Correlations between SIDIS azimuthal asymmetries in target and current fragmentation regions

1Transversity 2011, Lošinj, August 30, 2011 Aram Kotzinian

Correlations between SIDIS azimuthal asymmetries in target and current fragmentation regions

Polarized SIDISIntroduction: CFR TFR, STMD fracture functions

M.Anselmino, V.Barone and AK, arXiv:1102.4214; PLB 699 (2011) 108

DSIDIS: TFR+CFRAK, DIS2011; arXiv:1107.2292

M.Anselmino, V.Barone and AK, articles in preparation

Aram KotzinianUni&INFN, Torino & YerPhI, Armenia

2

Introduction

DIS: only lepton used as a probeNo access to quark kT and transverse polarization at LO

SIDIS: lepton and produced hadron used as a probeWe need excellent understanding of hadronization and factorization

Exploited in CFR: xF > 0For better understanding one have to exploit also TFR: xF < 0

New nonperturbative objects – Spin and Transverse Momentum Dependent Fracture Functions

Transversity 2011, Lošinj, August 30, 2011 Aram Kotzinian

3

STMD DFs


Quark polarization

U L T

Nucleon Polarization

U

L

T

21 ( , )q

Tf x k

12( , )qT

TT f x kM

k S

21 ( , )q

L L TS g x k

21 ( , )qTT

TTg x k

Mk S

21 ( , )q

T T Th x kS 2

1 ( , )T qTT

TT

T h x kM M

k Sk

21 ( , )q

L L TTS h x k

Mk

21 ( , )q

ij jT T

Thk x kM

ò

4

SIDIS: CFR


0Fx

1

( , ) ( , ) ( ) ( ) ( , ) ( , ) ( ) ( , )

, / , ,2 2 2

Nl N P S l h P X l q k s l q k sh

q s N S q sS T

d df DdxdQ d dzd P dQ

1 2 2, 1 1

'( , ) ( , ) ( , )h T Tq s T T T

h

D z D z p H z pm

p sp

5

LO cross section in SIDIS CFR


( , ) ( , ) ( ) ( ) 2

22 2 2

cos2,

sin 2

sin( )sin( )

, sin(

( 1

cos(2 )

sin(2 )

(1 )

( )

0)

( ) ( )

(sin( )

( )sin )

N

h

h

h S

h S

l N P S l h P X

S T

UU T nn UU

L nn UL L ll

h

h

h

LL

UTT

Sh SUT T nn

UT

Fd x x ydxdQ d dzd P yQ

F D y F

S D y F S D y F

FS F D y

F

3 )

cos( )

sin(3 )

c) o )( s(

h S

h S

h S

h ST ll LTS D y F

8 terms out of 18 Structure Functions, 6 azimuthal modulations

6

SIDIS: TFR


M.Anselmino, V.Barone and A.K., arXiv:1102.4214; PL B699 (2011) 108

0Fx

( , ) ( , ) ( ) ( ) ( , ) ( , ) ( ) ( , )

, / ,2 2 2

Nl N P S l h P X l q k s l q k shq s N S

S T

d dMdxdQ d d d P dQ

(1 )FN

P x xP

Probabilistic interpretation at LO

7

Quark correlator


2 3i

5

( · )[ ]6 3

5

1( , , , )4 (2 ) (2 ) 2

, | (0) | , ; , ; | ( ,0, ) | ,

, ,

Bx P k XB h

X X

h

i

h h h

d Px k PE

P S P S X P S X P S

d de

i

M

At LO 16 STMD fracture functions

8

Decomposition of quark correlator


5

5

[ ]1 1 1 1

2 2

[ ]1 1 1 1

2 2

[i ]1 1 1

2

1 12 22

( )ˆ ˆ ˆ ˆ

· ·ˆ ˆ ˆ ˆ

ˆ ˆ ˆ

( · ) ( · )ˆ ˆ

( ·

i

h hT TT T L

N N

h hT TL T T

N N

i ii hT

T L LN

i ihhT TT

T T T L T

T T T TL

L L TT

T T T TT

N

T T

Su u u um m m m

S l l l lm m m m

S P S kS t t tm m

P kt tm m

P S k S k P

P S k S k P

P S k S

k S

M

M

M

1 1 12 2

) ( · ) ˆ ˆ ˆij iji i

Tj Tjh hT TT

N

T T

N

P kP k t t tm m m m

P S ò ò

2 2

·

STMD fracture functions depend on

azimuthal dependence in fracture functi

, , , , ·

cos ) n( o s T T T T

T T T T h q

x k P

k P

k P

k P

9

STMD Fracture Functions for spinless hadron production


Quark polarization

U L T

Nucleon Polarization

U

L

T1

1

ˆ

ˆ

hT

TT

T T

h

T

Nm

u

u

m

k S

P S

1̂u

1( ) ˆL T T

N h

hL

Sm m

uk P1̂L LS l

1̂T

h

hT

Nm ml k P

1

1

ˆ

·

·

ˆ

hT

T

T

N

T

TT

h

l

l

m

m

k S

P S 1

22 1 1

1

ˆ ˆ( · )

