Single transversely polarized DY, observables, TMDs

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


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


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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


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


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

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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

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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.”

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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

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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

qq

q q

q q

qq

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

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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

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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?

Single transversely polarized DY, observables, TMDs

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