Surprises: Transverse Single Spin Asymmetries
(unintegrated polarized parton distributions)
 Single Spin Asymmetries (SSA) in elastic processes
 Single Spin Asymmetries in pQCD and in data
 Parton intrinsic motion and spin
 Spin - Transverse Momentum Dependent distribution and
fragmentation functions
 Phenomenology of SSA in inclusive processes
 Mysteries: spin asymmetries in pp → pp processes
Fisica dello Spin
27 Settembre 2005, Otranto
1
Transverse single spin asymmetries in elastic scattering
S
p'
θ
p
y
x
PT
z
–p
– p'


d   d    
AN 
 S  p  PT  sin θ


d  d
Example:
pp  pp
5 independent helicity amplitudes

AN  Im  5 (1   2   3   4 )
Fisica dello Spin

27 Settembre 2005, Otranto
M   ;    1
M  ;     2
M  ;     3
M ;    4
M ;    5
2
y
ΦS
S
Φ
x
p'
θ
p
PT
–p
z
– p'
for a generic configuration:


d   d    
AN 
 S  p  PT  PT sin( S  )


d  d
Fisica dello Spin
27 Settembre 2005, Otranto
3
Single spin asymmetries at partonic level. Example:
AN  0
qq '  qq '
needs helicity flip + relative phase
–
+
Im
+
+
+
+

x
+
+
QED and QCD interactions conserve helicity, up to corrections
AN 
mq
E
s
O ( mq / E )
at quark level
but large SSA observed at hadron level!
Fisica dello Spin
27 Settembre 2005, Otranto
4
Helicity conservation in pQCD or QED
q ' ,  'q
q, q ' four - momenta
q ,  'q helicities
q, q
uq (q)  ... u 'q   q  'q
odd numbers of
gamma matrices
Fisica dello Spin
 mq 
 O   q ,  'q
 E 
27 Settembre 2005, Otranto
5
BNL-AGS √s = 6.6 GeV
0.6 < pT < 1.2
p p   X
E704 √s = 20 GeV
0.7 < pT < 2.0
observed transverse Single
Spin Asymmetries
E704 √s = 20 GeV
0.7 < pT < 2.0
d   d 
AN 
d   d 
p p   X
experimental
data on SSA
Fisica dello Spin
27 Settembre 2005, Otranto
6

S
PT
p
X
y
– PT
x
–p
z




d ( PT )  d ( PT )








d ( PT )  d ( PT )  d ( PT )  d ( PT )

AN= simple left-right asymmetry
Fisica dello Spin
27 Settembre 2005, Otranto
7
STAR-RHIC √s = 200 GeV
1.1 < pT < 2.5
AN stays at high
energies ….
Fisica dello Spin
27 Settembre 2005, Otranto
8
l N  l  X
“Sivers moment”
d   d 
AN 
d   d 
sin(   S )
2 sin(    S )  AUT
2
Fisica dello Spin


d

d

(
d


d

) sin(    S )
S

27 Settembre 2005, Otranto


d

d

(
d


d

)
S

9
l N  l  X
“Collins moment”
d   d 
AN 
d   d 
sin(   S )
2 sin(    S )  AUT
2
Fisica dello Spin


d

d

(
d


d

) sin(    S )
S

27 Settembre 2005, Otranto


d

d

(
d


d

)
S

10
Transverse Λ polarization in unpolarized p-Be scattering at Fermilab
p N   X
Fisica dello Spin
27 Settembre 2005, Otranto
11


p p p p
Fisica dello Spin

p p pp
27 Settembre 2005, Otranto
12
Spin is challenging .......
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
Fisica dello Spin
27 Settembre 2005, Otranto
13
Transverse single spin asymmetries in SIDIS
y
ΦS



Φπ
x
S
PT
p
z
X


  
A N  S  p  PT  PT sin(   S )
 *  p c.m. frame
if partons inside p are all collinear there cannot be (at LO) any PT
needs k┴ dependent quark distribution in p↑ (Sivers mechanism) or
p┴ dependent fragmentation of polarized quark (Collins mechanism)
Fisica dello Spin
27 Settembre 2005, Otranto
14

f a ,s / p ,S ( x, k , Q 2 )
k┴ dependent parton distributions (TMD)
There must be a primordial intrinsic k┴ due to quark confinement:
x  1 fm  p  0.2 GeV/c
±1
There is intrinsic k┴ generated by QCD evolution
k┴
±
±
d lplhX   f q ( x, k  , Q 2 )  dˆ lqlq  Dqh ( z , p , Q 2 )
q
The elementary interaction depends on
 k (cos  , sin  ,0)
Azimuthal dependence in unpolarized SIDIS cross section
(Cahn effect)
Fisica dello Spin
27 Settembre 2005, Otranto
15
EMC data
Fisica dello Spin
27 Settembre 2005, Otranto
16
k   0.8 GeV
no k 
0
F. Murgia, U. D’Alesio
BNL data, PLB 73 (1978)
p p  0 X
s  20 GeV
original idea from Feynman-Field
Fisica dello Spin
c
X
f
a
ˆ
b
D
f
X
non collinear configurations
27 Settembre 2005, Otranto
17
Brodsky, Hwang, Schmidt model for Sivers function


