The transverse structure of the nucleon (resolving the quark motion inside a nucleon)

The transverse structure of the nucleon

(resolving the quark motion inside a nucleon)

Mauro Anselmino, Torino University and INFN, JLab, December 15, 2006

 the need for parton intrinsic motion

 transverse momentum dependent distribution

and fragmentation functions (TMD)

spin and k_┴: transverse Single Spin Asymmetries SSA in SIDIS

Spin effects in unpolarized e⁺e^– → h₁h₂ X at BELLE what do we learn from TMD’s?

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   



x p

uncertainty principle

gluon radiation

±1

± ±

k_┴

q_T distribution of lepton pairs in D-Y processes

Partonic intrinsic motion

p p

Q² = M²

q_T q_L

l⁺

l^–

 *

p_T distribution of hadrons in SIDIS

Hadron distribution in jets in e⁺e^– processes

Large p_T particle production in

pN  hX

Transverse motion is usually integrated, but there might be important

spin-k_┴ correlations

p xp

k_┴

frame c.m.

* p  hX



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

) ,

ˆ ( d )

, (

d f x Q

^{2 lq lq}

D

_q^h

z Q

²

q

q lhX

lp^

   ^

^





Intrinsic motion in unpolarized SIDIS, ^[ O ⁽ 

s⁰

^)]

q p x Q

  2

2

2

2 q

Q  

p l

q y p



  in collinear parton model

2 2

2

ˆ 1 ( 1 )

ˆ

d  ˆ

^lqlq

 s  u    y

thus, no dependence on azimuthal angle Ф_h at zero-th order in pQCD

the experimental data reveal that

h h

h X

lh

lq

/ d Φ A B cos Φ C cos 2 Φ

d  ˆ

^{ }

  

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

These modulations of the cross section with azimuthal angle are

assuming collinear fragmentation, φ = Φ_h

) 0 , sin

, cos

(

_



_





 k k

k

 

 



 

 

 



  

 2

^

1 cos

^2₂

ˆ 1

Q y k

Q sx k

s  O

 

 



 

 





 





 





^

cos

^2₂

1 1 2

) 1

ˆ (

Q k y

Q y k

x s

u  O

h h

h lhX lq

Φ C

Φ B

A u

Φ s ˆ ˆ cos cos 2 d

d  ˆ

^



₂



₂

  

SIDIS with intrinsic k

_┴

kinematics

according to Trento conventions (2004)

)

; ,

( )

; ,

ˆ ( d )

; ,

(

d ^

^lplhX

 

_q

f

_q

x k

_

Q

²

 ^

^lqlq

y ^k

_

Q

²

 D

_q^h

z p

_

Q

²

factorization holds at large Q², and

P

_T

 k

_

 

_QCD Ji, Ma, Yuan

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

k



k

z

p



P

T

p

h

 ^k

²

^{/ Q}

²



neglecting terms of order one has







 p k

P

_T

z

assuming:

one finds:

with

clear dependence on and (assumed to be constant)

Find best values by fitting data on Φ_h and P_T dependences

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

Large P_T 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

S p

p '

– p

P_T

θ

  ^{sin θ}

d d

d

d    





_^



^_{ T}

A

N

S p P









– p '

5

;

4

;

3

;

2

;

1

;





























































M M M M M

5 independent helicity amplitudes

 ^{       }

^



 Im

₅

(

_{1 2 3 4}

)

AN

pp pp 

z y

x

Example:

Transverse single spin asymmetries: elastic scattering

 0

A

N needs helicity flip + relative phase + –

+ +

Im

x

+

+ + +

s q

N

E

A  m 

at quark level but large SSA observed at hadron level!



QED and QCD interactions conserve helicity, up to corrections

O ( m

_q

/ E )

Single spin asymmetries at partonic level. Example:

qq '  qq '

BNL-AGS √s = 6.6 GeV 0.6 < p_T < 1.2

E704 √s = 20 GeV 0.7 < p_T < 2.0

E704 √s = 20 GeV 0.7 < p_T < 2.0

experimental data on SSA

X p

p

^

 

X p

p

^

 

observed transverse Single Spin Asymmetries











 









d d

d d

A

N

and A_N stays at high energies ….

