GPDs and spin effects in light mesons production. - IPN
GPDs and spin effects in light mesons production. - IPN
GPDs and spin effects in light mesons production. - IPN
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<strong>GPDs</strong> <strong>and</strong> <strong>sp<strong>in</strong></strong> <strong>effects</strong> <strong>in</strong> <strong>light</strong> <strong>mesons</strong><br />
<strong>production</strong>.<br />
S.V. Goloskokov<br />
Bogoliubov Laboratory of Theoretical Physics, Jo<strong>in</strong>t Institute for Nuclear Research,<br />
Dubna 141980, Moscow region, Russia<br />
• Factorization of Vector meson lepto<strong>production</strong> .<br />
• Model for <strong>GPDs</strong>.<br />
• Modified PA<br />
• Cross section for vector meson <strong>production</strong> <strong>in</strong> a wide energy range.<br />
• Results for A UT asymmetry at HERMES <strong>and</strong> COMPASS<br />
• Some new results on <strong>production</strong> of π <strong>mesons</strong>.<br />
In collaboration with P. Kroll, Wuppertal<br />
arXiv:0906.0460; Euro. Phys. J. C59, 809 (2009); C53, 367 (2008);C50, 829 (2007)<br />
———————————————————————————<br />
<strong>IPN</strong> ORSAY January 26, 2010
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H<strong>and</strong>bag factorization of Vector Mesons <strong>production</strong> amplitude<br />
• Large Q 2 - factorization <strong>in</strong>to a hard meson photo<strong>production</strong> off partons, <strong>and</strong> <strong>GPDs</strong>. (LL )<br />
Radyushk<strong>in</strong>, Coll<strong>in</strong>s,Frankfurt Strikman<br />
k = ((x + ξ)p + , ....)<br />
¢ §<br />
¤¥<br />
k ′ = ((x − ξ)p + , ....)<br />
k ≠ k ′ ; ξ ∼<br />
x B<br />
2 − x B<br />
L → L transition - predom<strong>in</strong>ant. Other amplitudes are suppressed as powers 1/Q<br />
The process of VM <strong>production</strong><br />
• φ <strong>production</strong> (gluon&strange sea)<br />
• ρ, ω <strong>production</strong> (gluon&sea&valence quarks)<br />
• Charge <strong>mesons</strong>- transition <strong>GPDs</strong>
Generalized Parton Distributions<br />
D.Mueller, 1994; Ji, 1997; Radyushk<strong>in</strong>, 1997<br />
ξ = (p − p′ ) +<br />
(p + p ′ ) + ∼ x b<br />
2 , x = ¯k + /¯p + , t<br />
×<br />
〈p ′ ν ′ | ∑ a,a ′ A aρ (0) A a′ ρ ′ (¯z)|pν〉 ∝<br />
{ū(p ′ ν ′ ) n/u(pν)<br />
2¯p · n<br />
+ ū(p′ ν ′ ) n/γ 5 u(pν)<br />
2¯p · n<br />
∫ 1<br />
0<br />
dx<br />
(x + ξ − iε)(x − ξ + iε) e−i(x−ξ)p·¯z<br />
H g (x, ξ, t) + ū(p′ ν ′ ) i σ αβ n α ∆ β u(pν)<br />
E g (x, ξ, t)<br />
4m ¯p · n<br />
˜H g (x, ξ, t) + ū(p′ ν ′ }<br />
) n · ∆ γ 5 u(pν)<br />
Ẽ g (x, ξ, t) .<br />
4m ¯p · n<br />
H g (x, 0, 0) = x g(x); ˜Hg (x, 0, 0) = x ∆g(x) (1)<br />
(Ẽ) determ<strong>in</strong>e ma<strong>in</strong>ly proton <strong>sp<strong>in</strong></strong>-flip – Some contribution for unpo-<br />
Distributions E g (x, ξ, t),<br />
larized case at moderate x.
Modell<strong>in</strong>g the <strong>GPDs</strong><br />
The double distributions for <strong>GPDs</strong> Radyushk<strong>in</strong> ’99 .<br />
– simple for the double distributions. Regge form: α i = α i (0) + α ′ t<br />
f i (β, α, t) = h i (β, t)<br />
⋆ Gluon contribution (n=2).<br />
Γ(2n i + 2)<br />
2 2n i+1<br />
Γ 2 (n i + 1)<br />
[(1 − |β|) 2 − α 2 ] n i<br />
(1 − |β|) 2n i+1<br />
, (2)<br />
h g (β, 0) = |β|g(|β|) (3)<br />
⇓<br />
H i (x, ξ, t) =<br />
∫ 1<br />
−1<br />
dβ<br />
∫ 1−|β|<br />
−1+|β|<br />
dα δ(β + ξ α − ¯x) f i (β, α, t) (4)<br />
⋆ h q sea (β, 0) = q sea(|β|) sign(β) - sea quark contribution (n=2).<br />
⋆ h q val (β, 0) = q val(|β|) Θ(β) –valence contribution (n=1).
