GPDs and spin effects in light mesons production. - IPN

GPDs and spin effects in light mesons
production.
S.V. Goloskokov
Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,
Dubna 141980, Moscow region, Russia
• Factorization of Vector meson leptoproduction .
• Model for GPDs.
• Modified PA
• Cross section for vector meson production in a wide energy range.
• Results for A UT asymmetry at HERMES and COMPASS
• Some new results on production of π mesons.
In collaboration with P. Kroll, Wuppertal
arXiv:0906.0460; Euro. Phys. J. C59, 809 (2009); C53, 367 (2008);C50, 829 (2007)
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IPN ORSAY January 26, 2010

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Handbag factorization of Vector Mesons production amplitude
• Large Q 2 - factorization into a hard meson photoproduction off partons, and GPDs. (LL )
Radyushkin, Collins,Frankfurt Strikman
k = ((x + ξ)p + , ....)
¢ §
¤¥
k ′ = ((x − ξ)p + , ....)
k ≠ k ′ ; ξ ∼
x B
2 − x B
L → L transition - predominant. Other amplitudes are suppressed as powers 1/Q
The process of VM production
• φ production (gluon&strange sea)
• ρ, ω production (gluon&sea&valence quarks)
• Charge mesons- transition GPDs

Generalized Parton Distributions
D.Mueller, 1994; Ji, 1997; Radyushkin, 1997
ξ = (p − p′ ) +
(p + p ′ ) + ∼ x b
2 , x = ¯k + /¯p + , t
×
〈p ′ ν ′ | ∑ a,a ′ A aρ (0) A a′ ρ ′ (¯z)|pν〉 ∝
{ū(p ′ ν ′ ) n/u(pν)
2¯p · n
+ ū(p′ ν ′ ) n/γ 5 u(pν)
2¯p · n
∫ 1
0
dx
(x + ξ − iε)(x − ξ + iε) e−i(x−ξ)p·¯z
H g (x, ξ, t) + ū(p′ ν ′ ) i σ αβ n α ∆ β u(pν)
E g (x, ξ, t)
4m ¯p · n
˜H g (x, ξ, t) + ū(p′ ν ′ }
) n · ∆ γ 5 u(pν)
Ẽ g (x, ξ, t) .
4m ¯p · n
H g (x, 0, 0) = x g(x); ˜Hg (x, 0, 0) = x ∆g(x) (1)
(Ẽ) determine mainly proton spin-flip – Some contribution for unpo-
Distributions E g (x, ξ, t),
larized case at moderate x.

Modelling the GPDs
The double distributions for GPDs Radyushkin ’99 .
– simple for the double distributions. Regge form: α i = α i (0) + α ′ t
f i (β, α, t) = h i (β, t)
⋆ Gluon contribution (n=2).
Γ(2n i + 2)
2 2n i+1
Γ 2 (n i + 1)
[(1 − |β|) 2 − α 2 ] n i
(1 − |β|) 2n i+1
, (2)
h g (β, 0) = |β|g(|β|) (3)
⇓
H i (x, ξ, t) =
∫ 1
−1
dβ
∫ 1−|β|
−1+|β|
dα δ(β + ξ α − ¯x) f i (β, α, t) (4)
⋆ h q sea (β, 0) = q sea(|β|) sign(β) - sea quark contribution (n=2).
⋆ h q val (β, 0) = q val(|β|) Θ(β) –valence contribution (n=1).

PDF t-dependence —Regge paramererization
.
h i (β, t) = e b 0t β −(δ i(Q 2 )+α ′ t) (1 − β) 2n i+1
3∑
c i β i/2 (5)
i=0
α i (t) = α i (0) + α ′ it
Q2
δ g (Q 2 ) = α P (0) − 1 = .1 + .06 ln
4GeV 2
α ′ g ≃ 0.15GeV −2
δ sea = α P (0) = 1 + δ g
δ val = α val (0) ∼ 0.48; α ′ val = 0.9GeV−2
σ L ∼ W 4δ g(Q 2 )
–High energies –HERA

⋆ Results for gluon and valence quark GPDs
-CTEQ6 parameterization of PDFs
6.0
5.0
4.0
3.0
2.0
0.1
0.01
0.001
H g
10.0
8.0
6.0
4.0
0.05
0.1
0.2
H u val
1.0
2.0
10 −4 10 −3 10 −2 10 −1 1 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5
¯x
¯x
H g GPD at t = 0 and Q 2 = 4GeV 2 . Linesfor
ξ = 10 −3 , 10 −2 , 10 −1 .
H u val GPD at t = 0and Q2 = 4GeV 2 . Linesfor
ξ = 0.05, 0.1, 0.2.

