transversity and transverse-momentum-dependent partonic functions

Transversity andTransverse-Momentum-

Dependent Partonic Functions

Alessandro Bacchetta

Alessandro Bacchetta - Transverse-momentum-dependent functions

2/58

Outline

• Theoretical framework

• Transversity

• Sivers function

• Boer-Mulders function

Theoretical framework


4/58

Semi-inclusive Deep Inelastic Scattering

proton

pionlepton

l p l X2 2 virtuality ( ') of photonk k Q

k

'k

transverse momentum of pionhP

hP


5/58

Feynman diagrams & Factorization

proton

lepton lepton

pion

SIDIS

Partonic scattering amplitude

Fragmentation amplitude

Distribution amplitude


6/58


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


7/58


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


8/58

Factorization proofsFactorization proofs for involving TMD functions LEADING TWIST (1/Q0), NOT for pp to pions

– Collins, Soper, NPB 193 (81)– Ji, Ma, Yuan, PRD 71 (04)– Ji, Ma, Yuan, PLB 597 (04)– Collins, Metz, PRL 93 (04)

QCD hP Q

See talk by A. Metz


9/58

Nonperturbative elements

2

=

4

4 [0, ]( , , ) , (0) ( ) ,(2 )

i kij j i

dk P S e P S P S

U

,P S

k


10/58

Origin of the gauge link

P

k ......


11/58

Different processes

SIDIS

q

q

Drell-Yan

q

q

q

q

pp to hadrons

Bacchetta, Bomhof, Mulders, Pijlman, hep-ph/0505268

See talk by C. Bomhof


12/58

[0, ] , ]

2

[

( ) ( , , )

, (0) ( ) ,2

ij T ij k x P

i Pj a

xi

x dk d k k P S

de P S a P S

U U

The correlation functionintegrated over kT

P

( ,0 ,0 )a

k

axis 0 a

SIDIS


13/58

[0, ] [ , ]

2( ) ( , , )

, (0) ( ) ,2

ij T ij k x P

ia

x Pj i

x dk d k k P S

de P S a P S

U U

The correlation functionintegrated over kT

P

( ,0 ,0 )a

k

axis0 a

Drell-Yan


14/58

[0,

2

3 ] [ , ]

( , ) ( , , )

(0) ( )(2 )

ij T ij k x P

i kTj a i

x k dk k P S

d de P a P

U U

The correlation functionUNintegrated over kT

( ,0 , )Ta

( )

[0, ] [0, ] [ ,

2

3 ]

( , ) ( , , )

(0) ( )(2 )

ij T ij k x P

i kT Taj ia

x k dk k P S

d de P a P

U U U

P

kaxis

0a

Belitsky, Ji, Yuan, NPB656 (03)

SIDIS


15/58

The correlation functionUNintegrated over kT

( ,0 , )Ta

( )

[0, ] [0, ] [ , ]

2

3

( , ) ( , , )

(0) ( )(2 )

ij T ij k x P

i Ta a

kTj i

x k dk k P S

d de P a P

U U U

P

k

0a

axis

Ji, Yuan, PLB 543 (02); Belitsky, Ji, Yuan, NPB656 (03)

Drell-Yan


16/58

Practical consequences

• Some TMD functions will not be sensitive to the differences in the structure of the gauge link (i.e. they will be connected to standard partonic cross-sections)

• Some TMD functions will be sensitive to it and will be multiplied by prefactors to be computed for each partonic subprocess (i.e. they will be connected to gluonic-pole cross-sections)

See talk by C. Bomhof


17/58

Decomposition of the correlation function integrated over kT

1 5151( ) ( )1

( )2

() Lqq q

Thx f x g x S x S

Leading twist only

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 )Q


18/58

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)


19/58

Parton distribution functions with transverse spin

-Sivers

-Boer-Mulders

-Transversity

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

Transversity


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Definition of transversity distribution function

Probability to find a quark with momentum xP+ and transverse spin sq in a proton with transverse spin S

11

1( ) ( )

2( )q q

qq pf x f x h S sx

-

NOTE: transverse momentum is integrated over!