( · )

ˆ

·(

( · )

· ˆ)

hhT T

T T T

h

T T T

T

h

T

T T T

T

N

T

NT

T

h

t t

tm m

t

mm

P P S

P k S k P S

Sk k S

1 1ˆ ˆh LL

N

TL

TL

h

SS tm

tm

kP

1 1ˆ ˆh

i ij jT

j jT T

h

T

N

P t k tm m

òò

10

Sum Rules


1

2 21 1

2 21 12

2 21 1

2 21 12

1

1

22

ˆ (1 )

·ˆ ˆ (1 )

(1 )

· (1 )

·

( , )

( , )

ˆ ( , )

ˆ

ˆ ˆ ( , )

ˆ

T

Th

N TT T T

h h T

T

hT

L L Th

N TT T T T

h h T

N T

h T

T

hT

T L

P u x f k

P u u x f kk

P

d d x

md d xm

x g k

P x g kk

d d l x

md d l l

P

xm

md dm

t tk

k P

k P

k P

1

1

1

1

2

2

2 21 12

2 2 22 2

4

2 22

1 1

1 12

12 2

(1 )

·ˆ ˆ (1 )

2 ·

( , )

( , )

( , )

2 2

ˆ ˆ (1 )

ˆ ˆ ˆ

L

T

hL

T

hhNT T T

h

hhT T

Th

hN TT

h h T

T T TT

h T

T TT T

N h

x

md d xm

md d xm

d dm m

x h k

P t t x h kk

k PP t t x h k

k

k PP t t t

k P

k P

21 )(1 ( ,) T

h

x h x k

Nonzero fracture functions u,l,t. Useful for modeling.

11

LO cross-section in TFR


( , ) ( , ) ( ) ( ) 22

2 2

2 21

2 21 1

24

1

( 1 (1 )

, ,

(2 )

si

0)

, ,

, ,

n( )

co ), s(,

Nl N P S l h P XF

S T

hTT T T T

h

hTL L T T T

h S

h

qq

T Sh

ddxdQ d d d P yQ

P P

y

x x y e

Pu x S u xm

Py S l x x PSm

P l

At LO only 4 terms out of 18 Structure Functions,

2 azimuthal modulationsNo access to quark

transverse polarization

2 21 2 2 1

2 2 2 21 2 2 1 12

2

2 21 2 2 1

2 2 2 21 2 2 1 12

2

ˆ( , , )

·ˆ ˆ( , , )

ˆ( , , )

·ˆ ˆ( , , )

T

TT

T

T

B T

h h TT B T T T

N T

L B T L

h h TT B T T T

N TT

u x P k u

mu x P k u um P

l x P k l

ml x P k l lm P

d

d

d

d

k P

k P

12

Quark spin in hard l-q scattering


2

2(1 )( ) , ( ) , 1 (1 )T nn T nn s s

ys D y s D yy

s

s

sT

ˆTl

CFR

TFR

'Ts

( , ) ( , ) ( ) ( , ) 22 2 2 2 2

2 2 4

2 1 (

( · ' ) ( · )( ' · ) ( · )( '

)(1 ) ( ) ( )

2 4 4 · )

(

)

l q k s l q k s

L L

T

q

T T T T T T T

L

T T

Lss sd e udQ Q

s s sus

su u s

s s s l s l s l s l

and are usual Mandelstam variabless u

13

DSIDIS: TFR & CFR


1 1 2 22 1

( , ) ( , ) ( ) ( ) ( ) ( , ) ( , ) ( ) ( , )

