A sideways look into the proton

Transverse momentum and transverse spin in QCD

Alessandro Bacchetta

Alessandro Bacchetta A sideways look at the proton

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Outline

• Introduction• Transverse spin• Transverse momentum• Transverse spin and transverse

momentum• Factorization and hadron collisions• Conclusions

Introduction

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Deep Inelastic Scattering - DIS

target proton

electron

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Inclusive DIS

proton

lepton l

'l

tran

sver

se

longitudinal

( ) ( ) ( ) l p P l X

2 2 virtuality ( ' of pho o) t n l l Q2

2 ( ')

Q

xP l l

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Semi-inclusive DIS (SIDIS)

proton

pion

transverse momentum of pionhP

hP lepton l

'l

( ) ( ) ( ) ( ) hl p P l h P X

2 2 virtuality ( ' of pho o) t n l l Q2

2 ( ')

Q

xP l l

( ')

hP P

zP l l

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Vectors and angles involved

DIS

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Vectors and angles involved

SIDIS

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Transverse momentum effects

hP

SIDIS

Tq

Drell-Yan

TK

e-e+ to pions

TR

3-Dp-p to pions

Whenever we measure transverse-momentum effects, we need kT-factorization and we need transverse momentum dependent (or unintegrated) parton distributions

Collins, Soper, NPB 193 (81)

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Feynman diagrams & Factorization

proton

lepton lepton

pion

SIDIS

Partonic scattering amplitude

Fragmentation amplitude

Distribution amplitude

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Feynman diagrams & Factorization

Partonic scattering amplitude

Fragmentation amplitude

Distribution amplitude

proton

lepton lepton

pion

SIDIS

electron

positron

pion

pion

e–e+ to pions

proton

proton lepton

antilepton

Drell-Yan

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Feynman diagrams & Factorization

proton

lepton lepton

pion

SIDIS

proton

proton lepton

antilepton

Drell-Yan

proton

proton

pion

pion

p-p to pions

electron

positron

pion

pion

e–e+ to pions

??

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Nonperturbative elements

2

,

0

[0 ]( ; ) , (0) ( ) ,2

i x Pij j i

dx S e P S P SU

,P S

k

=

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Decomposition of the correlation function

1 1 5 1 5( ) ( )1

)2

(( ); Lq

Tq qf x g x h xx S S S

1

1

1

( )

(

(

)

( ) (

)

) )

(

(

)

q

q

qT

q x

q x

q x

f x

g x

q xh x

Unpolarized distr. func.

Helicity distr. func.

Transversity distr. func.

0(1 ), leading twistO Q

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Longitudinal vs transverse momentum

target

parton

Tk

P

partonk

longk xP

photon

Long. view

long. spinphoton

Trans. view Photon moves into the screen/ proton moves out of the screen

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Longitudinal vs transverse momentum

Tkpartonk

longk xP

photon

Long. view

trans. spinphoton

Trans. view Photon moves into the screen/ proton moves out of the screen

target

partonP

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Probabilistic interpretation

( ) ( )qTh x q x 1

_

( ) ( )qf x q x 1

( ) ( )qg x q x 1_

Photon moves into the screen/ proton moves out of the screen

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Decomposition of the correlation function UNintegrated over kT

12

1

21

2 ( , )1

( , ( , ))2

..( , ) .

qT T

qT

qT

T T TT

T

f x kS k

x kM

ki

Mh

k

k

f x

x

Boer-Mulders

Sivers

Sivers, PRD 43 (91)

Boer, Mulders, PRD 57 (98)

Mulders, Tangerman, NPB 461 (96)Goeke, Metz, Schlegel, PLB 618 (05)

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Probabilistic interpretation

-Sivers

-Boer-Mulders

Photon moves into the screen/ proton moves out of the screen

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Unpolarized distribution functions q(x)

ZEUS Coll, EPJ C42 (05)

7 groups are working on the extraction of these PDFs (see www-spires.dur.ac.uk/hepdata/pdf.html)

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Helicity distribution function q(x)

AAC, Hirai et al. PRD69 (04)

6 groups are working on the extraction of these PDFs (see www-spires.dur.ac.uk/hepdata/pdf.html)

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• What about the transverse spin distribution h1(x) ?

• What about the transverse momentum dependence, e.g. f1(x,kT

2)?

Transverse spin

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Transversity in SIDIS

12

12(... sin( ) ( , ) ., .) .T h S TTh x pd S H z k

see e.g. A.B., Diehl, Goeke, Metz, Mulders, Schlegel, JHEP 0702,093

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Data from Hermes and Compass

HERMES, hep-ex/0507013

HERMES, PRL 94 (05)

COMPASS, NPB 765 (07)

COMPASS, PRL 94 (05)

PROTON DEUTERON

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Data from BELLE

BELLE, PRL 96 (06)

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First extraction of transversity

Anselmino et al., PRD 75, 054032 (07)

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Comparison with some models

[1] Soffer et al. PRD 65 (02)

[2] Korotkov et al. EPJC 18 (01)

[3] Schweitzer et al., PRD 64 (01)

[4] Wakamatsu, PLB 509 (01)

[5] Pasquini et al., PRD 72 (05)

[6] Anselmino et al., PRD 75 (07)

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Tensor charge from Lattice QCD

1 1

1 1

( ) 0.84, ( ) 0.64,

( ) 0.23, ( ) 0.35

u u

d d

u h x dx u g x dx

d h x dx d g x dx

S. Aoki et al., PRD 56 (1997)M. Göckeler et al. [QCDSF/UKQCD], PLB (05)

Compared to axial charge

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Comparison of tensor charges