STAR-RHIC √s = 200 GeV 1.2 < p_T < 2.8

) ˆ (

d )

( )

(

d

_{/ /}

, , , , ,

/

x f x D z

f

_{a b p b} ^{ab cd} _c

g q q d c b a

p

a





   

^







PDF FF

pQCD elementary interactions a

b c



0

D

X

X

 ^ˆ

f

f

(collinear configurations)

X pp  

⁰

factorization theorem

p

p

X p

p  

⁰

RHIC data s  200 GeV

X p

p  

Transverse Λ polarization in unpolarized p-Be scattering at Fermilab

X N

p  

^{  }

















 









d d

d

P d

p p p

p

^

 p

^

p

^

 p p











 









d d

d d

AN











 









d d

d d

ANN

“Sivers moment”

X l

N

l

^

 











 









d d

d A d

 















 





) d d

( d d

) sin(

) d d

( d 2 d

) sin(

2 ^{sin( )}









S

S S

Φ Φ UT S

Φ Φ

Φ Φ

Φ Φ

A Φ

Φ ^S

“Collins moment”











 









d d

d A d

X l

N

l

^

 

 















 





) d d

( d d

) sin(

) d d

( d 2 d

) sin(

2 ^{sin( )}









S

S S

Φ Φ UT S

Φ Φ

Φ Φ

Φ Φ

A Φ

Φ ^S

and now ….?

Polarization data has often been the graveyard of fashionable theories.

If theorists had their way, they might

just ban such measurements altogether out of self-protection.

J.D. Bjorken

St. Croix, 1987

  ^{P sin (Φ Φ ) * p c.m. frame}

A  S  p  P

_T



_{T π}



_S

 

needs k_┴ dependent quark distribution in p^↑ and/or p_┴ dependent fragmentation of polarized quark

z y

Φ

_S

Φ

_π x



 X



p

S

P_T

Transverse single spin asymmetries in SIDIS

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

spin-k

_┴

correlations? orbiting quarks?

Transverse Momentum Dependent distribution functions Space dependent distribution functions (GPD)

) ,

( x k

_

q

) , ( b x q

k 



?

b 

q p T

q

N

f

M

f   k

^{ }



_{ 1}

/

2

p

k_┴φ S q

p

_q



φ _S

p_┴ q

ˆ ) ( ˆ

) ,

2 ( 1

) ,

( )

, (

/ / /























p p

S p

q q q

h N

q q h

h

p z D

p z D

z D

ˆ ) ( ˆ

) ,

2 ( 1

) , ( )

, (

/ / /























k p S

k

k x f

k x f

x f

p q N

p p q

q

Amsterdam group notations

q q h

h

N H

M z

D  p^{ }

 _{ 1}

/ 2

spin-k

_┴

correlations

Sivers function Collins function

q p

q

N

h

M f   k

^{ }



_{ 1}

/

p

S_q k_┴φ

q

p

_q



φ _S p_┴ Λ

ˆ ) ( ˆ

) ,

2 ( 1

) ,

2 ( ) 1 ,

(

/ / /





 



 















p p

S p

q q N

q q h

p z D

p z D

z D

ˆ ) ( ˆ

) ,

2 ( 1

) , 2 (

) 1 ,

(

/ / /























k p S

k

p q q N

p p q

q

k x f

k x f

x f

Amsterdam group notations

q q T

N D

M z

D p ^





 

 _{ 1}

/ 2

spin-k

_┴

correlations

Boer-Mulders function polarizing F.F.

8 leading-twist spin-k

_┴

dependent distribution functions

Sivers function

dependence on S

_T

in cross section

The Sivers mechanism for SSA in SIDIS processes

) ,

ˆ ( ) ,

2 ( 1

/ 2 2

6 





 



 D z p

z J z dQ

x d f

d

d

_q^h

h lq

lq p

q q



  k k

Sivers angle

SSA:

after integration over k

_┴

one obtains:

) sin(

_{h S}

A    

) 1

,

( _   _ 

h

T z

J z zk

P p

 











 

] [

d

sin ] [

2 d

sin









d d

Φ

Φ d

d A

^Φ

Φ

data are presented for the sinΦ moment of the analyzing power





 P  k

p

_T

z

Parameterization of the Sivers function

1 1  

 N

_q

q = u, d.

The Sivers function for sea quarks and antiquarks is assumed to be zero.

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

from Sivers mechanism

) sin( _S

A

UT ^

Deuteron target

  ^

d^h

^

h

p u d N p

u N

UT f f D D

A ^{h S}       





 ⁾ _{/ /} 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 (

 

















 ^{d k} _m ^{f x k}

f f

p q N p

q T q

N

k

hep-ph/0511017

1

?

1

d T u

T

f

f

^

 

^

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.

