The total intrinsic k ┴ carried by quarks

The total intrinsic k┴ carried by quarks

Extraction of the Sivers function

What do we learn from it?

k┴ and orbital angular momentum

Orbiting quarks … ?

Mauro Anselmino, Parton Orbital Angular Momentum, RBRC-UNM Workshop, Albuquerque, 24-

26/02/2006

?

Plenty of theoretical and experimental evidence for transverse motion of partons within nucleons

GeV/c 0.2 fm 1 pxuncertainty principle

gluon radiation

±1

± ±k┴

qT distribution of lepton pairs in D-Y processes

p p

Q2 = M2

qT

qL

l+

l–γ*

Partonic intrinsic motion

pT distribution of hadrons in SIDIS

hXp*

22 (GeV/c) 25.0 kestimates of

)ˆ(

),(),(

)ˆˆ( ),(2

1),(),(

1/

///

M

kpSkxfkxf

kpSkxfkxfkxf

aTpa

pa

Npapa

aTpa

N fM

kf 1/

2

The Sivers distribution function

S

kp

ˆ

k

number density of partons with longitudinal momentum fraction x and transverse

momentum k┴, inside a proton with spin S

0),( /

2 a pakxfkkddx

M. Burkardt, PR D69, 091501 (2004)

frame c.m. * )sin(PPpSA STTN p

needs k┴ dependent quark distribution in p↑ (Sivers mechanism)

z

y

xΦSΦπ

X

p

S

PT

Transverse single spin asymmetries in SIDIS

in collinear configurations there cannot be (at LO) any PT

)dd( d

sin )dd( d2

dd

dd sin

UTN AA

p┴ = PT – z k┴

+ O(k┴2/Q2)

Sivers mechanism in SIDIS

q

hqpqhS

q ShhqSpq

NhS

TUT

pzDdkxfkddd

pzDdkxfkddd

PzyxA S

),( ˆ ),(

)sin( ),( ˆ )sin( ),(

),,,(

/2

/

2

)sin(

TPddzdQdx

dd

22

5

2

ˆˆ

dQ

dd

lqlq

Other approaches, with some simplifying assumptions:

q

hqpqq

q

hq

qTq

M

P

UT zDxfxe

zDzxfxezxA

TS

)( )(

)( )( 2),(

/2

)1(1

2)sin(

),( 2

)( 12

22)1(

1 kxf

M

kkdxf q

Tq

T

dzdydx

ddzdy

dzdydxd

dzdyxA

UU

SiversUT

N

)(

)( )( )2

1(4

)2/1(

1

2

4

2

zDxfxy

yQ

s

dzdydx

d hq

qT

emSiversUT

),( )( 12)2/1(

1

kxf

M

kkdxf q

Tq

T

J.C. Collins et al.

W. Vogelsang, F. Yuan

“Sivers moment”

XlNl

dd

ddNA

)dd(d d

)sin()dd(d d2

)sin(2

S

S

)sin(

S

UTSSA

M.A., U.D’Alesio, M.Boglione, A.Kotzinian, A Prokudin

from Sivers mechanism)sin( S

UTA

Deuteron target hd

hupd

N

pu

NUT DDffA Sh

4 //

)sin(

first p┴ moments of extracted Sivers

functions, compared with models

M.A, M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A. Prokudin

data from HERMES and COMPASS

),( 4

/

2

)1(1

)1(

kxfM

kkd

ff

pq

N

qTq

N

comparison of different extractions: hep-ph/0511017

0/ pq

N f

The first and 1/2-transverse moments of the Sivers quark distribution functions, defined in Eqs. (3, 9), as extracted in Refs. [20, 21, 23]. The fits were constrained mainly (or solely) by the preliminary HERMES data in the indicated x-range. The curves indicate the 1-σ regions

of the various parameterizations.

),( )( 12)2/1(

1

kxf

M

kkdxf q

Tq

T

M. Anselmino, M. Boglione, J.C. Collins, U. D’Alesio, A.V. Efremov, K. Goeke, A. Kotzinian, S. Menze, A. Metz, F. Murgia, A. Prokudin, P. Schweitzer, W. Vogelsang, F. Yuan

),( 2

12

22)1(

1

kxf

M

kkdf q

Tq

T

),( )( 1

2)2/1(1

kxfM

kkdxf q

Tq

T

Predictions for K production at HERMES

Predictions for COMPASS, hydrogen target

Q2 > 1 (GeV/c)2 W2 > 25 GeV2 PT > 0.1 GeV/c Eh > 4 GeV 0.2 < zh < 0.9 0.1 < y < 0.9

predictions for JLab, proton target

E704 data, E = 200 GeV

fit to AN with Sivers effect

U. D’Alesio, F. Murgia

SSA in p↑p → π X

dd

ddNA

SSA in p↑p → l+ l- X

//

2pqpq

N

qqN ffeA

dd

ddNA

p p

Q2 = M2

qT

qL

l+

l–γ*

TqddMdy

dd 22

4

YD

qTSIDIS

qT ff

11QCD test:

SSA in p↑p → D X

only Sivers effect: no transverse spin transfer in

dominance of gluonic channel, access to gluon Sivers function

QQggQQqq ,

maxNA

assuming saturated Sivers

function

papa

N ff //2

) :lines thin , :lines(thick QQqqQQgg

),(ˆ 2

ˆ cosˆ sin

)ˆˆ( ),(ˆ2

1),(ˆ

/

2

//2

kxfkdkdxji

kpSkxfkxfkkddxk

pa

NSS

pa

Npa

a

What do we learn from Sivers distribution?

total amount of intrinsic momentum carried by partons of flavour a

for a proton moving along the +z-axis and polarization vector

jiS SSˆ sinˆ cos

S

ak

)sin()ˆˆ( SkpS

Numerical estimates

),( )( )(2),( // kxfkhxΝkxf pqqpq

N

qq

qq

qq

qq

qqqq xxCxΝ

)()(

)1( )(

22

2 )(

Mk

Mkkh

22 /2/

1)( ),( kk

qpq ek

xfkxf

22 (GeV/c) 25.0 k from fitting data on Cahn effect

MduqC qqq ),,( , , from fitting HERMES and COMPASS data on AN

M.A., M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia and A. Prokudin

GeV/c ˆ sinˆ cos 128.0

GeV/c ˆ sinˆ cos 141.0

jik

jik

SSd

SSu

uk

dk? 0

du kk

Sivers functions extracted from AN data inXpp give also opposite results, with

036.0 032.0 du kk

k┴ and orbital angular momentum (case of 2 quarks)

021 kk

02 1 LL

021 kk

02 1 LL

021 kk

02 1 LL

021 kk

Does It depends on space distribution …… to be continued ……

02 1 LL

021 kk

imply

02 1 LL

?