Simple model for u and d sea .
H u sea = H d sea = κ s H s sea
The flavor symmetry breaking factor is from the fit if CTEQ6M PDFs
κ s = 1 + 0.68/(1 + 0.52 ln(Q 2 /Q 2 0))
0.5
0.4
0.3
0.001
¯xH s sea
0.2
0.01
H s sea GPD at t = 0 and Q2 = 4GeV 2
0.1
0.1
10 −4 10 −3 10 −2 10 −1 1
¯x

Modified PA for vector meson production
We calculate the L → L and T → T amplitudes – important in analyses of spin observales.
⋆ Wave function
The k- dependent wave function is used
ˆΨ V = ˆΨ 0 V + ˆΨ 1 V .
ˆΨ 0 V = ( /V + m V ) /ɛ V φ V (k, τ).
ˆΨ 1 V = [ 2
/V /ɛ V /K − 2 ( /V − m V )(ɛ V · K)]φ ′ V (k, τ).
M V M V
• ˆΨ 0 V
• ˆΨ 1 V
- leading twist wave function -L polarized meson
- higher twist wave function -T polarized meson
J. Bolz, J. Körner and P. Kroll, 1994
• V is a vector meson momentum and m V is its mass
• ɛ V is a meson polarization vector and K is a quark transverse momentum
• M V is a scale in the ˆΨ 1 V . We use M V = m V /2

⋆ Structure of the amplitudes of vector meson production
γ ∗ µ → V ′
µ
quark & gluons contributions
M µ ′ +,µ+ ∼ ∑ e a Ba
V
a
∫ 1
[
]
×
{C F dxĤa (x, ξ, t) Fµ a ′ ,µ (x, ξ) 1
−1
x + ξ − iˆε + 1
x − ξ + iˆε
∫ 1 Ĥ g (x, ξ, t) F g µ
+ 2 (1 + ξ) dx
′ ,µ (x, ξ )
}
. (6)
0 (x + ξ)(x − ξ + iˆε)
√
Ĥ i = [H i −
ξ2
] + [
1 − ξ ˜H −t
i
2Ei (Ẽi ) contributions]. M µ ′ −,µ+ ∼ [< E > +...ξ < Ẽ >]
2m
The hard scattering amplitudes-transverse quark motion
F a(g)
µ ′ ,µ (x, ξ) = 8πα ∫
s(µ R ) 1 ∫ d 2 k
√ ⊥
dτ 2Nc 16π φ 3 V µ ′(τ, k2 ⊥ ) fa(g) µ ′ ,µ (k ⊥, x, ξ, τ)D . (7)
0
φ V (k ⊥ , τ) = 8π 2√ 2N c f V a 2 V exp [−a 2 V
k⊥
2 ]
. (8)
τ ¯τ

The product of propagator denominators reads
D −1 = ( k 2 ⊥ + ¯τ Q 2) ( k 2 ⊥ + τ Q 2) ( k 2 ⊥ + ȳ ¯τ Q 2 − iˆε )
× ( k 2 ⊥ + y ¯τ Q2 − iˆε ) ( k 2 ⊥ + τ y Q2 − iˆε ) ( k 2 ⊥ + τ ȳ Q2 − iˆε ) .
The y and ȳ are defined by y = (x + ξ)/(2ξ) , ȳ = (ξ − x)/(2ξ) .
At k 2 = 0 divergences may appear at τ = 0, (ξ − x) = 0 -happened in TT
D ∼
1
(k 2 ⊥ +τQ2 )... - contains power corrections ∼ k2 ⊥ /Q2 .
The model leads to the following form of helicity amplitudes
L → L :
T → T :
M V (g)
0 ν,0 ν ∝ 1
M V (g)
+ν,+ν ∝
k2 ⊥
QM V
(9)