) , ( )

(

² ₁

) 2 / 1 (

1   

 



 f x k

M d k

x

f

_T ^q

k

_T^q

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

^{2 1}

(

2 2

) 1 (

1   

 



 f x k

M d k

f

_T ^q

k

_T^q

predictions for JLab, proton target, 6 GeV

0.4 ≤ z_h ≤ 0.7 0.02 ≤ P_T ≤ 1 GeV/c 0.1 ≤ x_B ≤ 0.6

0.4 ≤ y ≤ 0.85 Q² ≥ 1 (GeV/c)² W² ≥ 4 GeV² 1 ≤ E_h ≤ 4 GeV

predictions for JLab, proton target, 12 GeV

0.4 ≤ z_h ≤ 0.7 0.02 ≤ P_T ≤ 1.4 GeV/c 0.05 ≤ x_B ≤ 0.7

0.2 ≤ y ≤ 0.85 Q² ≥ 1 (GeV/c)² W² ≥ 4 GeV² 1 ≤ E_h ≤ 7 GeV

ˆ ) ) (

, ( )

, (

ˆ ) ( ˆ

) ,

2 ( ) 1

, ( )

, (

1 /

/ / /

k M x

f k

x f

k x f

k x f

x f

a T p

a

p a N p

p a a

 

 











 













_



k p

S

k p

S k



 



 _   ^ _T^a

p a

N f

M

f k ₁

/

2

S

 k



pˆ k



number density of partons with longitudinal momentum fraction x and

transverse momentum k_┴, inside a proton with spin S

0 ) ,

/

(

2



 

^a

^{dx d k}

^

^k

^

^f

^{a p}

^{x k}

^

M. Burkardt, PR D69, 091501 (2004)

What do we learn from the Sivers distribution?

  ^{ˆ ( , )}

2 cos ˆ

sin ˆ

ˆ ) ( ˆ

) ,

ˆ ( 2

) 1 ,

ˆ (

/ 2

/ / 2

































 

     







k x f

k dk dx

k x f

k x f

d dx

p a N S

S

p a N p

a a

j  i

k p

S k

k k

Then we can compute the total amount of intrinsic momentum carried by partons of flavour a

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

 ^{i j} 

S  cos 

_S

ˆ  sin 

_S

ˆ

S

a

) k sin(

ˆ )

( ˆ     

 p k

_{ S}

S

 

 ^{sin ˆ cos ˆ}  ^GeV/c

13

. 0

GeV/c

cos ˆ sin ˆ

14

. 0

03 . 0

02 . 0 0.05 0.06 -

j i

k

j i

k

S S

d

S S

u



























 

u

k



d

k



?

 0

 _

 u k d

k

Sivers functions extracted from A_N data in

X p

p  give also opposite results, with

036 .

0

032 .

0  



_

u

k

d

k

Numerical estimates from SIDIS data

U. D’Alesio

Collins mechanism for SSA

q

q’ initial quark transverse spin is transmitted to the final

quark





 q l

q

l 1 ( 1 )

2

) 1

( 2

y y d

d

d d

q q

q q

q q

q q





 





































(in collinear configuration, here ↑,↓ means perpendicular to the leptonic plane)

 ^ˆ



 d

ˆ ) ( ˆ

) ,

2 ( ) 1 ,

( )

,

(

/ /

/ _ 





 

_  S_q



p_q



p

q h N q

q h

h z p D z p D z p

D



C h

S q

q

 (

p

ˆ 

p

ˆ

_

)  sin(

Φ



Φ

)  sin

Φ S

The polarization of the fragmenting quark q' can be computed in QED

depolarization factor

 ^{    }





^



q

N

T

q d D

d

d    ^ˆ

x y

 

^C

Φ

Φ

S

Φ

_h

S

q

S

q

l l  

S h

C

Φ Φ

Φ     sin Φ

_C

 sin( Φ

_h

 Φ

_S

)

y q y

q

x q x

q

y S S y

y S S y

) ) (

1 ( 1

) 1

( ) 2

(

) ) (

1 ( 1

) 1

( ) 2

(

2 2





 







 





 















 

] [

) sin(

] d

2 [d

) sin(









d d

dΦ dΦ

Φ Φ

dΦ A dΦ

S h

S h

S Φ h

Φ UT ^{h S}

Collins effect in SIDIS

)

, ( )

, ˆ (

) sin(

) ,

ˆ (

) ,

(

/ 2 /

2

2 / 1

2 )

sin(

 

 





















 





q q p q p

lq lq S

h

q h q h S

lq N lq q

S h

Φ Φ UT

p z D

k x dQ f

d d dΦ dΦ

Φ Φ

z dQ D

k d x h

d dΦ dΦ

A

^{h S}



 k

p k

:

