Transcript Document

The transverse spin
structure of the
nucleon
Mauro Anselmino, Torino
University and INFN,
Vancouver, July 31, 2007
The longitudinal structure of nucleons is “simple”
It has been studied for almost 40 years
xP
P
Q2


essentially x and Q2 degrees of freedom ….
Very good or good knowledge of q(x,Q2), g(x,Q2) and Δq(x,Q2)
Poor knowledge of Δg(x,Q2)
1
  S q    S g    Lq    Lg  ?
2
(Research Plan for Spin Physics at RHIC)
Talk by L. De Nardo
The transverse structure is much more
interesting and less studied
spin-k┴
correlations?
orbiting quarks?

k

b
Transverse Momentum Dependent
distribution functions
2
q( x, k ; Q )
Space dependent
distribution functions
2
q( x, b;Q )
The mother of all functions
M. Diehl, Trento workshop, June 07
GPD’s
TMD’s
Wigner (Belitsky, Ji, Yuan)
function
TMDs in SIDIS
SSA in SIDIS: Sivers functions
Collins function from e+e- unpolarized processes
(Belle) and first extraction of transversity
SSA in hadronic processes
Future measurements and transversity
(Trento workshop on “Transverse momentum, spin, and
position distributions of partons in hadrons”, June 07)
Main source of information
on transverse nucleon
structure
SIDIS kinematics
according to
Trento
conventions
(2004)
Polarized SIDIS cross section, up to subleading order in 1/Q
0
1
d  d UU
 cos 2Φh d UU

1
1
2
3
cosΦh d UU
 e sin Φh d LU
Q
Q


 6
1
1
4
5
7 
 S L sin 2Φh d UL  sin Φh d UL  e d LL  cosΦh d LL  
Q
Q




8
9
10
 ST sin(Φh  ΦS ) d UT
 sin(Φh  ΦS ) d UT
 sin(3Φh  ΦS ) d UT

1
11
12
 sin(2Φh  ΦS ) d UT
 sinΦS d UT
Q



1
15
 e cos(Φh  ΦS ) d 13

cosΦS d 14
LT
LT  cos(2Φh  ΦS ) d LT
Q






Kotzinian, NP B441 (1995) 234
Mulders and Tangermann, NP B461 (1996) 197
Boer and Mulders, PR D57 (1998) 5780
Bacchetta et al., PL B595 (2004) 309
Bacchetta et al., JHEP 0702 (2007) 093
SIDIS in parton
model with
intrinsic k┴
factorization holds at large Q2, and
PT  k  QCD
Ji, Ma, Yuan
d lplhX  q f q ( x, k  ; Q 2 )  dˆ lqlq ( y, k  ; Q 2 )  Dqh ( z, p ; Q 2 )
Azimuthal dependence induced by quark intrinsic motion
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
Sivers function
1 N
f q / p , S ( x, k  )  f q / p ( x, k  )   f q / p  ( x, k  ) S  ( pˆ  kˆ )
2
k q
 f q / p ( x, k  ) 
f1T ( x, k  ) S  ( pˆ  kˆ )
M
Boer-Mulders function
1
1 N
f q , sq / p ( x, k  )  f q / p ( x, k  )   f q  / p ( x, k  ) sq  ( pˆ  kˆ )
2
2
1
1 k q
 f q / p ( x, k  ) 
h1 ( x, k  ) sq  ( pˆ  kˆ )
2
2M
8 leading-twist spin-k┴ dependent distribution functions
Courtesy of Aram Kotzinian
Collins function
1 N
Dh / q , sq ( z , p )  Dh / q ( z, p )   Dh / q  ( z , p ) sq  ( pˆ q  pˆ  )
2
p
 Dh / q ( z, p ) 
H1 q ( z, p ) sq  ( pˆ q  pˆ  )
z Mh
Polarizing fragmentation function
1
1 N
D , S  / q ( z , p )  Dh / q ( z, p )   D / q ( z , p ) S   ( pˆ q  pˆ  )
2
2
1
p
 Dh / q ( z , p ) 
D1Tq ( z, p ) S   ( pˆ q  pˆ  )
2
z M
sin( Φ ΦS )
2 sin(Φ  ΦS )  AUT
2


d
Φ
d
Φ
(
d


d

) sin(Φ  ΦS )
S



d
Φ
d
Φ
(
d


d

)
S

 q eq2 N f q / p  ( x, k  )  D / q ( z , p )
Large K+
asymmetry!
sin( Φ ΦS )
2 sin(Φ  ΦS )  AUT
2