S

p
X
q
+
diquark, Q
q
–
diquark, Q
 * p  q Q cannot be forward in order to have a SSA
intrinsic k┴ of quark
Fisica dello Spin
27 Settembre 2005, Otranto
18
Sivers mechanism in SIDIS
q
φ
k┴
S
p

f q / p  ( x, k  )  f q / p ( x, k  ) 

1 N
 f q / p  ( x, k ) S  ( pˆ  kˆ )
2
p┴ = PT – z k┴ + O(k┴2/Q2)
Fisica dello Spin
27 Settembre 2005, Otranto
19
sin(   S )
UT
A
from Sivers mechanism
M.A., U.D’Alesio, M.Boglione, A.Kotzinian, A Prokudin
Fisica dello Spin
27 Settembre 2005, Otranto
20
Deuteron target
Fisica dello Spin


sin( h   S )
AUT
 N f u / p   N f d / p  4 Duh  Ddh
27 Settembre 2005, Otranto

21
HERMES + E704: Sivers function is not zero
Spin-k┴ partonic correlations in nucleons
Theory: not quite universal
f1Tq
SIDIS
  f1Tq
D Y
J. Collins, a “QCD theorem”
Models: few, in fair agreement
u
1T
d
1T
f
chiral models
f1Tq   q
M. Burkardt
f
Fisica dello Spin
27 Settembre 2005, Otranto
22
Collins mechanism for SSA

Asymmetry in the fragmentation of a
transversely polarized quark
φ S
q
p┴
pq
(Fundamental QCD property? D. Sivers)
q’

Dh / q  ( z , p )  Dh / q ( z , p ) 
q

1 N
 Dh / q  ( z , p ) S q  ( pˆ q  pˆ  )
2
y
initial q spin is transferred to
final q', which fragments

S q '  ( pˆ q '  pˆ  )  sin(  h   S )
Sq
Sq’
p┴
Φh
ΦS
x
Fisica dello Spin
27 Settembre 2005, Otranto
l  l'
23
neglecting intrinsic motion in partonic distributions:
d   d 
Collins function
A 



d  d
2
2
N
e
h
(
x
)
(
1

y
)
/(
xy
)

Dh / q  ( z , p )
q  q 1q
sin(  h   S )
2
2
2
q  eq f q / p ( x) [1  (1 - y) ] /( xy ) Dq / p ( z, p )
h
N
sin( h   S )
AUT
2


d

d

[d


d

] sin(  h   S )
 h S


d

d

[d


d

]
 h S
some data available from HERMES, first extraction of Collins functions:
W. Vogelsang and F. Yuan (assuming Soffer-saturated h1)
Fisica dello Spin
27 Settembre 2005, Otranto
24
fit to HERMES data on
Fisica dello Spin
sin( h   S )
AUT
27 Settembre 2005, Otranto
25
spin-k┴ correlations
q
φ
k┴

S
p
φ S
q
p┴
pq
Sivers function
Collins function

f q / p  ( x, k  )  f q / p ( x, k  ) 

Dh / q  ( z , p )  Dh / q ( z , p ) 

1 N
 f q / p  ( x, k ) S  ( pˆ  kˆ )
2

1 N
 Dh / q  ( z , p ) S q  ( pˆ q  pˆ  )
2
Amsterdam group notations
 fq/ p
N
2k   q

f1T
M
Fisica dello Spin
 Dh / q 
N
27 Settembre 2005, Otranto
p
2
H1 q
z Mh
26
spin-k┴ correlations
q

φ S
q
k┴
φ
p┴
p
SΛ
pq
Boer-Mulders function
polarizing f.f.

1
f q  / p ( x, k  )  f q / p ( x, k  ) 
2

1 N
 f q  / p ( x, k  ) S q  ( pˆ  kˆ )
2

1
D / q ( z , p )  Dh / q ( z , p ) 
2

1 N
 D / q ( z , p ) S   ( pˆ q  pˆ  )
2
Amsterdam group notations
 f q / p
N
Fisica dello Spin
k q
  h1
M
 D / q
N
27 Settembre 2005, Otranto
p
2
D1Tq
z M
27