Barone, Drago, Ratcliffe, PR 359 (2002)

lattice

quark soliton 1

spectator

bag

quark soliton 2

quark

non-relativistic

2.4 GeV2

0.3 GeV2

Wakamatsu, 0705.2917[hep-ph]

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Evolution equations

Barone, Drago, Ratcliffe, PR 359 (2002)

Hayashigaki, Kanazawa, Koike, PRD56 (97)

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Other observables

12

12(... sin( ) ( , ) ., .) .T h S TTh x pd S H z k

Efremov, Mankiewicz, Tornquist, PLB 284 (92)Collins, Heppelmann, Ladinsky, NPB 420 (94)Jaffe, Jin, Tang, PRL 80 (98)

12

1(... sin( ) ( , , ) .) ..T h R h xd S H z M

Dihadron fragmentation

11 1 212 ( )... ( ) ...T Td S h xS h x

Doubly polarized Drell-YanRalston, Soper, NPB 152 (79)

Transverse momentum

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Semi-inclusive DIS

proton

pion

l p l X 2 2 virtuality of p( ') hotonl l Q

transverse momentum of pionhP

hP lepton l

'l

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Example of model calculation

diquarkproton

quark

(GeV)xk

yk

21 ( , )q

Tf x k

0.5x

(GeV)xk

yk

21 ( , )q

Tf x k

0.01x

2 20.13 GeVTk 2 20.48 GeVTk

Jakob, Mulders, Rodrigues, NPA 626 (1997)

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Example of a measurement

H1 Coll, NPB485(97)

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Transverse momentum distribution

q

q

h

intrinsic

pQCD

Koike, Nagashima, Vogelsang, NPB744 (06)

fragmentation

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Differences between flavors?

(GeV)xk (GeV)xk

yk yk

21 ( )u

Tf k 21 ( ) 2d

Tf k x

ud diquarkproton

u

uu diquarkproton

d

Assuming ud uuM M

Up and down quark can have different

transverse momentum distributions

Transverse momentum and transverse spin

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right

left

lepton l

'l

Semi-inclusive Deep Inelastic Scattering

proton

pion

l p l X

R L

R L

N N

N N

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Measured asymmetries in DIS

HERMES Coll, hep-ex/0507013

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Parton distribution functions with transverse spin

-Sivers

-Boer-Mulders

-Transversity

Photon moves into the screen/ proton moves out of the screen

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right

left

Chromodynamic lensing

proton

quarks

Long. view Trans. view

Burkardt, PRD 66 (02)

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right

left

Chromodynamic lensing

photon

Long. view Trans. view

NOTE: QCD tells us that the FSI has to be attractive, since quark and remnants form a color antisymmetric state

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right

left

Chromodynamic lensing

photon

Long. view Trans. view

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right

left

Distortions in transverse space

proton

quarks

Long. view Trans. view

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right

left

Distortions in transverse space

proton

quarks

Long. view Trans. view

A distortion in the distribution of quarks in transverse space can give rise to a nonzero Sivers function

The presence of spin can distort the distribution of quarks in transverse space (orbital angular momentum of quarks is required)

Spin-orbit correlations

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Results from Lattice QCD

QCDSF/UKQCD Collab. (see e.g. hep-ph/05110032)

Sivers function expected to be:

•NEGATIVE for up quarks

•POSITIVE for down quarks

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Data from Hermes and Compass

HERMES, hep-ex/0507013

HERMES, PRL 94 (05)COMPASS, PRL 94 (05)

PROTON

DEUTERON

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Different fits

Anselmino et al., hep-ph/0511017

[20] Anselmino et al., PRD72 (05)

[21] Vogelsang, Yuan, PRD72 (05)

[23] Collins et al., hep-ph/0510342

Factorization and hadron-hadron

collisions

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Feynman diagrams & Factorization

proton

lepton lepton

pion

SIDIS

proton

proton lepton

antilepton

Drell-Yan

proton

proton

pion

pion

p-p to pions

electron

positron

pion

pion

e–e+ to pions

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right

left

Sivers effect in SIDIS

proton

quarks

Long. view Trans. view

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right

left

Sivers effect in Drell-Yan

proton

Long. view Trans. view

photon

antiquark

The Sivers effect gets an extra minus sign due to the difference between initial and final state interactions. Impossible to predict

it in parton model.

Clear-cut prediction of QCD Collins, PLB 536 (02)

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Final/initial state interactions

SIDIS

q

q

q

q

pp to hadrons

A.B., Bomhof, Mulders, Pijlman, hep-ph/0505268

Drell-Yan

q

q

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kT -factorization is broken

My personal opinion: what is shown by Collins and Qiu is not sufficient to make such a statement. Maybe new concepts (i.e. “advanced” factorization and “generalized” universality) can work. Bomhof, Mulders, Pijlman, PLB596

A.B., Bomhof, Mulders, Pijlman, PRD72

Is kT -factorization broken ?

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An example

jet atRHICp p X

A.B., D’Alesio, Bomhof, Mulders, Murgia, hep-ph/0703153

“Standard” factorization & universality

“Generalized” factorization & universality

Broken factorization?

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Conclusions

• Transverse spin and transverse momentum observables uncover a new dimension of the structure of the nucleon

• Intense progress is taking place both from the theoretical and experimental point of view

• Impact also on hadron colliders and unpolarized physics