25(1

1( ) T ( , )

4)r ,Tq p

kq

x P

f x dk d k k Ps S

Notation of

Anselmino et al.


22/58

Transversity: birth and growth

Citations

050100150200250300350400

1980-1985

1986-1990

1991-1995

1996-2000

2001-2005

Ralston, Soper, NPB 152 (79)


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Helicity and transversity

1qh q

1qg q


24/58

Transversity vs helicity• Different evolution

h1@ 0.079 GeV2

g1h1@ 0.75 GeV2

g1

S. Scopetta, V. Vento,PLB 424 (1997)


25/58

Transversity vs helicity• Different evolution• Different integrals (axial and tensor charge of the

nucleon). E.g. 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)


26/58

Transversity vs helicity• Different evolution• Different integrals (axial and tensor charge of the

nucleon). E.g. from lattice QCD:

• Different sum-rules Bakker, Leader, Trueman, PRD 70 (04)

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)

,1 1

2 2q g

q

q G L ,1 1

2 2q gT

q

q L


27/58

Comparison of tensor charges

Barone, Drago, Ratcliffe, PR 359 (2002)

lattice

quark soliton 1

spectator

bag

quark soliton 2

quark

non-relativistic


28/58

First data: Collins asymmetry

1

1 1

1

sin( )sin( ) h S

h S UT

H

f D

h

hadron plane

lepton plane

S

h

Airapetian et al. [HERMES], PRL 94 (05)

Alexakhin et al. [COMPASS], PRL 94 (05)

See talk by G. Schnell, A. Martin

hP

'kk


29/58

First attempt to combine BELLE and HERMES dataEfremov, Goeke, Schweitzer, PRD 73 (06)

1. Get Collins function from BELLE

Talk by R. Seidl

Talks by P. Schweitzer and M. Boglione


30/58

First attempt to combine BELLE and HERMES data

2. Check size of transversity at HERMES

See talk by P. Schweitzer


31/58

Where to observe itProcess Experiment Observable Grade

Hermes, Compass, Clas, EIC

Hermes, Compass, EIC

Compass, EIC

Rhic, Pax, JPARC

Panda Compass

Rhic

l p l X

p p l l X

p l l X

p p l l X

l p l X

l p l X

1 1

1

h H

h H

1 1h H

1 1h H

1 1h h

1 1

(1)1 1

h h

h h

p p X

p p X

1 1 1

1 1 1

f h H

f h H

Schnell,Martin, Avakian,

Elschenbroich

Martin, Radici, Giordano

Heppelman, Dalpiaz

Talks by

Koch, Fischer

Heppelman, Aidala

The Sivers function


33/58

Introducing the Sivers function

1( , ) Tr ( , , )

2Tq pk x P

f x k dk k P S

Probability to find a quark with momentum xP+ and kT in a proton

with transverse spin S

21

21

ˆ( )( , ) ( , )( , ) q

Tq T

T Tq Tp

P k Sf x k f x kf x k

M

see e.g. Bacchetta, D’Alesio, Diehl, Miller, PRD 70 (04)

2Tk

M

-

T-odd


34/58

Sivers function: birth and growth

Citations

0

50

100

150

200

1990-1993 1994-1997 1998-2001 2001-2005

D. Sivers, PRD41 (90)


35/58

Two ingredients

• Final-state interactions (included in the gauge link) Ji, Yuan, PLB 543 (02); Belitsky, Ji, Yuan, NPB656 (03)

• Transverse-spin dependent distribution of quarks in transverse spaceBurkardt, PRD 66 (02); Diehl, Hägler, EPJ C44 (05)


36/58

down

up

Final-state interactions

proton

quarks

Side view Front view


37/58

down

up


photon


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


38/58

down

up


photon

Burkardt, PRD 66 (02)

Chromodynamic lensing


See talk by M. Burkardt


39/58

down

up

Change of sign in Drell-Yan

proton

photon

quarks


antiquark

Clear-cut prediction of QCD

Collins, PLB 536 (02)


