Mauro Anselmino, Torino University and INFN,

Vancouver, July 31, 2007

The transverse spin structure of the

nucleon

xP

P

The longitudinal structure of nucleons is “simple” It has been studied for almost 40

years

Q2

essentially x and Q2 degrees of freedom ….

Very good or good knowledge of q(x,Q2), g(x,Q2) and Δq(x,Q2) Poor knowledge of Δg(x,Q2)

? 2

1 gqgq LLSS

(Research Plan for Spin Physics at RHIC)

Talk by L. De Nardo

spin-k┴ correlations? orbiting quarks?

Transverse Momentum Dependent distribution

functions

Space dependent distribution functions

);,( 2Qxq k ),( 2;Qxq b

k

?

b

The transverse structure is much more interesting and less studied

The mother of all functions M. Diehl, Trento

workshop, June 07

TMD’s

GPD’sWigner function

(Belitsky, Ji, Yuan)

TMDs in SIDIS

SSA in SIDIS: Sivers functions

Collins function from e+e- unpolarized processes (Belle) and first extraction of

transversity

SSA in hadronic processes

Future measurements and transversity

(Trento workshop on “Transverse momentum, spin, and position distributions of partons in hadrons”, June

07)

SIDIS kinematics according to

Trento conventions

(2004)

Main source of information on

transverse nucleon structure

Polarized SIDIS cross section, up to subleading order in 1/Q

Kotzinian, NP B441 (1995) 234

Mulders and Tangermann, NP B461 (1996) 197

Boer and Mulders, PR D57 (1998) 5780

Bacchetta et al., PL B595 (2004) 309

Bacchetta et al., JHEP 0702 (2007) 093

SIDISLAND

SIDIS in parton model with intrinsic k┴

);,();,(ˆd);,(d 222 QzDQyQxf hq

lqlq

q qlhXlp

pkk

factorization holds at large Q2, and

QCDT kP Ji, Ma, Yuan

EMC data, µp and µd, E between 100 and 280 GeV

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

Azimuthal dependence induced by quark intrinsic motion

)ˆˆ( ),(),(

)ˆˆ( ),(2

1),(),(

1/

//,/

kpS

kpSkS

kxfM

kkxf

kxfkxfxf

qTpq

pq

Npqpq

Sivers function

Boer-Mulders function

)ˆˆ( ),(2

1),(

2

1

)ˆˆ( ),(2

1),(

2

1),(

1/

///,

kps

kpsks

qq

pq

qpq

Npqpq

kxhM

kkxf

kxfkxfxfq

8 leading-twist spin-k┴ dependent distribution functions

Courtesy of Aram Kotzinian

)ˆˆ( ),(

),(

)ˆˆ( ),(2

1 ),(),(

1/

//,/

pps

ppsps

qqq

hqh

qqqh

Nqhqh

pzHMz

ppzD

pzDpzDzDq

Collins function

)ˆˆ( ),(

),(2

1

)ˆˆ( ),(2

1 ),(

2

1),(

1/

///,

ppS

ppSpS

qq

Tqh

qq

Nqhq

pzDMz

ppzD

pzDpzDzD

Polarizing fragmentation function

),(),(

)dd(d d

)sin()dd(d d2

)sin(2

//

2

)sin(

pzDkxfe

ΦΦ

ΦΦΦΦ

AΦΦ

qpq

N

q q

S

SS

ΦΦUTS

S

Large K+ asymmetr

y!

),(),(

)dd(d d

)sin()dd(d d2

)sin(2

/12

)sin(

pzDkxhe

ΦΦ

ΦΦΦΦ

AΦΦ

qh

N

q qq

S

SS

ΦΦUTS

S

Talk by G.

Schnell

COMPASS measured Collins and Sivers asymmetries for positive (●) and negative (○) hadrons

dhuhpd

N

pu

NUT DDffA Sh

////

)sin( 4

small values due to

deuteron target:

cancellation between u and d contributions

dh

N

uh

NduUT DDhhA Sh

//11)sin( 4

talk by T. Iwata

M. Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A. ProkudinA fit of HERMES + COMPASS pion

data, information on u and d Sivers funtions

no sea contribution

sea contribution?