Amplitudes of vector meson leptoproduction (leading twist)
M φ = e 8πα s
N c Q f −1
φ〈1/τ〉 φ
3
M ρ = e 8πα s
N c Q f ρ〈1/τ〉 ρ
1 √2
{ 1
M ω = e 8πα s
N c Q f 1
ω〈1/τ〉 ω
3 √ 2
The integrals:
I g = 2
I sea = 2
I a val = 2
{ 1
2ξ I g + C F I sea
}
∫ 1
0
∫ 1
0
∫ 1
,
2ξ I g + κ s C F I sea + 1 3 C F Ival u + 1 6 C F Ival
d
{ 1
2ξ I g + κ s C F I sea + C F Ival u − 1 }
2 C F Ival
d
−ξ
}
,
(10)
ξH g (x, ξ, t ′ )
dx
(x + ξ)(x − ξ + iɛ) ,
xH s
dx
sea(x, ξ, t ′ )
(x + ξ)(x − ξ + iɛ) ,
xHval a dx
(x, ξ, t′ )
(x + ξ)(x − ξ + iɛ) . (11)
Different combinations of valence quark GPDs. Valence quark contribution u val ∼ 2d val
φ: val=0;
ρ: ∼ 5/6 d val ;
ω: ∼ 9/6 d val

Combinations of quark contribution to different reactions
Uncharged VM production - Standard p → p GPDs for ρ, ω: Transition p → Σ + GPDs for K ∗0 :
ρ :
∼ e u H u − e d H d = 2 3 Hu + 1 3 Hd
ω :
∼ 2 3 Hu − 1 3 Hd
K ∗0 : ∼ H d − H s
Charged VM production - Transition p → n GPDs :
Pseudoscalar mesons production :
ρ + : ∼ H (3) = H u − H d ;
π + : ∼ ˜H (3) = ˜H u − ˜H d ;
π 0 : ∼ 2 3 ˜H u + 1 3 ˜H d
These combinations can be tested in mentioned reactions. GPDs in amplitudes are in integrated
form. Direct GPDs extraction is impossible .
Our way: parameterization of GPDs and comparison with experiment.

Modified PA amplitudes
The hard scattering amplitudes-transverse quark motion
F a(g)
µ ′ ,µ (x, ξ) = 8πα ∫
s(µ R ) 1
√ 2Nc
0
dτ
∫ d 2 k ⊥
16π 3 φ V µ ′(τ, k2 ⊥ ) fa(g) µ ′ ,µ (k ⊥, x, ξ, τ) ˆD(τ, Q, b) . (12)
φ V (k ⊥ , τ) = 8π 2√ [
2N c f V a 2 V exp −a 2 k 2 ]
⊥
V . (13)
τ ¯τ
We consider Sudakov suppression of large quark-antiquark separations. These effects suppress
contributions from the end-point regions , where factorization breaks down . Since the Sudakov
factor is exponentiate in the impact parameter space- we have to work in this space.
The helicity amplitudes for vector-meson electroproduction read
M H µ ′ +,µ+ = √ e ∫
∫
C V dxdτ f µ ′ µ H(x, ξ, t)× d 2 b φ V (τ, b 2 ) ˆD(τ, Q, b) α s (µ R ) exp [−S(τ, b, Q)] .
2Nc
The renormalization scale µ R is taken to be the largest mass scale appearing in the hard scattering
amplitude, i.e. µ R = max (τQ, ¯τQ, 1/b).
(14)

Cross sections of VM production
Q 2 dependence of cross sections of ρ andφ production at W = 75GeV. H1 and ZEUS data.
10 3
10 3
σ(ρ) [nb]
10 2
10 1
σ(φ) [nb]
10 2
10 1
10 0
10 -1
10 0
4 6 10 20 40 60 100
5 10 20 30 40
10 -1
Q 2 [GeV 2 ]
Q 2 [GeV 2 ]
Cross sections of ρ production with errors
from uncertainty in parton distributions
at W = 75GeV/10 and W = 90GeV.
Dashed line leading twist results.
Cross sections of φ production with errors
from uncertainty in parton distributions
at W = 75GeV. Dashed line leading
twist results.
⋆ Power corrections ∼ k 2 ⊥ /Q2 in propagators are important at low Q 2 –1/10 suppression at Q 2 ∼
3GeV 2

