Transcript The transverse structure of the nucleon

The transverse structure of the nucleon
(resolving the quark motion inside a nucleon)
 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?
Mauro Anselmino, Torino University and INFN, JLab, December 15, 2006
Partonic intrinsic motion
Plenty of theoretical and experimental evidence for transverse motion
of partons within nucleons and of hadrons within fragmentation jets
uncertainty principle
x  1 fm  p  0.2 GeV/c
±1
k┴
gluon radiation
±
±
qT distribution of lepton pairs in D-Y processes
l+
*
p
qL
Q2 = M2
l–
qT
p
pT distribution of
hadrons in SIDIS
 * p  hX c.m. frame
Hadron distribution in jets in e+e– processes
Large pT particle production in pN  hX
k┴
p
xp
Transverse motion is
usually integrated, but
there might be important
spin-k┴ correlations
0
Intrinsic motion in unpolarized SIDIS, [O ( s )]
d lplhX   f q ( x, Q 2 )  dˆ lqlq  Dqh ( z , Q 2 )
q
in collinear parton model
dˆ lqlq  sˆ 2  uˆ 2  1  (1  y) 2
thus, no dependence on azimuthal angle Фh at
zero-th order in pQCD
the experimental data reveal that
dˆ
lqlh X
/ dΦh  A  B cos Φh  C cos 2Φh
M. Arneodo et al (EMC): Z. Phys. C 34 (1987) 277
Q2
x
2pq
Q 2  q 2
pq
y
l p
Cahn: the observed azimuthal dependence is related to
the intrinsic k┴ of quarks (at least for small PT values)
k  (k cos , k sin , 0)
 k2 
 2k 

sˆ  sx 1 
1  y cos    O 2 
Q


Q 


 k 2 
2k 
uˆ  s x (1  y ) 1 
cos    O 2 
 Q 1  y

Q 
assuming collinear fragmentation, φ = Φh
dˆ lqlhX
 sˆ 2  uˆ 2  A  B cos Φh  C cos 2Φh
dΦh
These modulations of the cross section with azimuthal angle are
denoted as “Cahn effect”.
SIDIS with intrinsic k┴
kinematics
according to Trento
conventions (2004)
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 )
ph
PT
p
zk 
k
The full kinematics is complicated as the produced hadron has also
intrinsic transverse momentum with respect to the fragmenting parton


2
2
neglecting terms of order k / Q one has
PT  p  zk
assuming:
one finds:
with
clear dependence on
and
(assumed to be constant)
Find best values by fitting data on Φh and PT dependences
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
Large PT 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
Transverse single spin asymmetries: 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 )

M   ;    1
M  ;     2
M  ;     3
M ;    4
M ;    5
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
at quark level
but large SSA observed at hadron level!
O ( mq / E )
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 ….
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 )  f b / p ( xb )  dˆ abcd  D / c ( z )
PDF
FF
pQCD elementary
interactions
RHIC data
p p  0 X
s  200 GeV
p p  X
Transverse Λ polarization in unpolarized p-Be scattering at Fermilab

pN  X
P 
d

d

 d

 d

p p  p p
d   d 
AN 
d   d 
p p  p p
ANN
d   d 

d   d 
l N  l  X
“Sivers moment”
d   d 
A
d   d 
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

l N  l  X
“Collins moment”
d   d 
A
d   d 
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

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
Transverse single spin asymmetries in SIDIS
y
ΦS



Φπ
x
S
PT
p
z
X
A  S   p  PT   PT sin (Φπ  ΦS )
 *  p c.m. frame
in collinear configurations there cannot be (at LO) any PT
needs k┴ dependent quark distribution in p↑ and/or p┴ dependent
fragmentation of polarized quark
spin-k┴ correlations?
orbiting quarks?

k

b
Transverse Momentum Dependent distribution functions
Space dependent distribution functions (GPD)
q( x, k )
q ( x, b )
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
 Dh / q 
N
p
2
H1 q
z Mh
spin-k┴ correlations
q