1 or q

h _q _T transversity distribution

ˆ ) ( ˆ

) , ( )

,

( /

/       

 _ p _ S_q p_q p

q h N q

h

ND z D z p

fit to HERMES data on ^{sin(Φh ΦS )}

A

UT ^

 ^{2 | |} 

h

₁

  q  q

assuming

W. Vogelsang and F. Yuan Phys. Rev. D72, 054028 (2005)

A. V. Efremov, K. Goeke and P. Schweitzer

(h₁ from quark-soliton model)

Collins function from e

⁺

e

^–

processes

(spin effects without polarization)

e⁺e^- CMS frame:

2 , 10 . 52 GeV



 s

s z E

^h

X h h q

q e

e

^{ }

 

^{ }

BELLE @ KEK

e⁺



_

1

P 

h 2

P 

h



_



^e-

e⁺

thrust-axis

z y

x

1

p

2

p 

single quark or antiquark are not polarized, but there is a strong correlation between their spins

 









² _{2 2}

4 sin 3 cos

ˆ cos

ˆ

e

q

s d

d d

d

^



^



) ,

( )

, cos (

ˆ

cos

_, ^{/ 1 1 / 2 2}

2 2 1 2 2

1 ^{1 2}

2 1





















 p p

p

p D z D z

d d d

d d

dz dz

d

q h q

spins h q

spins X

h h e e









cross section for detecting the final hadrons inside the jets

contains the product of two Collins functions

 

^{   }

 





q q h q h q

q h q

N q

h N q

z D

z D

e

z D

z D

e

d dz dz

d d d

dz dz

d

A ( ) ( )

) ( )

) ( cos (cos

1 sin 8

1 1

cos 2

1

cos

2 /

1 / 2

/ 2 / 1

2 2 0

2

2 1

0 2

1 1

2 1

2



1



















^{  }



 



^N

D ( z ) d

²

p

^N

D ( z , p ), etc. 

₀

 

₁

 

₂

Cosine modulations clearly visible

0 2

1 0

0

) cos

(  

P N P

N

 

2

1 

 

0 0) ( N N 

M. Grosse Perdekamp, A. Ogawa, R. Seidl

Talk at SPIN2006

BELLE data

P1

) , ( ) cos(

1 _{1 2 1 1 2}

1

1 P z z

A A

L

U  







U = unlike charged pions

L = like charged pions





















 

















 

) ( )

( 2

) ( )

( 5

) ( )

( 5

) ( ) ( 7 ) ( ) ( 5

) ( )

( 7

) ( )

( 5

cos 1

sin 8

) 1 , (

2 1

2 1

2 1

2 1

2 1

2 1

2 1

2 2 2

1 1

z D z

D z

D z

D z

D z

D

z D z D z

D z D

z D z

D z

D z

z D z P

U N U

N F

N U

N U

N F

N

U U

F F

U N U

N F

N F

N





etc.

,

, _/

/ U

N u

N F

N u

ND   D  D   D

  

fit of HERMES data on

A

_UT^{sin(hS)}

Collins functions and transversity distributions from a global best

fit of HERMES, COMPASS and BELLE data

M.A., M. Boglione, U.D’Alesio, A.Kotzinian, F. Murgia, A Prokudin, C. Türk, in preparation

fit of COMPASS data on

A A

_UTUT^{sin(sin(hhSS))}

fit of BELLE data on

P

₁

( z

₁

, z

₂

)

Extracted favoured and unfavoured Collins functions

positivity bound Efremov et al.

Vogelsang, Yuan

Extracted transversity distributions

Soffer bound

Predictions for Collins asymmetry

at JLab, proton target, 6 GeV

Predictions for Collins asymmetry

at JLab, proton target, 12 GeV

Sivers function and orbital angular momentum

Sivers mechanism originates from

S  L

_q

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 /

q y p

q

N

L

k x f

dx  



^{ }

Open issues and wild ideas …

Sivers function and proton anomalous magnetic momentum

M. 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 /

2

p q q

N

f x k C

d

dx   

 ^k

^{  }

in qualitative agreement with large z data:

d u Φ

Φ UT

Φ Φ

UT

S S

A A







 _









) sin(

) sin(

Conclusions

Towards the transverse spin and momentum structure of the nucleon via azimuthal and P

_hT

dependence in SIDIS

Information on quark intrinsic motion Spin-k

_┴

correlation from Sivers function

Orbiting quarks?

Extracting Collins function and accessing transversity

Much more data needed …