Q 2 dependence of ρ and φ production cross sections.
10 2
σ L
(γ * p->ρp) [nb]
10 2
10 1
σ L
(γ * p->φp) [nb]
10 1
10 0
2 4 6 8
Q 2 [GeV 2 ]
2 4 6 8
Q 2 [GeV 2 ]
The longitudinal cross section for ρ electroproduction
at W = 5 GeV-full line
and W = 10 GeV -dashed line. Open
triangle- E665
The same for φ electroproduction.
Solid circles-HERMES data.
• All data are described fine. • Predictions for COMPASS should be fine too.

W dependence of ρ production cross section at high energies.
10 3 13.5
σ(ρ) [nb]
10 2
10 1
Q 2 [GeV] 2
3.7
6.0
8.3
10 0
32
50 100 150 200
W [GeV]
Cross sections of ρ production at different Q 2 . New ZEUS data.

Ratio of cross sections of φ/ρ production.
σ L
(φ)/σ L
(ρ)
0.30
0.25
0.20
0.15
0.10
2/9
Show strong violation of σ φ /σ ρ = 2/9 at
HERA energies and low Q 2 is caused by
the flavor symmetry breaking between ū
and ¯s
ū(x) = ¯d(x) = κ s s(x)
0.05
0.00
2 4 6 8 10 20 40
The flavor symmetry breaking factor is
from the fit of CTEQ6M PDFs
Q 2 [GeV 2 ]
Q 2 dependence of σ φ /σ ρ at HERA is determined by κ s factor completely.
At HERMES energies we have valence quarks contribution which gives additional suppression of
σ φ /σ ρ ratio.

Analyses of different contributions to the ρ production.
σ L
(γ * p->ρp) [nb]
10 2
10 1
Full line- cross section.
Red-gluon contribution
Green- gluon+ sea contribution.
Blue -Valence- gluon+sea interference
4 6 8 10 20 40 60 80100
W[GeV]
At HERMES energies valence quarks contribution is essential. At energies higher 5GeV 2 it
decreases rapidly with energy growing . As a result, COMPASS (unpolarized cross sectoin) is
closed to HERA physics- gluon and sea is essential .

Cross section of ρ and φ production cross
σ L
(γ * p->φp) [nb]
10 2
10 1
σ L
(γ * p->ρp) [nb]
10 2
10 0
10 1
4 6 810 20 40 60 100
4 6 810 20 40 60 100
W[GeV]
W[GeV]
The longitudinal cross section for φ at
Q 2 = 3.8 GeV 2 . Data: HERMES (solid
circle), ZEUS (open square), H1 (solid
square), open circle- CLAS data point
The longitudinal cross section for ρ at
Q 2 = 4.0 GeV 2 . Data: HERMES
(solid circle), ZEUS (open square), H1
(solid square), E665 (open triangle), open
circles- CLAS, CORNEL -solid triangle
Such problem appears in all the cases when valence quark distributions are essential at low W : ρ 0 ,
ρ + , ω production.- Break in DD, handbag, other effects???

Ratio of longitudinal and transverse cross sections
R(V ) = σ L(γ ∗ p → V p)
σ T (γ ∗ p → V p) , (15)
R(ρ)
12
10
8
6
4
2
0
4 6 8 10 20 40
Q 2 [GeV 2 ]
The ratio of longitudinal and transverse
cross sections for ρ production versus Q 2
at W ≃ 75 GeV.
R(φ)
10
8
6
4
2
0
5
10 20 30
Q 2 [GeV 2 ]
The ratio of longitudinal and transverse
cross sections for φ production versus Q 2
at W ≃ 75 GeV.

5
4
R(ρ)
4
3
2
1
R(φ)
3
2
1
0
2 3 4 5 6 7 8 9 10
Q 2 [GeV 2 ]
The ratio of longitudinal and transverse
cross sections for ρ production at low Q 2
Blue line- HERA , red-COMPASS and
black- HERMES.
0
2 3 4 5 6 7 8 9 10
Q 2 [GeV 2 ]
The ratio of longitudinal and transverse
cross sections for φ production at low Q 2
Blue line- HERA , black- HERMES.
Squares- H1 and ZEUS data, open diamond- COMPASS, filled circles- HERMES.