, / , ,2 2 2 21 2

Nl N P S l h P h P X l q k s l q k sh hq s N S q s

S T T

d dM DdxdQ d dzd P d d P dQ

2 10, 0F Fx x

1 2 2, 1 1

'( , ) ( , ) ( , )h T Tq s T T T

h

D z D z p H z pm

p sp ( ) R

T nn TD y s s

14

Integrated over one hadron transverse momentum cross-sections


M.Anselmino, V.Barone and AK, article next week to ArXive

5

5

2 [ ]2 1 1

[ ]22 1 1

2

[i ]22 1 1 1 12

·

1( )2i

T TN

T L L TN

i j jiji ij

TjiT L T

N N

T

T T

T T T TL T

TN

d P u um

d P S l lm

k k S kS kd P S t t t tm m m

k S

k S

k

M

M

òM

Only 8 kT-dependent

“capless” fracture functions.Same prefactors as in TMDs

PT-integrated fracture functions

2 22 2 2

1 2 2 1 1 12 22

ˆ ˆ ˆ( , , )2 2

T hhTB T T T T

NT

k Pt x k P t t tm m

d

15

DSIDIS cross section integrated over PT2


1 1 2 21

11

( , ) ( , ) ( ) ( ) ( ) 2 2cos2em

, 12 21

2sin( )s

1

in 21 1

1 (1 )cos 22

(1 )sin 2 1 1 sin( )2 2

(1 )s

N

S

l N P S l h P h P X

UU T UUS T B

L UL L T

T

LL S UT

d yy ydxdyd dz d d dP x yQ

y yS y S y S y

S y

F F

F F F

1 1

1

sin( ) sin(3 )1 1

cos( )1

in( ) (1 )sin(3 )

1 cos( )2

S S

S

S UT s U

T

T

S LT

TS y

yS y

F F

F

1

1

1

1 1cos211 1

1

1sin( )1 1

1

1sin( )1 1

1

2 2 2 21

1

1

2( · )( · ) · ,

· ,

· ,

[ ] ( / ) ( , )

S

S

T T T

T

TUU

N

UT TN

TUT

a B T Ta

T

T T T T Td

t Hm m

u Dm

t Hm

wuD e x k k z w ud

PP

P k P k k k P

P k

P

k P

k

k k k

F C

F C

F C

C 2 22 1( , , ) ( , )a a

TB Tx k D z k

16

DSIDIS cross section integrated over PT1


1 1 2 2

2

( , ) ( , ) ( ) ( ) ( ) 22em

2 22

22 22

1 2 2 1 2 2 22

2 21 2 2 1 2

2

1 1

1 ( , , ) ( , , )sin( )2

1 ( , ,

( )

) ( ,2

Nl N P S l h P h P X

aaS T

hTB T T T B T S

hTL L B T T BT

d edxdyd dz d d dP yQ

Pyy u x P S u x Pm

Pyy S l x Pm

z

xS l

D

22 2, )cos( )T SP

Compared with one hadron production in TFR,

presence of known integrated fragmentation functions D1(z)

allows quark flavor separation of “tilded” fracture functions

As in SIDIS in TFR only kT-integrated fracture functions contribute

17

Unintegrated DSIDIS cross-section


1

2 1

1

1 2 2

22 1 1

( , ) ( , ) ( ) ( ) ( )

2 2 21 2

22

4

2

4

1

2

11ˆ

'ˆ1 (1 )

1 (1 )

( )ˆ

N

h hT

l N P S l h P h P X

S T T

T

UU L UL T UT

ll LU L L

h

L T L

hll

T

h h

h

ddxdQ d dzd P d d P

x yD y

x

u

Q y

S Sy

D S S

t

y

l

Q

H

DD

m

p s

2

(2 )( )1 (1 )ll

y yD yy

18

Examples


AK @ DIS2011; arXiv:1107.2292

1

1 1

1 1 1

1

1

1 ˆ ·1 21

1 2

2 2ˆ· ·2 2

2 21

2ˆ ·1

1 11

ˆ·0 2

22

ˆ

1

1cos( )

cos(2 )

cos(2 )

h

h

t HTkp

t HT TUU nn p

H t HT Tkpt

pN

N

u D P PD Fm m

P P

P

F Fm m m

m

m

Fm

F

1 1ˆ ·1 21

21 2 s in( )

hl DT TLU k

N

P P Fm m

2ˆ · 1 2 2 1 2

1 2 2 21 2 1 2

( · )( · ) ( · )ˆ ··

M D T T T T Tk

T T T T

F C M D

P P P k P k PP P P P

19

ALU asymmetry


12 21 2...