d
Φ
d
Φ
(
d


d

) sin(Φ  ΦS )
S



d
Φ
d
Φ
(
d


d

)
S

 q eq2 h1q ( x, k  )  N Dh / q  ( z , p )
Talk by
G. Schnell
COMPASS measured Collins and Sivers asymmetries for
positive (●) and negative (○) hadrons
small values due
to deuteron
target:
cancellation
between u and d
contributions
talk by T. Iwata


sin(  h   S )
AUT
 N f u / p   N f d / p  4 Dh / u  Dh / d 

sin(  h   S )
AUT
 h1u  h1d  4N Dh / u   N Dh / d 

M. Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A. Prokudin
A fit of HERMES + COMPASS pion data,
information on u and d Sivers funtions
sea contribution?
no sea contribution
(Kretzer fragmentation functions)
Fragmentation functions
DSS = de Florian, Sassot, Stratmann
KRE = Kretzer
HKNS = Hirai, Kumano, Nagai, Sudoh
M. Anselmino, M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A. Prokudin
fit of HERMES + COMPASS pion and kaon data, with new set
of fragmentation functions (de Florian, Sassot, Stratmann)
Predictions for JLab
Present knowledge of Sivers function (u,d)
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
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.
f1T(1) q
2
k
  d 2 k   2 f1T q ( x, k  )
2M
f1T(1/ 2 ) q ( x)   d 2 k 
k q
f1T ( x, k  )
M
What do we learn from the Sivers
distribution?
number density of partons
with longitudinal momentum
fraction x and transverse
momentum k┴, inside a proton
with spin S
  dx d
a
2
k  k  f a / p  ( x, k  )  0
M. Burkardt, PR D69, 091501 (2004)
S
k
pˆ  k
Total amount of intrinsic momentum carried by
partons of flavour a
1 Nˆ
ˆ

ˆ
ˆ
 k    dx d k  k   f a / p ( x, k  )   f a / p  ( x, k  ) S  ( p  k  )
2


a

2



2
N ˆ
ˆ
ˆ
 sin  S i  cos S j
dx
dk
k

f a / p  ( x, k  )
 

2
for a proton moving along the +z-axis and
polarization vector

S  cosS iˆ  sin S ˆj

S  ( pˆ  kˆ )  sin(S  )
 ka 
S
Numerical estimates from SIDIS data
U. D’Alesio


0.05
 k u   0.14-0.06
sin  S iˆ  cos S
 k d   0.130.03 sin  iˆ  cos

 0.02
S
S
ˆj
ˆj


GeV/c
GeV/c
Sivers functions extracted from AN data in
p p   X give also opposite results, with
ku   0.032
kd   0.036
k   k   0 ?
u

d

 ku 
 kd 
Sivers function and orbital angular momentum
D. Sivers
Sivers mechanism originates from S  Lq
then it is related to the quark orbital angular momentum
For a proton moving along z and polarized along y
1

0
dx N f q / p ( x, k ) 
Lqy
2
?
Sivers function and proton anomalous magnetic moment
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
dx d 2 k N f q / p ( x, k )  C q
?
in qualitative agreement with large z data:
sin( Φ
 ΦS )
sin( Φ
 ΦS )
AUT
AUT


related result (M. Burkardt)
u

d
Collins function from e+e– processes
(spin effects without polarization, D. Boer)
2 

Ph 2
thrust-axis

p 2
e-

Ph1
y

p1
x
e+
z
e+
1
e e  q q  h h X
e+e-
CMS frame:
BELLE @ KEK
2 Eh
z
,
s
s  10.52 GeV
Fit of BELLE data
A12  N D / q  ( z1 )  N D / q  ( z 2 )
Extraction of
Collins functions and
transversity
distributions from
fitting HERMES +
COMPASS + BELLE
data
M. Anselmino,
M. Boglione,
U. D’Alesio,
A. Kotzinian,
F. Murgia,
A. Prokudin
C. Türk
What do we learn (if anything yet) from the
transversity distributions and the Collins functions ?
h1u and h1d have opposite signs
They are both smaller than Soffer bound
Still large uncertainties
Tensor charges appear to be smaller than
results from lattice QCD calculations
M. Wakamatsu, hep-ph/0705.2917
Collins functions well below the positivity
bound. Favoured and unfavoured ones have
opposite signs, comparable magnitudes
Only the very beginning …
TMDs and SSAs in hadronic collisions
(abandoning safe grounds ….)
pp   0 X
(collinear configurations)
factorization theorem
(?)