(Kretzer fragmentation functions)

DSS = de Florian, Sassot, Stratmann KRE = Kretzer HKNS = Hirai, Kumano, Nagai, Sudoh

Fragmentation functions

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

fit of HERMES + COMPASS pion and kaon data, with new set of fragmentation functions (de Florian, Sassot,

Stratmann)

Predictions for JLab

The first and 1/2-transverse moments of the Sivers quark distribution functions. 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

kdxf q

Tq

T k

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

kdf q

Tq

T k

Present knowledge of Sivers function (u,d)

S

kp

k

number density of partons with longitudinal

momentum fraction x and transverse momentum k┴, inside a proton with spin S

0),( /

2 a paxfddx kkk

M. Burkardt, PR D69, 091501 (2004)

What do we learn from the Sivers distribution?

),(ˆ 2

ˆ cosˆ sin

)ˆˆ( ),(ˆ2

1),(ˆ

/

2

//2

kxfkdkdx

kxfkxfddx

pa

NSS

pa

Npa

a

ji

kpSkkk

Total amount of intrinsic momentum carried by partons of flavour a

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

jiS ˆ sinˆ cos SS

S

ak

)sin()ˆˆ( SkpS

GeV/c ˆ cosˆ sin 13.0

GeV/c ˆ cosˆ sin 14.003.002.0

0.050.06-

jik

jik

SSd

SSu

uk

dk? 0

du kk

Sivers functions extracted from AN data in Xpp give also opposite results,

with

036.0 032.0 du kk

Numerical estimates from SIDIS dataU. D’Alesio

Sivers function and orbital angular momentum

Sivers mechanism originates from

qLS then it is related to the quark orbital angular momentum

D. Sivers

For a proton moving along z and polarized along y

? 2

),( 1

0 /

qy

pq

NL

kxfdx

Sivers function and proton anomalous magnetic momentM. Burkardt, S. Brodsky, Z. Lu, I. Schmidt

Both the Sivers function and the proton anomalous magnetic moment are related to correlations of proton wave functions with opposite helicities

? ),( 1

0 /

2qpq

N Ckxfddx k

in qualitative agreement with large z data:

d

uΦΦ

UT

ΦΦ

UT

S

S

A

A

) sin(

) sin(

related result (M. Burkardt)

Collins function from e+e–

processes (spin effects without polarization, D. Boer)

e+e- CMS frame: GeV 52.10 ,2

ss

Ez h

Xhhqqee

BELLE @ KEK

e+

1hP

2hP e-

e+

thrust-axis

z

y

x

1p

2p

)()( 2/1/12 zDzDAq

N

q

N

Fit of BELLE data

Extraction of Collins functions and transversity distributions from fitting HERMES +

COMPASS + BELLE data

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

A. Prokudin C.

Türk

What do we learn (if anything yet) from the transversity distributions and the Collins

functions ? h1u and h1d have opposite signs

They are both smaller than Soffer bound

Still large uncertainties

Tensor charges appear to be smaller than results from lattice QCD calculations M. Wakamatsu, hep-ph/0705.2917

Collins functions well below the positivity bound. Favoured and unfavoured ones have opposite signs, comparable magnitudes

Only the very beginning …

)(ˆd)()( d //,,,,,

/ zDxfxf ccdab

bpbagqqdcba

pa

PDF FF pQCD elementary

interactions

(collinear configurations)Xpp 0factorization theorem

(?)

ab

c

0DX

X

f

fp

p

TMDs and SSAs in hadronic collisions (abandoning safe

grounds ….)

