Spin and k┴ dependent parton distributions

Mauro Anselmino, ECT* Trento, 04/07/2006

- a closer look at the nucleon structure -

integrated partonic distributions

the missing piece, transversity

partonic intrinsic motion (TMD)

spin and k┴: transverse Single Spin Asymmetries

SSA in SIDIS and p-p, p-pbar inclusive processes

conclusions

(with the help of antiprotons)

K┴ integrated parton distributions

quark distribution – well known

quark helicity distribution – known

transversity distribution – unknown

all equally important

related to

positivity bound

qqq

qqq

qqqT

q qq 5

chiral-even

qT qq 5 related to chiral-odd

qqqT ||2

ggg gluon helicity distribution – poorly known

are fundamental leading-twist quarkqq , and 1h (or ) , qq T distributions depending on longitudinal momentum fraction x

),( 2Qxq ),( 2Qxq

de Florian, Navarro, Sassot

Research Plan for Spin Physics at RHICFebruary 11, 2005

Figure 11: Left: results for Δg(x,Q2 = 5GeV2) from recent NLO analyses [1, 2, 36] of polarizedDIS. The various bands indicate ranges in Δg that were deemed consistent with the scalingviolations in polarized DIS in these analyses. The rather large differences among these bandspartly result from differing theoretical assumptions in the extraction, for example, regarding theshape of Δg(x) at the initial scale. Note that we show xΔg as a function of log(x), in orderto display the contributions from various x-regions to the integral of Δg. Right: the “net gluonpolarization” Δg(x,Q2)/g(x,Q2) at Q2 = 5 GeV2, using Δg of [2] and its associated band,and the unpolarized gluon distribution of [82].

+–– –

++ +

+ +

+ =),( 2Qxq),( 2Qxq

+– =),( 2Qxq

),( 2QxqT

|| 2

1 iin helicity basis

–+

+ – ),( 2

1 Qxh decouples from DIS (no quark helicity flip)

The missing piece, transversity

h1 must couple to another chiral-odd function. For example

++ –

++

–

+ –

–– +

–

+ –

h1 x h1

h1 x Collins function

J. Ralston and D.Soper, 1979 J. Cortes, B. Pire, J. Ralston, 1992

J. Collins, 1993

XlplXllpp SIDIS, and , Y,D

–+

+1 –1

2 g

1

No gluon contribution to h1

simple Q2 evolution

Q2 = 25 GeV2 Q02

= 0.23 GeV2

V. Barone, T. Calarco, A. Drago

First ever “transversity”, according to Gary Goldstein, QCD-N06, Villa Mondragone, June 15, 2006

Elementary LO interaction:

3 planes:

)( )()( )(1

9

42121

2

212

2

2

2

xqxqxqxqexxsMdxdM

daaaa

aa

F

sqxsMxxxxx LFF /2 / 22121

llqq *

plenty of spin effects

p p

Q2 = M2

qT

qL

l+

l–

h1 in Drell-Yan processes

*

plane plane, * llp , on vectorspolarizati to plane

q q

q qqqqq

TTTTxqxqxqxqe

xhxhxhxheaA

)()()()(

)()()()(ˆ

dd

dd

21212

211121112

h1 from

)cos(2 cos1

sin

ˆdˆd

ˆdˆdˆ

2

2

TTa

RHIC energies:

small x1 and/or x2

h1q (x, Q2) evolution much slower than

Δq(x, Q2) and q(x, Q2) at small x

ATT at RHIC is very small

smaller s would help Martin, Schäfer, Stratmann, Vogelsang

Barone, Calarco, Drago

22 GeV 100 GeV 200 Ms

3102

Xllpp at RHIC

h1 from

)()(

)()(ˆ

)()()()(

)()()()(ˆ

21

2111

21212

211121112

xuxu

xhxha

xqxqxqxqe

xhxhxhxheaA uu

TT

q q

q qqqqq

TTTT

22 GeV 2 GeV 21030 MsGSI energies: large x1,x2

one measures h1 in the quark valence region: ATT

is estimated to be large, between 0.2 and 0.4

at GSIXllpp

PAX proposal: hep-ex/0505054

Energy for Drell-Yan processes

"safe region": /JMM

s

M J /2

QCD corrections might be very large at smaller values of M:

yes, for cross-sections, not for ATT K-factor almost spin-independent

Fermilab E866 800 GeV/c

H. Shimizu, G. Sterman, W. Vogelsang and H. Yokoya

M. Guzzi, V. Barone, A. Cafarella, C. Corianò and P.G. Ratcliffe

s=30 GeV2 s=45 GeV2

s=210 GeV2s=900 GeV2

s=30 GeV2 s=45 GeV2

s=210 GeV2s=900 GeV2

data from CERN WA39, π N processes, s = 80 GeV2

H. Shimizu, G. Sterman, W. Vogelsang and H. Yokoya

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

GeV/c 0.2 fm 1 pxuncertainty principle

gluon radiation

±1

± ±k┴

qT distribution of lepton pairs in D-Y processes

Partonic intrinsic motion

p p

Q2 = M2

qT

qL

l+

l–*

pT distribution of hadrons in SIDIS

Xhp *

Hadron distribution in jets in e+e– processes

Large pT particle production in hXpN

Transverse motion is usually integrated, but

there might be important spin-k┴ correlations

pxp

k┴

spin-k┴ correlations? orbiting quarks?

Transverse Momentum Dependent distribution functions

Space dependent distribution functions (GPD)

),( kxq

),( bxq

k

?

b

M. Arneodo et al (EMC): Z. Phys. C 34 (1987) 277

),(ˆd ),(d 22 QzDQxf hq

lqlq

qq

lhXlp

Unpolarized SIDIS, )]([ 0sO

qp

Qx

2

2

22 qQ

pl

qpy

in collinear parton model

222 )1(1ˆˆˆd yuslqlq thus, no dependence on azimuthal angle Фh at zero-th order in pQCD

the experimental data reveal that

hhhXlhlq ΦCΦBAΦ 2coscosd/ˆd

Cahn: the observed azimuthal dependence is related to the intrinsic k┴ of quarks (at least for small PT values)

These modulations of the cross section with azimuthal angle are denoted as “Cahn effect”.

assuming collinear fragmentation, φ = Φh

)0 ,sin ,cos( kkk

2

2

cos12

1ˆQ

ky

Q

ksxs O

2

2

cos1

21)1( ˆ

Q

k

yQ

kyxsu O

hhh

lhXlq

ΦCΦBAusΦ

2coscosˆˆd

ˆd 22

SIDIS with intrinsic k┴

kinematics according to Trento conventions (2004)

);,();,(ˆd);,(d 222 QpzDQkyQkxf hq

lqlq

q qlhXlp

factorization holds at large Q2, and QCDT kP Ji, Ma, Yuan

The situation is more complicated as the produced hadron has also intrinsic transverse momentum with respect to the fragmenting parton

k

kz

p

TP

px

hp

Qk /neglecting terms of order one has

kzpPT

assuming:

one finds:

with

clear dependence on (assumed to be constant)and

Find best values by fitting data on Φh and PT dependences

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

Large PT data explained by NLO QCD corrections

dashed line: parton model with unintegrated distribution and fragmentation functions solid line: pQCD contributions at LO and a K factor (K = 1.5) to account for NLO effects

ccdab

pbgqqdcba

pa Dff //,,,,,

/ ˆd d

PDF FF pQCD elementary

interactions

ab

c

0DX

X

f

f

(collinear configurations)Xpp 0factorization theorem

p

p

The cross section

uts ˆ ,ˆ ,ˆ elementary Mandelstam variables

uts , , hadronic Mandelstam variables

uxtxzsxx baba

Xpp 0

RHIC data GeV 200s

Xpp

FNAL data, PLB 73 (1978)

F. Murgia, U. D’Alesio

Xpp 0 GeV 20soriginal idea by Feynman-Field

GeV 8.0 k k no

a b

c

0D

X

X

ff

non collinear configurations

S

p

p'

– pPTθ

θsinPpSdd

ddT

NA

– p'

5;

4;

3;

2;

1;

M

M

M

M

M

5 independent helicity amplitudes

)(Im 43215NA

pppp

z

y

x

Example:

Transverse single spin asymmetries: elastic scattering

0NA needs helicity flip + relative phase

–+

++

Im x

+

++

+

sq

N E

mA at quark level

but large SSA observed at hadron level!