0
D
X
c
f
p
a
ˆ
b
f
p
X
d 

a ,b , c , d  q , q , g
f a / p ( xa )  fb / p ( xb )  dˆ abcd  D / c ( z)
PDF
FF
pQCD elementary
interactions
RHIC data
s  200 GeV
p p  X
excellent agreement with data for
unpolarized cross section, but no SSA
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
p p   X
d   d 
AN 
d   d 
experimental
data on SSA
STAR-RHIC √s = 200 GeV
1.2 < pT < 2.8
and AN stays at high energies ….
talk by L. Bland
SSA in hadronic processes: intrinsic k┴, factorization?
Two main different (?) approaches
Generalization of collinear scheme
(M. A., M. Boglione, U. D’Alesio, E. Leader, F. Murgia, S. Melis)
D
X
p
f
a

0
c
ˆ
b
f
p
X
d 

a ,b,c ,d q ,q , g
f a / p ( xa ,ka )  fb / p ( xb ,kb )  dˆ abcd (ka ,kb )  D / c ( z, p )
It generalizes to polarized case
d A,S A  B,S B  C  X   a /A' ,S A f a / A,S A ( xa ,ka )  b /B' ,S B f b / B,S B ( xb ,kb )
a a
b b
ab cd
ab cd
ˆ
ˆ
 M c ,d ;a ,b M ' , ;' ,' (ka ,kb )  DC,,' C ( z, p )
c
d
a
b
c
c
plenty of phases
main remaining contribution to SSA from Sivers effect
d p , S  p   X   N f q / p  ( xa ,k  a )  f b / p ( xb ,k b )
q
 dˆ ab cd (k  a ,k b )  D / c ( z , p )
U. D’Alesio, F. Murgia
E704 data
fit
STAR data
prediction
Higher-twist partonic correlations
(Efremov, Teryaev; Qiu, Sterman; Kouvaris, Vogelsang, Yuan)
contribution to SSA ( A B  h X )
d   Ta (k1 , k2 , S )  fb / B ( xb )  H abc (k1 , k2 )  Dh / c ( z)
a ,b,c
twist-3 functions
hard interactions
“collinear expansion” at order ki┴
a
a
Ta  Na x (1  x) f a / A ( x)
fits of E704 and STAR data
Kouvaris, Qiu, Vogelsang, Yuan
Gluonic pole cross sections and SSA in H1H 2  h1h2 X
Bacchetta, Bomhof, Mulders, Pijlman; Vogelsang, Yuan
factorization ? (Collins)
(1)
(
T

f
Sivers contribution to SSA a
1T )
d   f1T(1) ( x1 )  fb / H1 ( x2 )  dˆ[ a]bcd  Dh1 / c ( z1 )  Dh2 / d ( z2 )
a ,b,c
gluonic pole cross sections take into account gauge links
dˆ[ a ]bcd   CG[ D]dˆ abD cd
D
(breaking of factorization?)
CG[D] 
Diagram dependent Gauge
link Colour factors
Gluonic pole cross sections and SSA in H1H 2  h1h2 X
to be compared with the usual cross section
dˆ l[ q ]lq  dˆ lqlq
dˆ[ q ]q l l   dˆ qq l l 
Non-universality of Sivers Asymmetries:
Unique Prediction of Gauge Theory !
Simple QED
example:
DIS: attractive
Same in QCD:
As a result:
Drell-Yan: repulsive
TMDs and SSAs in Drell-Yan processes
(returning to safer grounds and looking
at future ….)
l+
l–
γ*
p
d
D Y
qT
qL
factorization holds, two scales, M2, and
Q2 = M 2
p
qT
 q f q ( x1 , k ; Q )  f q ( x2 , k ; Q ) dˆ
2
2
qq l  l 
Unpolarized cross section already very interesting
1 d
3 1 
 2

2
2

1


cos



sin

cos


sin

cos
2



 d 4   3 
2

Collins-Soper frame
(Polarized) Drell-Yan cross sections allow to
access many TMDs (Boer-Mulders, …) D-YLAND
q
f
verify whether 1T
SIDIS
  f1Tq
D Y
!!!
and offer the golden channel to measure the
transversity distribution
ATT   h1q ( x1 )  h1q ( x2 )
q
Talks by R. Kaiser, M. Contalbrigo
Conclusions
Increasing knowledge about quark transverse
motion, and spin-k┴ correlations (Sivers function)
First experimental information on h1
Work in progress on GPDs and quark space
distribution (Lq)
Towards resolving the orbital motion of quarks in
a proton …
More data from HERMES, COMPASS, JLab and
RHIC
Thanks