Spin density matrix elements
Some essential spin density matrix elements read
N L = 2|M V (g)
0 +,0 + |2 ,
N T = ∑ [
]
|M V +ν,+ν| (g) 2 + |M V (g)
0 ν,+ν |2 ,
ν
r00 04 = 1 ∑
(|M V (g)
0ν,,+ν
N T + εN |2 + ε |M V (g)
L
r 1 1−1 = −Im r 2 1−1 =
Re r 5 10 = −Im r 6 10 =
ν
1
1 + ɛ
√ ˜R
˜R
1 + ɛ ˜R
1
N T
∣ ∣M H
++,++
∣ ∣
2
,
1
√ 2NL N T
Re
[
0ν ,0ν |2 )
]
M H ++,++ M H∗
0 +,0 +
.........................................................................

Spin density matrix elements-ρ production at HERA
0.9
0.8
0.7
0.20
0.15
0.10
0.00
-0.05
-0.10
0.6
0.05
-0.15
0.5
4 6 810 20 40
0.00
4 6 810 20 40
-0.20
4 6 810 20 40
0.20
0.18
0.16
0.14
0.12
0.10
4 6 810 20 40
-0.10
-0.12
-0.14
-0.16
-0.18
4 6 810 20 40
The SDME sensitive to LL and TT transitions via Q 2 .

Spin density matrix elements-ρ production at energies from
HERMES to HERA
0.7
0.3
-0.1
0.6
0.5
0.2
-0.2
0.4
3 4 5 6 7 8
0.1
3 4 5 6 7 8
-0.3
3 4 5 6 7 8
0.22
0.20
0.18
0.16
0.14
0.12
3 4 5 6 7 8
-0.12
-0.14
-0.16
-0.18
-0.20
-0.22
3 4 5 6 7 8
The SDME sensitive to LL and TT transitions via Q 2 .
Blue- HERA , red- COMPASS , black- HERMES.
Open diamond- COMPASS, filled circles- HERMES.

Spin density matrix elements-φ production at HERMES
0.7
0.6
0.5
0.4
3 4 5 6 7 8
0.3
0.2
0.1
3 4 5 6 7 8
0.22
0.20
0.18
0.16
0.14
0.12
3 4 5 6 7 8
The SDME for φ meson via Q 2 at W = 5GeV .
Relative phase φ between TT and LL amplitudes for light VM:
φ model ∼ 3 o φ experim ∼ 22
(Results from HERA, HERMES).

Hierarchy of cross sections for various meson production.
σ (γ * p->Vp) [nb]
10 2
10 1
10 0
10 -1
2 3 4 5 6 7 8
Q 2 [GeV 2 ]
σ (γ * p->Vp) [nb]
10 2
10 1
10 0
10 -1
2 3 4 5 6 7 8
Q 2 [GeV 2 ]
Our predictions for HERMES energy
W = 5GeV.
Our predictions for COMPASS energy
W = 10GeV
• Red- dot-dashed line ρ 0 ; Black-full line ω. –gluon contribution here - growing with energy .
• Blue- dotted line ρ + ; Green-dashed line K ∗0 . K ∗0 - sea contribution .

Effects of ˜H in σ U & A LL asymmetry.
˜H determines amplitudes with unnatural parity and cross section
σ U ∝ |M ˜H| 2
The leading term in A LL is an interference between the H and the ˜H terms.
[
]
A LL [ep → epV ] = 2 √ Re M H
1 − ε 2 ++,++ M ˜H∗
++,++
ε|M H 0+,0+ |2 + |M H ++,++| . (16)
2
This ratio is of order 〈 ˜H g 〉/〈H g 〉 .
A LL asymmetry can be measured with longitudinally polarized beam and target.
Gluon and sea polarized distributions compensate each other essentially.
The dominant contribution comes from quark polarized distributions We construct ∆u and ∆d
distributions. The standard values δ val = 0.48; α ′ val
= 0.9GeV −2 are used. GPDs ˜H is estimated
using double distribution.