QED and QCD interactions conserve helicity, up to corrections )/( EmO q

Single spin asymmetries at partonic level. Example: '' qqqq

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

STAR-RHIC √s = 200 GeV 1.1 < pT < 2.5

AN stays at high energies ….

Transverse Λ polarization in unpolarized p-Be scattering at Fermilab

XNp

dd

ddP

pppp pppp

dd

ddNA

dd

ddNNA

“Sivers moment”

XlNl

dd

ddNA

)dd(d d

)sin()dd(d d2

)sin(2

S

S

)sin(

S

UTSSA

“Collins moment”

dd

ddNA

XlNl

)dd(d d

)sin()dd(d d2

)sin(2

S

S

)sin(

S

UTSSA

qTpq

N fM

kf

1/

2

pSk┴

φq

pq

φ Sqp┴

)ˆˆ( ),(2

1

),(),(

/

//

ppSpzD

pzDpzD

qqqh

N

qhqh

)ˆˆ( ),(2

1

),(),(

/

//

kpSkxf

kxfkxf

pq

N

pqpq

Amsterdam group notations

q

hqh

N HMz

pD 1/

2

spin-k┴ correlations

Sivers function Collins function

q

pq

N hM

kf 1/

p

Sqk┴

φq

pq

φSΛp┴

)ˆˆ( ),(2

1

),(2

1),(

/

//

ppSpzD

pzDpzD

qq

N

qhq

)ˆˆ( ),(2

1

),(2

1),(

/

//

kpSkxf

kxfkxf

qpq

N

pqpq

Amsterdam group notations

qTq

N DMz

pD

1/ 2

spin-k┴ correlations

Boer-Mulders function polarizing f.f.

8 leading-twist spin-k┴ dependent distribution functions

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

from Sivers mechanism)sin( S

UTA

Deuteron target hdhupd

N

pu

NUT DDffA Sh

4 //

)sin(

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

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

fit to HERMES data on)sin( Sh

UTA

W. Vogelsang and F. Yuan

Hadronic processes: the cross section with intrinsic k┴

factorization is assumed, not proven in general; some progress for Drell-Yan processes, two-jet production, Higgs

production via gluon fusion (Ji, Ma, Yuan; Collins, Metz; Bacchetta, Bomhof, Mulders, Pijlman)

intrinsic k┴ in distribution and fragmentation functions and in elementary interactions

The polarized cross section with intrinsic k┴

A

aa

SAa ,/'

badcM ,;,ˆ

CC

ccD

,

, '

helicity density matrix of parton a inside polarized hadron A

pQCD helicity amplitudes

product of fragmentation amplitudes

E704 data, E = 200 GeV

fit to AN with Sivers effects alone

maximized value of AN

with Collins effects alone

M.A, M. Boglione, U. D’Alesio, E. Leader, F. Murgia

SSA in p↑p → π X

Maximised (i.e., saturating positivity bounds) contributions to AN

quark Sivers contribution

gluon Sivers contribution

Collins contribution

Special channels available with antiprotons – Results from U. D’Alesio

Xpp

dd

ddNA

Xllpp

Predictions for AN in D-Y processes

Sivers function from SIDIS data, large asymmetry and cross section expected

XDpp at PAX, contrary to RHIC, dominates the

ccqq channel:

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

)rapidities denote 8.3 ,2 ,1 ,0

:lines thin , :lines(thick QQqqQQgg

predictions based on Sivers functions

extracted from fitting E704 data

maximised contributions from h1 times B-M, not suppressed by phases

channels dominating and gqqqqg

dd

ddNA

Xpp

predictions based on Sivers functions from E704 data Xpp ,0,

Conclusions

Polarized antiprotons are the only way to access directly the transversity distribution: optimum energy at s 200 ≈ GeV2

Unintegrated (TMD) distribution functions allow a much better description of QCD nucleon structure and hadronic interactions

(necessary for correct differential distribution of final state particles, (Collins, Jung, hep-ph/0508280)

k┴ is crucial to understand observed SSA in SIDIS and pp interactions; antiprotons will add new information and allow further

test of our understanding

Spin-k┴ dependent distribution and fragmentation functions: towards a complete phenomenology of spin asymmetries

Open issues: factorization, QCD evolution, universality, higher perturbative orders, …