1 2 1 2 1 2

2 1

ˆ·2

2

depend on

· cos( ), with One can choose as independent angles and ( )Inte

, , ,In general

grating and

structure functions

over

, and ·

T T

T T T

T T

l

D

T

UU U

u x

P P

F z P P

P PP P

2 we obtain

1 1

1

2 21 2

2 2

2

2

ˆ ·1 21

1

2

2ˆ·

0

cos( ) sin( )

co

, , , , ,

, , ), , s, (

h

LULU

UU

l DT Tk

Nu

TD

T T

T

A

x z P P

d

d

P P Fm m

F x z P P

20

Hints from LEPTO: h-q azimuthal correlation, a1(ζ)


2 2 2 2/ / 1 2

2 2

ˆ ( , , , , · ) ( , , , ) 1 cos( ) cos 2( )

( , , , )q p T T T T q p T T h q h q

i i T T

u x k P u x k P a a

a a x k P

k P

21

h-q azimuthal correlation, a2(ζ)


2 1a a

22

h1-h2 azimuthal correlation, a1(ζ)


23

ALU


2 21 2

2 21 2

2 2

1 12 2

1 2

( , , , ) 1 cos( ) cos(2 ) sin( )( , , , ) 1 cos( ) cos(2 )

( , , , ) 1sin( ) si

s

n(2 )( , ,

in( ), ) 2

sin(2 )

LU T T LU LU

UU T T UU UU

LU T TLU UU

U

L

U T T

U

x k P a ax k P a a

x k P a ax

p

A

k Pp

24

CONCLUSIONS New members appeared in the polarized TMDs family -- 16 LO STMD fracture functionsFor SIDIS in the TFR, only 4 kT-integrated fracture functions of unpolarized and longitudinally polarized quarks are probed.

SSA contains only a Sivers-type modulation sin(φh-φS) but no Collins-type sin(φh+φS) or sin(3φh-φS). The eventual observation of Collins-type asymmetry will indicate that LO factorized approach fails and long range correlations between the struck quark polarization and PT of produced in TFR hadron might be important.

DSIDIS cross section at LO contains 2 azimuthal independent and 20 azimuthally modulated terms. Integrated over one hadron transverse momentum DSIDIS cross-section expressions are rather simple. The measurements of these cross-sections allow to access transverse quark polarization and perform flavor separation.The ideal place to test the fracture functions factorization and measure these new nonperturbative objects are JLab12 and EIC facilities with full coverage of phase space.The role of STMF FracFuns in pp → μ+ μ- + h +X, pp → h +X. Factorization?


25

σUL

DIS2011, JLab, April 14, 2011 Aram Kotzinian

1 1

1 1

1 1 1 1

1 1

·1 21

2

·

1 2

11 2

11 2

2 2· ·2 2

2 2

2ˆ ·1

1 1

ˆ

ˆ

ˆ ˆ

1 2

2

2

1

1

sin( )

sin( )

sin(2 )

sin(2 )

hL

hL

L

L

hL

DT TUL k

N

HT Tnn p

t HTkp

u

t

H HT Tk p

N

t

N

tp

P F

P P Fm m

P PD Fm m

P PF Fm m m m

m m

26

σUT


1 1 1 1

1 1

1 1

1

1

1 1

1

1

ˆ ˆ· ·2 20 2

2

2 2· ·1 2 1 2

1 321 2 1 2

3

ˆ

ˆ ˆ

·11 1

ˆ ·11

1

21 2

421

·112

1

2

ˆ

2 2

2

sin( )

2

(

(

s

)

in )hT T

hh h

T

T TT

T

u DTk S

N

t H

UT

u D u DT Tk

N

H HTp

T T T Tp kp

N

HTk

T Tkkp

N

S

kpN

t t

nn

t

P PF Fm m

P P P PF Fm m m m m

P Fm m

P Fm

P Fm

P P Fm

D y

m

1 1

1 1 1 1 1 1 1 1

1 1 1

1

1

1 ·152

1

3 2· · · ·2 2 1 2

ˆ

ˆ ˆ ˆ ˆ22 2 1 42

1 1 2 1 2 1 2

2 3ˆ ˆ· ·1 2 2 2

2 32 21 1

1ˆ ·

2 2

2 2

sinT

hh h hT T T T

T T

T

St H HT

kkpN

H H H HT T T T Tp p kp kp

N N

t

t t t t

t tH HT T T Tkkp kkp

N N

P Fm m

P P P P PF F F Fm m m m m m m m m

P P P PF Fm m m m

1 1

1 1

1 1 1 1

1 1 1 1

·62

1

3 3· ·2 2

2 32 21 2 1

2 2

3ˆ

· ·1 2 1 21 42 2

1 2 1

ˆ

ˆ ˆ

·11 12

1

ˆ

2

ˆ

2sin

sin 32 2

2 2

i

si

s n 32

T

hhT T

hh

T

T T

Hkkp

N

H HT Tp kk

t HTkkp S

St

tp

N

H HT T T Tp kkp

N

N

St

t t

P

Fm m

P PF Fm

Fm m

m m m

P P P PF Fm m m m

1 1

1 1

1 1

2·1 2

11

ˆ

ˆ

2

2·1 2

31 2

1 2

1 2

1 2

1

2·1 2

221

ˆ2

2

2

2

n 2

sin 2

sin 2

sin 2

hT

hT

T

S

S

S

S

HT Tkp

N

HT Tkp

N

HT Tkkp

N

t

t

t

P P Fm m m

P P Fm m m

P P Fm m

27

σLU, σLL, σLT


1 1ˆ ·1 21

21 2 s in( )