0.20
0.2
0.15
0.1
σ u
/σ (ρ)
0.10
A LL
(ρ)
0.0
0.05
-0.1
0.00
2 3 4 5 6 7 8
Q 2 [GeV 2 ]
-0.2
2 3 4 5 6 7 8
Q 2 [GeV 2 ]
The ratio of σ U and σ for ρ production versus Q 2 at W = 5 GeV. Data taken from HERMES .
The solid (dashed, dash-dotted) line represents our estimate (obtained with e u ˜Hu val − e d ˜Hd val , with
e u ˜Hu val
+ e d ˜Hd val
). .
Larger result is expected for this ratio for ω cross section ratio.
The A LL asymmetry for ρ production at W = 5 (10) GeV dashed (dash-dotted) line.
Data taken from COMPASS and HERMES

⋆ Spin-flip contribution. Effects of E GPDs.
The proton spin-flip amplitude is associated with E GPDs.
√ ∫ −t 1
M µ ′ −,µ+ ∝ dx E a (x, ξ, t) Fµ a 2m
′ ,µ(x, ξ)
−1
Double distribution model is used to construct E GPDs. E parameters- from Pauli form factor.
Standard connection with ordinary distribution :
E a (x, 0, 0) = e a (x)
M. Diehl, ...,P.Kroll ’04
And ∫ 1
dxe a val(x) = κ a , κ u ∼ 1.67, κ d ∼ −2.03
κ a -anomalous magnetic moments of valence quarks.
0
This means that E a ∝ κ a and E u and E d have different signs.
⋆ ρ production- M ρ ∝ e u Eval u − e dEval d = 2 3 Eu val + 1 3 Ed val
different signs- compensation .
⋆ ω production- M ω ∝ e u Eval u + e dEval d = 2 3 Eu val + 1 3 Ed val
same signs- enhancement.

A UT asymmetry for ρ production.
A UT = −2 Im[M ∗ +−,++ M ++,++ + εM ∗ 0−,0+M 0+,0+ ]
∑ ′
ν [|M +ν ′ ,++| 2 + ε|M 0ν ′ ,0+| 2 ]
∝ Im < E >∗ < H >
| < H > | 2
E distributions needed to calculate M +−,++ – from Pauli FF of nucleon. +DD form for GPD.
A UT
(ρ 0 )
0.1
0.0
-0.1
-0.2
A UT
(ρ 0 )
0.20
0.15
0.10
0.05
0.00
-0.05
-0.10
W=8 GeV
Q 2 =2 GeV 2
-0.3
0.0 0.2 0.4 0.6
-t' [GeV 2 ]
-0.15
-0.20
0.0 0.2 0.4 0.6
-t'[GeV 2 ]
Predictions for HERMES energy W =
5GeV, Q 2 = 3GeV 2 . Preliminary HER-
MES data are shown.
Predictions for COMPASS energy W =
10GeV. Preliminary COMPASS data at
W = 8GeV Sandacz, Photon 09, Bradamante, TPSH 09

A UT asymmetry for ω and ρ + production.
A UT
(ω)
0.05
0.00
-0.05
-0.10
-0.15
2 4 6 8
Q 2 [GeV 2 ]
A UT
(ρ + )
0.6
0.5
0.4
0.3
0.2
0.1
0.0
2 4 6 8
Q 2 [GeV 2 ]
Our predictions for ω at HERMES energy
W = 5GeV.
Not small negative asymmetry: no compensation
in e u E u + e d E d .
Our predictions for ρ + at HERMES
energy W = 5GeV .
Large positive asymmetry- no compensation
in E u − E d .

Hierarchy of A UT asymmetry for various meson production.
A UT
(V)
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
0.0 0.2 0.4 0.6
-t' [GeV 2 ]
A UT
(V)
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
2 3 4 5 6 7 8
Q 2 [GeV 2 ]
Predictions for HERMES energy W =
5GeV, Q 2 = 3GeV 3 . ρ 0 , ω. ρ + , K ∗0 Predictions for COMPASS energy W =
10GeV.
At small −t A UT ∝ √ −t. For ρ + flatness caused by large spin-flip contribution E to cross section
∝ t. (No compensation between u and d quarks here) .