PDF t-dependence —Regge paramererization<br />
.<br />
h i (β, t) = e b 0t β −(δ i(Q 2 )+α ′ t) (1 − β) 2n i+1<br />
3∑<br />
c i β i/2 (5)<br />
i=0<br />
α i (t) = α i (0) + α ′ it<br />
Q2<br />
δ g (Q 2 ) = α P (0) − 1 = .1 + .06 ln<br />
4GeV 2<br />
α ′ g ≃ 0.15GeV −2<br />
δ sea = α P (0) = 1 + δ g<br />
δ val = α val (0) ∼ 0.48; α ′ val = 0.9GeV−2<br />
σ L ∼ W 4δ g(Q 2 )<br />
–High energies –HERA
⋆ Results for gluon <strong>and</strong> valence quark <strong>GPDs</strong><br />
-CTEQ6 parameterization of PDFs<br />
6.0<br />
5.0<br />
4.0<br />
3.0<br />
2.0<br />
0.1<br />
0.01<br />
0.001<br />
H g<br />
10.0<br />
8.0<br />
6.0<br />
4.0<br />
0.05<br />
0.1<br />
0.2<br />
H u val<br />
1.0<br />
2.0<br />
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<br />
¯x<br />
¯x<br />
H g GPD at t = 0 <strong>and</strong> Q 2 = 4GeV 2 . L<strong>in</strong>esfor<br />
ξ = 10 −3 , 10 −2 , 10 −1 .<br />
H u val GPD at t = 0<strong>and</strong> Q2 = 4GeV 2 . L<strong>in</strong>esfor<br />
ξ = 0.05, 0.1, 0.2.
Simple model for u <strong>and</strong> d sea .<br />
H u sea = H d sea = κ s H s sea<br />
The flavor symmetry break<strong>in</strong>g factor is from the fit if CTEQ6M PDFs<br />
κ s = 1 + 0.68/(1 + 0.52 ln(Q 2 /Q 2 0))<br />
0.5<br />
0.4<br />
0.3<br />
0.001<br />
¯xH s sea<br />
0.2<br />
0.01<br />
H s sea GPD at t = 0 <strong>and</strong> Q2 = 4GeV 2<br />
0.1<br />
0.1<br />
10 −4 10 −3 10 −2 10 −1 1<br />
¯x
Modified PA for vector meson <strong>production</strong><br />
We calculate the L → L <strong>and</strong> T → T amplitudes – important <strong>in</strong> analyses of <strong>sp<strong>in</strong></strong> observales.<br />
⋆ Wave function<br />
The k- dependent wave function is used<br />
ˆΨ V = ˆΨ 0 V + ˆΨ 1 V .<br />
ˆΨ 0 V = ( /V + m V ) /ɛ V φ V (k, τ).<br />
ˆΨ 1 V = [ 2<br />
/V /ɛ V /K − 2 ( /V − m V )(ɛ V · K)]φ ′ V (k, τ).<br />
M V M V<br />
• ˆΨ 0 V<br />
• ˆΨ 1 V<br />
- lead<strong>in</strong>g twist wave function -L polarized meson<br />
- higher twist wave function -T polarized meson<br />
J. Bolz, J. Körner <strong>and</strong> P. Kroll, 1994<br />
• V is a vector meson momentum <strong>and</strong> m V is its mass<br />
• ɛ V is a meson polarization vector <strong>and</strong> K is a quark transverse momentum<br />
• M V is a scale <strong>in</strong> the ˆΨ 1 V . We use M V = m V /2
⋆ Structure of the amplitudes of vector meson <strong>production</strong><br />
γ ∗ µ → V ′<br />
µ<br />
quark & gluons contributions<br />
M µ ′ +,µ+ ∼ ∑ e a Ba<br />
V<br />
a<br />
∫ 1<br />
[<br />
]<br />
×<br />
{C F dxĤa (x, ξ, t) Fµ a ′ ,µ (x, ξ) 1<br />
−1<br />
x + ξ − iˆε + 1<br />
x − ξ + iˆε<br />
∫ 1 Ĥ g (x, ξ, t) F g µ<br />
+ 2 (1 + ξ) dx<br />
′ ,µ (x, ξ )<br />
}<br />
. (6)<br />
0 (x + ξ)(x − ξ + iˆε)<br />
√<br />
Ĥ i = [H i −<br />
ξ2<br />
] + [<br />
1 − ξ ˜H −t<br />
i<br />