RHIC data GeV 200s

Xpp

excellent agreement with data for unpolarized cross section, but no

SSA

BNL-AGS √s = 6.6 GeV 0.6 < pT < 1.2

E704 √s = 20 GeV 0.7 < pT < 2.0

E704 √s = 20 GeV 0.7 < pT < 2.0

experimental data on

SSA

Xpp

Xpp

observed transverse Single Spin

Asymmetries

dd

ddNA

and AN stays at high energies ….

STAR-RHIC √s = 200 GeV 1.2 < pT < 2.8

talk by L. Bland

SSA in hadronic processes: intrinsic k┴, factorization?

Two main different (?) approachesGeneralization of collinear scheme (M. A., M. Boglione, U. D’Alesio, E. Leader, F. Murgia, S. Melis)

ab

c

0DX

X

ffp p

),()(ˆd)()( d //,,,,,

/

pkkkk zD,,xf,xf cbacdab

bbpbaagqqdcba

pa

It generalizes to polarized case

),()(ˆˆ

)()(d

,

,,;,,;,

,/,/

,/,/ ,,

''''

''

pkk

kk

zD,MM

,xf,xf

CC

cc

cdab

badc

cdab

badc

B

B

bbA

A

aa

BA

ba

bbSBbSBb

aaSAaSAaXCSBSA

plenty of phases

main remaining contribution to SSA from Sivers effect

),()(ˆd

)()( d

/

//

,

pkk

kk

zD,

,xf,xf

cbacdab

bbpbaaq

pq

NXpSp

E704 data

STAR data

U. D’Alesio, F. Murgia

fit prediction

Higher-twist partonic correlations (Efremov, Teryaev; Qiu, Sterman; Kouvaris, Vogelsang, Yuan)

)(),()(),,( d /21/21,,

zDkkHxfkkT chcab

bBbcba

a S

twist-3 functions

hard interactions

contribution to SSA

) ( XhBA

“collinear expansion” at order ki┴

)()1( / xfxxNT Aaaaaa

fits of E704 and STAR data

Kouvaris, Qiu, Vogelsang, Yuan

Gluonic pole cross sections and SSA in

XhhHH 2121

Bacchetta, Bomhof, Mulders, Pijlman; Vogelsang, Yuan

)()(ˆ)()( d 2/1/][2/1,,

)1(1 211

zDzDdxfxf dhchcdbaHbcba

T

gluonic pole cross sections take into account gauge links

Sivers contribution to SSA

D

Dcdab

DGcdba dCd ˆˆ ][

][][D

GC Diagram dependent Gauge link Colour factors

factorization ? (Collins)

)( )1(1 Ta fT

(breaking of factorization?)

Gluonic pole cross sections and SSA in

XhhHH 2121

to be compared with the usual cross section

llqqllqqlqlqlqql dddd ˆˆ ˆˆ

][][

Simple QEDexample:

DIS: attractive Drell-Yan: repulsiveSame in QCD:

As a result:

Non-universality of Sivers Asymmetries:Unique Prediction of Gauge Theory !

TMDs and SSAs in Drell-Yan processes (returning to safer

grounds and looking at future ….)

p p

Q2 = M2

qT

qL

l+

l–γ*

llqqqq q

YD QxfQxf ˆd );,();,(d 22

21 kk

factorization holds, two scales, M2, and

Tq

2cossin

2cossincos1

3

1

4

31 222

d

d

Unpolarized cross section already very interesting

Collins-Soper frame

(Polarized) Drell-Yan cross sections allow to

access many TMDs (Boer-Mulders, …) D-YLAND

verify whether

and offer the golden channel to measure the transversity distribution

q

qqTT xhxhA )()( 2111

YD

qTSIDIS

qT ff

11 ! ! !

Talks by R. Kaiser, M. Contalbrigo

ConclusionsIncreasing knowledge about quark transverse motion, and spin-k┴ correlations (Sivers function)

First experimental information on h1

Work in progress on GPDs and quark space distribution (Lq)

Towards resolving the orbital motion of quarks in a proton …

More data from HERMES, COMPASS, JLab and RHIC Thanks