Parton angular momenta.
Evaluation from Ji’s sum rule
< J a > = 1 2 [ea v
20 + ha v
20 ] + eā20 + hā20
< J g > = 1 2 [eg 20 + hg 20 ] (17)
e(h) a 20 - second moments of corresponding parton distributions.
In our model we find the angular momenta for valence quarks:
< J u v >= 0.222, < Jd v >= −0.015, < Jg >= 0.214. (18)
These results are closed to other estimations (Lattice) ...
Orbital angular momenta:
Polarized distributions from BB fit.
< L i >=< J i > − 1 2˜h i 10 (19)
< L u v >= −0.241, < L d v >= 0.155, < L g >=< J g > . (20)

π + production at HERMES.
Pion pole and handbag contributions to π + production.
Calculation of M 0−,++ is a special case.
M µ ′ ν ′ ,µν ∝ √ −t ′|µ−ν−µ′ +ν ′ |
from angular momentum conservation.
M 0−,++ ∝ √ −t ′ 0 ∝ const but handbag amplitude ∝ t ′
M 0−,++ -is determined by twist 3 contribution → const .

Cross section of π + production at HERMES.
300
250
W=3.83 GeV
300
250
W=3.99 GeV
Q 2 =2.45 GeV 2
dσ/dt (π + ) [nb/GeV) 2 ]
200
150
100
50
dσ/dt (π + ) [nb/GeV) 2 ]
200
150
100
50
0
LT
T
TT
L
0
0.2 0.4 0.6
Q 2 =3.44 GeV 2 0.0 0.2 0.4 0.6
-50
-t'[GeV 2 ]
-t'[GeV 2 ]
Cross section of π + production at
HERMES with HERMES data.
The partial cross section
dσ L /dt, dσ T /dt, dσ LT /dt, dσ TT /dt .
Information about different amplitudes
contribution.

Asymmetry in π + production at HERMES.
A UT
sin (φ- φ S
)(π + )
1.0
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
-0.6
-0.8
-1.0
0.0 0.2 0.4 0.6
-t'[GeV 2 ]
A UL
(π + )
0
-1
0.0 0.2 0.4 0.6
-t'[GeV 2 ]
A UT asymmetry of π + production at
HERMES with preliminary HERMES
data.
A UL asymmetry of π + production at
HERMES with preliminary HERMES
data. Dashed line- twist 3 is omitted.

π 0 production at HERMES.
10
8
W=3.99 GeV
Q 2 =2.45 GeV 2
0.6
0.5
sin φ S
W=3.99 GeV
Q 2 =2.45 GeV 2
dσ/dt (π 0 ) [nb/GeV) 2 ]
6
4
2
0
T
LT
0.2 0.4 0.6
A UT
(π 0 )
0.4
0.3
0.2
0.1
0.0
sin (φ- φ S
)
0.2 0.4 0.6
-t'[GeV 2 ]
-t'[GeV 2 ]
Cross section of π 0 production. Unseparated
cross section- black, red-transverse,
blue-longitudinal-transverse interference
cross section.
sinφ S and sinφ − φ S moments of A UT
asymmetry for π 0 production at
HERMES
No pion pole contribution here → small cross section . At COMPASS π 0 cross section smaller
then 1nb/GeV 2

Conclusion
• We analyse meson electroproduction within handbag approach.
• Modified PA is used to calculate hard subprocess amplitude.
• PDF:
–gluon, sea, valence PDFs -from CTEQ6 parameterization.
–polarized ˜H from BB parameterization.
– E distributions -from Pauli nucleon FF analyses.
• GPDs are calculated using these PDF on the bases of DD representation.
• Direct GPDs extraction is impossible .
Our way: parameterization of GPDs , amplitudes calculation in handbag approach ;
comparison with experiment.
• We describe fine :
–cross section of light meson production: corrections ∼ k 2 ⊥ /Q2 in propagators are important.
–spin observables: R -ratio , SDME and different asymmetries for various meson production.
• Vector meson electroproduction-can be an excellent object to study GPDs.
In different energy ranges, information about quark and gluon GPDs can be extracted from the
cross section and spin observables of the vector meson electroproduction.