation

IntroductionPositivity bounds for TMD DFsStatus of Lorentz Invariance relations

Predictions for azimuthal dependence of ALT

Cahn effect and predictions for azimuthal dependence of ALL

Conclusions


TMD quark DFs of Nucleon

Twist-two:

6 T-even & 2 T-

odd

Use this DFs and

put soft and hard

factors equal to 1

At twist-three:

8 T-even & 8 T-

odd

Collins, Soper (81,82); Ji, Ma, Yuan (03, 04); Collins, Metz (04)

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


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:


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)


g1T(1)

GRV98+GRSV2000 LO, std DFs Kretzer FFs


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: COMPASS


PT-weighted asymmetries: JLab


Transverse momentum dependence (1)

N is fixed by

holds if 21 0.246 (GeV/c)2

From analysis

of unpolarized

PhT dependence

and Cahn effect

Predictions for


Transverse momentum dependence (2)

pT-unintegrated asymmetries


Transverse momentum dependence (Deuteron)


x,y, z dependence of asymmetries


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


ST correction

0S

l’l

γ*

SST

x

z


Gaussian parameterization of DF

x- and z-dependencies are known

within this approach !


cosφh-weighted ALL


Results PT-dependence, Proton target

2 21

2 21

2 21

0.10 (GeV/c)

0.15 (GeV/c)

0.20 (GeV/c)

LT

Influence of

ΔσLT

is negligible


Results x,y,z-dependence at COMPASS


Results x,y,z-dependence at HERMES


Results x,y,z-dependence at JLab6


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 the transverse momentum dependence of g1T and g1L (and xBj dependence of Sivers function) taking into account the positivity bounds for TMF DFs.

Data are already available at COMPESS, HERMES & JLab !


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 the transverse momentum dependence of g1T and g1L (and xBj dependence of Sivers function) taking into account the positivity bounds for TMF DFs.

Data are already available at COMPASS, HERMES & JLab !


Additional slides


Spin physics “golden rule”

Minimize relative error:

Maximize:Constant acceptance, N


Contributions from long. & tran. components of target polarization


Transverse momentum dependence (Proton)






PT-weighted asymmetries: HERMES


Results PT-dependence, Deuteron target

2 21

2 21

2 21

0.10 (GeV/c)

0.15 (GeV/c)

0.20 (GeV/c)

Influence of

ΔσLT

is negligible

Documents

1 Frascati, June 14, 2006 Aram Kotzinian Double spin azimuthal asymmetries A LT and A LL in semi-inclusive DIS Aram Kotzinian Torino University & INFN