40/58

down

up

Distortions in transverse space

proton

quarks



41/58

down

up

Distortions in transverse space

proton

quarks


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)


42/58

Relation to GPDs

• The transverse-space deformation of unpolarized quarks in a transversely polarized nucleon is described by the generalized parton distribution function E

22

1 22(

1( , ) ( , )

(2 ),0, ) T T

q Tibq T

T Tq py

df x b f x b e

ME x

b

anomalous( ,0 mag,0 netic moment) qqE kxdx Work of Burkardt


43/58

Lattice-QCD studies

Preliminary results by QCDSF Collab. (see e.g. hep-ph/05110032)

up quarks

Sivers function for up quarks expected to be NEGATIVE

See talk by P. Hägler


44/58

Model calculations

spectator

proton

1 2 4 5F Se e C

1e

2e

x

(1 2)1Txf

up

down

Bacchetta, Schäfer, Yang, PLB 578 (04)

Brodsky, Hwang, Schmidt PLB530 (02)

Gamberg, Goldstein, Oganessyan, PRD 67 (03)

See talk by L. Gamberg


45/58

Fits to HERMES and COMPASS data

Airapetian et al. [HERMES], PRL 94 (05) Anselmino et al., PRD72 (05)


46/58

Comparison with COMPASS data

Alexakhin et al. [COMPASS], PRL94 (05), Anselmino et al., PRD72 (05)

deut

eron


47/58

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


48/58

New preliminary kaon data

See talk by U. Elschenbroich


49/58

Future experimental data• HERMES@DESY: 0 and weighted asymmetries

(this year?)• COMPASS@CERN: proton data (this year?)• CLAS@Jlab• STAR,PHENIX@RHIC: jet-production in pp collisions

• EIC (DIS with polarized protons)• FAIR@GSI (Drell-Yan with polarized protons)

See talk by U. D’Alesio

The Boer-Mulders function


51/58

Definition of the Boer-Mulders function

5,

1( , ) Tr ( , , 0)

4(1 )

T

q pk x P

T qk

k sf x dk k P S

Probability to find a quark with momentum xP+ and kT and spin sq

12

12 ( ,

ˆ( )1( , ) ( , ) )

2T qq

T Tq

q

p Th x kP k s

f x k f x kM

T-odd

-Tk

M


52/58

Boer-Mulders function: birth and growth

Citations

01020304050607080

1998-1999 2000-2001 2002-2003 2004-2005

Boer, Mulders, PRD57 (98)


53/58

down

up

Relation to transverse space

proton

quarks



54/58

down

up

Relation to transverse space

proton

quarks


A distortion in the distribution of quark spin in transverse space can give rise to a Boer-Mulders function Burkardt, hep-ph/0510408

Quark spin can be unevenly distributed in transverse space (orbital angular momentum of quarks is required)


55/58

Lattice-QCD studies

Preliminary results by QCDSF Collab. (see e.g. hep-ph/05110032)

up quarks

Boer-Mulders function for up quarks expected to be NEGATIVE


56/58

Model calculations

x

(1 2)1xh

up

down

Bacchetta, Schäfer, Yang, PLB 578 (04)

See talk by L. Gamberg, G. Goldstein


57/58

Where to look for it

• Unpolarized semi-inclusive DIS (HERMES, COMPASS, H1, ZEUS, EIC, LHEC)

• Unpolarized Drell-Yan (GSI)• Jet-production in unpolarized pp collisions (RHIC,

LHC)


58/58

Conclusions

Many progresses from the theoretical, experimental, phenomenological sides…

...Pay attention to the rest of the workshop!


59/58

down

up

Semi-inclusive Deep Inelastic Scattering

proton

pion

lepton

l p l X

up down

up down

N N

N N

transversity and transverse-momentum-dependent partonic functions

Documents

transverse spin photon

ktx axis0axtbelitsky

ktx axis0adrell

partonic subprocess

kt leading twist

gluonicpole crosssectionssee

screen proton