hl DT TLU k

N

P P Fm m

1 1ˆ ·

0 lL

DL F

1 1 1

1 1

1

ˆ ·

· ·2 20 2

2

ˆ ˆ

11

2

1cos( )

cos( )hT

T

Tl

l DTkLT

D DT Tkl

S

SN

N

P PF

P F

Fm m

m

28

Convolutions & tensorial decomposition


ˆ ·0

ˆ ˆ· ·1 1 2 2

ˆ ˆ· ·1 1 2 2

ˆ ˆ ˆ· · ·1 1 1 2 2 2 3

ˆ ˆ· ·1 1 1 2 2 2 1 2 1 2

ˆ[ · ]ˆ[ · ]ˆ[ · ]

ˆ[ · ]ˆ[ · ]

M D

i i M D i M DT k T k

i i M D i M DT p T p

i j i j M D i j M D ij M DT T kk T T kk kk

i j i j M D i j M D i j j iT T kp T T kp T T T T

C M D F

C M Dk P F P F

C M D p P F P F

C M Dk k P P F P P F F

C M Dk p P P F P P F P P P P

ˆ ˆ· ·3 4

ˆ ˆ ˆ· · ·1 1 1 1 1 1 2 2 2 2 2 3

ˆ ˆ ˆ· · ·2 2 1 4 1 5 2 6

ˆ · 2 21. 2.. 1

ˆ[ · ]

where , depend on , , , , ·

M D ij M Dkp kp

i j k i j k M D i j k M D i j k M DT T T kkp T T T kkp T T T kkp

i j k M D k ij M D k ij M DT T T kkp T kkp T kkp

M DT T T

F F

C M Dk k p P P P F P P P F P P P F

P P P F P F P F

F x z P P

P 2TP

2 2 2 (2) 2 2 21 2 2

ˆ ˆ[ · ] ( ) ( , , , , · ) ( , )a T T T T T a T T T T a Ta

C M Dw e d k d z M x k P Dp z p w k p P k P

29

Structure functions


2ˆ · 1 2 2 1 2

1 2 2 21 2 1 2

2ˆ · 1 1 2 2 12 2 2 2

1 2 1 2

2 2 22 2 4 2 2 2 21 1 2 2 2 1 2 1 2ˆ ·

1

( · )( · ) ( · )ˆ ··

( · )( · ) ( · )ˆ ··

2 · 2 · 2 ·ˆ ·

4

M D T T T T Tk

T T T T

M D T T T T Tk

T T T T

T T T T T T T T TM Dkk

F C M D

F C M D

F C M D

P P P k P k PP P P P

P k P P P k PP P P P

P k k P P P k k P P P P P

P

2 2 2 21 2 1 2 1 2

2 2 2 22 2 2 2 2 2 41 2 1 2 1 1 1 2 1 2 2 1ˆ ·

2 2 2 2 21 2 1 2 1 2

2 2 22 2 2 2 21 2 1 2 2 1 1 2ˆ ·

3

· ·

2 · · · ·ˆ ·

2 · ·

· · ·ˆ ·

2

T T T T T T

T T T T T T T T T T T TM Dkk

T T T T T T

T T T T T T T TM Dkk

F C M D

F C M D

P P P P P

P P P P P k P P P P P k P k P

P P P P P P

P P P P k P k P P k P

P

21 2

22 2 22 42 2 2 1 2 1 2ˆ 1 1 1 2·

1 2 4 2 42 2 2 21 2 1 2 1 2 1 2 1 2 1 2

·

2( · )( · ) ( · ) 2 ·2( · )( · ) ( · )ˆ ·4 · · 4 · ·

T T

T T T T T T TT T T TM Dkp

T T T T T T T T T T T T

F C M D

P

P k P p k p P P P P PP k P p k p P P

P P P P P P P P P P P P

Documents

Correlations between SIDIS azimuthal asymmetries in target and current fragmentation regions