φ S
q
k┴
p┴
p
1
f q / p ( x, k  ) 
2
1 N
 f q  / p ( x, k  ) S q  ( pˆ  kˆ )
2
SΛ
pq
Boer-Mulders function
f q  / p ( x, k  ) 
φ
polarizing F.F.
D / q ( z , p ) 
1
Dh / q ( z , p ) 
2
1 N
 D / q ( z , p ) S   ( pˆ q  pˆ  )
2
Amsterdam group notations
 f q / p
N
k q
  h1
M
 D / q
N
p
2
D1Tq
z M
8 leading-twist spin-k┴ dependent distribution functions
Courtesy of Aram Kotzinian
The Sivers mechanism for SSA in SIDIS processes
Sivers function
dependence on ST in cross section
1
d 
2
6

dˆ lqlq z h
q  d k f q / p ( x, k ) dQ 2 J z Dq ( z, p )
h
2
SSA:
( p  PT  zk  , J
z
 1)
zh
Sivers angle
after integration over k┴ one obtains:
A  sin(  h   S )
data are presented for
the sinΦ moment of
the analyzing power
AsinΦ  2


d
Φ
[
d


d

] sin Φ



d
Φ
[
d


d

]

p  PT  zk
Parameterization of the Sivers function
1  Nq  1
q = u, d. The Sivers function for sea quarks and antiquarks
is assumed to be zero.
sin(   S )
UT
A
from Sivers mechanism
M.A., U.D’Alesio, M.Boglione, F. Murgia, A.Kotzinian, A Prokudin
Deuteron target


sin( h   S )
AUT
 N f u / p   N f d / p  4 Duh  Ddh

M.A, M. Boglione, U. D’Alesio, A. Kotzinian, F. Murgia, A. Prokudin
hep-ph/0511017
First p┴ moments of
extracted Sivers
functions, compared
with models
data from HERMES and
COMPASS
N f q(1)   f1T(1) q 
k N
 d k 4m p  f q / p ( x, k )
2
f1Tu   f1T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.
(1) q
1T
f
k2
q
  d k
f
( x, k  )
2 1T
2M
2
f1T(1/ 2 ) q ( x)   d 2 k 
k q
f1T ( x, k  )
M
predictions for JLab, proton target, 6 GeV
0.4 ≤ zh ≤ 0.7
0.02 ≤ PT ≤ 1 GeV/c
0.1 ≤ xB ≤ 0.6
0.4 ≤ y ≤ 0.85
Q2 ≥ 1 (GeV/c)2 W2 ≥ 4 GeV2
1 ≤ Eh ≤ 4 GeV
predictions for JLab, proton target, 12 GeV
0.4 ≤ zh ≤ 0.7
0.02 ≤ PT ≤ 1.4 GeV/c
0.05 ≤ xB ≤ 0.7
0.2 ≤ y ≤ 0.85
Q2 ≥ 1 (GeV/c)2 W2 ≥ 4 GeV2
1 ≤ Eh ≤ 7 GeV
What do we learn from the Sivers distribution?
1 N
f a / p  ( x, k  )  f a / p ( x, k  )   f a / p  ( x, k  ) S  ( pˆ  kˆ )
2
S  ( pˆ  k  )
a
 f a / p ( x, k  )  f1T ( x, k  )
M
2k
 N
a 
  f a / p     f1T 
M


number density of partons with
longitudinal momentum fraction x and
transverse momentum k┴, inside a
proton with spin S
2
dx
d
k k f a / p  ( x, k )  0
a 
M. Burkardt, PR D69, 091501 (2004)
S
k
p̂  k
Then we can compute the total amount of intrinsic momentum
carried by partons of flavour a
1 Nˆ
ˆ

ˆ
k   dx d k  k   f a / p ( x, k  )   f a / p ( x, k  ) S  ( pˆ  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

S  cos  S iˆ  sin  S ˆj

S  ( pˆ  kˆ )  sin(  S   )
k a
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
Collins mechanism for SSA

1
Dh / q  ( z , p )  Dh / q ( z , p )  N Dh / q  ( z, p ) S q  ( pˆ q  pˆ  )
2
S q  ( pˆ q  pˆ  )  sin( ΦS  Φh )  sin ΦC
q’
q

l q l q

initial quark transverse spin is
transmitted to the final quark
 d̂
d
q  q 
d
q  q 
 d
q  q 
 d
q  q 
2(1  y )

1  (1  y ) 2
(in collinear configuration, here ↑,↓ means perpendicular to the leptonic plane)
y
The polarization of the fragmenting
quark q' can be computed in QED
Sq
S q

l  l
ΦC

ΦS
Φh
ΦC    Φh  ΦS
x
 2(1  y )
( S q ) x 
(Sq ) x
2
1  (1  y )
2(1  y )
( S q ) y 
(Sq ) y
2
1  (1  y )
depolarization factor
sin ΦC  sin( Φh  ΦS )
d   d   q T q  dˆ  N D
Collins effect in SIDIS
sin(Φh ΦS )
AUT
2


dΦ
dΦ
[d


d

] sin( Φh  ΦS )
 h S


dΦ
dΦ
[
d


d

]
 h S
sin(Φh ΦS )
AUT

lqlq
ˆ
d


2
N
dΦ
dΦ
d
k
h
(
x
,
k
)

Dh / q  ( z , p ) sin( Φh  ΦS )
q  h S  1q  dQ 2
dˆ lqlq
2
q  dΦh dΦS d k dQ 2 f q / p ( x, k ) Dq / p ( z, p )
h1q or  T q : transversity distribution
N Dh / q  ( z , p )  N Dh / q  ( z , p ) S q  ( pˆ q  pˆ  )
fit to HERMES data on
sin(Φh ΦS )
UT
A
assuming 2 | h1 |  q  q
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)
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
single quark or antiquark are not polarized, but
there is a strong correlation between their spins
dˆ 
dˆ 
3 2 2 2


eq sin 
d cos  d cos 
4s
cross section for detecting the final hadrons inside the jets
e  e   h1h2 X
d
dˆ spins
 
Dh / q  ( z1 , p1 ) Dh / q  ( z 2 , p 2 )
2
2
2
dz1 dz 2 d p1 d p 2 d cos  q , spins d cos  1
contains the product of two Collins functions
d
2
N
N
2
e

D
(
z
)

Dh / q  ( z 2 )


q
1
dz1 dz 2 d 0 d cos 
1 sin 
q
h1 / q
2
A1 
 1
(cos 0 )
2
2
1
d
8 1  cos 
q eq Dh1 / q ( z1 ) Dh2 / q ( z2 )
2 dz1 dz 2 d cos 
N Dh / q ( z)   d 2 p N Dh / q ( z, p ), etc.
0  1  2
N ( 0 )
N0
BELLE data
M. Grosse Perdekamp,
A. Ogawa, R. Seidl
Talk at SPIN2006
Cosine modulations
clearly visible
1  2
N ( 0 )
 P1  P2 cos  0
N0
P1
A1U
 1  cos(1   2 ) P1 ( z1 , z2 )
L
A1
U = unlike charged pions
L = like charged pions
N D  / u  N DF , N D  / u  N DU , etc.
1 sin 2   5N DF ( z1 )N DF ( z 2 )  7 N DU ( z1 )N DU ( z 2 )
P1 ( z1 , z 2 ) 

8 1  cos 2  
5 DF ( z1 ) DF ( z 2 )  7 DU ( z1 ) DU ( z 2 )
5N DF ( z1 )N DU ( z 2 )  5N DU ( z1 )N DF ( z 2 )  2N DU ( z1 )N DU ( z 2 ) 


5DF ( z1 ) DU ( z 2 )  5DU ( z1 ) DF ( z 2 )  2 DU ( z1 ) DU ( z 2 )

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 HERMES data on
sin( h   S )
AUT
fit of COMPASS data on
sin( h   S )
AUT
fit of BELLE data on
P1 ( z1 , z2 )
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
Open issues and wild ideas …
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 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
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


u

d
?
Conclusions
Towards the transverse spin and momentum structure of
the nucleon via azimuthal and PhT 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 …