2Ei (Ẽi ) contributions]. M µ ′ −,µ+ ∼ [< E > +...ξ < Ẽ >]<br />
2m<br />
The hard scatter<strong>in</strong>g amplitudes-transverse quark motion<br />
F a(g)<br />
µ ′ ,µ (x, ξ) = 8πα ∫<br />
s(µ R ) 1 ∫ d 2 k<br />
√ ⊥<br />
dτ 2Nc 16π φ 3 V µ ′(τ, k2 ⊥ ) fa(g) µ ′ ,µ (k ⊥, x, ξ, τ)D . (7)<br />
0<br />
φ V (k ⊥ , τ) = 8π 2√ 2N c f V a 2 V exp [−a 2 V<br />
k⊥<br />
2 ]<br />
. (8)<br />
τ ¯τ
The product of propagator denom<strong>in</strong>ators reads<br />
D −1 = ( k 2 ⊥ + ¯τ Q 2) ( k 2 ⊥ + τ Q 2) ( k 2 ⊥ + ȳ ¯τ Q 2 − iˆε )<br />
× ( k 2 ⊥ + y ¯τ Q2 − iˆε ) ( k 2 ⊥ + τ y Q2 − iˆε ) ( k 2 ⊥ + τ ȳ Q2 − iˆε ) .<br />
The y <strong>and</strong> ȳ are def<strong>in</strong>ed by y = (x + ξ)/(2ξ) , ȳ = (ξ − x)/(2ξ) .<br />
At k 2 = 0 divergences may appear at τ = 0, (ξ − x) = 0 -happened <strong>in</strong> TT<br />
D ∼<br />
1<br />
(k 2 ⊥ +τQ2 )... - conta<strong>in</strong>s power corrections ∼ k2 ⊥ /Q2 .<br />
The model leads to the follow<strong>in</strong>g form of helicity amplitudes<br />
L → L :<br />
T → T :<br />
M V (g)<br />
0 ν,0 ν ∝ 1<br />
M V (g)<br />
+ν,+ν ∝<br />
k2 ⊥<br />
QM V<br />
(9)
Amplitudes of vector meson lepto<strong>production</strong> (lead<strong>in</strong>g twist)<br />
M φ = e 8πα s<br />
N c Q f −1<br />
φ〈1/τ〉 φ<br />
3<br />
M ρ = e 8πα s<br />
N c Q f ρ〈1/τ〉 ρ<br />
1 √2<br />
{ 1<br />
M ω = e 8πα s<br />
N c Q f 1<br />
ω〈1/τ〉 ω<br />
3 √ 2<br />
The <strong>in</strong>tegrals:<br />
I g = 2<br />
I sea = 2<br />
I a val = 2<br />
{ 1<br />
2ξ I g + C F I sea<br />
}<br />
∫ 1<br />
0<br />
∫ 1<br />
0<br />
∫ 1<br />
,<br />
2ξ I g + κ s C F I sea + 1 3 C F Ival u + 1 6 C F Ival<br />
d<br />
{ 1<br />
2ξ I g + κ s C F I sea + C F Ival u − 1 }<br />
2 C F Ival<br />
d<br />
−ξ<br />
}<br />
,<br />
(10)<br />
ξH g (x, ξ, t ′ )<br />
dx<br />
(x + ξ)(x − ξ + iɛ) ,<br />
xH s<br />
dx<br />
sea(x, ξ, t ′ )<br />
(x + ξ)(x − ξ + iɛ) ,<br />
xHval a dx<br />
(x, ξ, t′ )<br />
(x + ξ)(x − ξ + iɛ) . (11)<br />
Different comb<strong>in</strong>ations of valence quark <strong>GPDs</strong>. Valence quark contribution u val ∼ 2d val<br />
φ: val=0;<br />
ρ: ∼ 5/6 d val ;<br />
ω: ∼ 9/6 d val
Comb<strong>in</strong>ations of quark contribution to different reactions<br />
Uncharged VM <strong>production</strong> - St<strong>and</strong>ard p → p <strong>GPDs</strong> for ρ, ω: Transition p → Σ + <strong>GPDs</strong> for K ∗0 :<br />
ρ :<br />
∼ e u H u − e d H d = 2 3 Hu + 1 3 Hd<br />
ω :<br />
∼ 2 3 Hu − 1 3 Hd<br />
K ∗0 : ∼ H d − H s<br />
Charged VM <strong>production</strong> - Transition p → n <strong>GPDs</strong> :<br />
Pseudoscalar <strong>mesons</strong> <strong>production</strong> :<br />
ρ + : ∼ H (3) = H u − H d ;<br />
π + : ∼ ˜H (3) = ˜H u − ˜H d ;<br />
π 0 : ∼ 2 3 ˜H u + 1 3 ˜H d<br />
These comb<strong>in</strong>ations can be tested <strong>in</strong> mentioned reactions. <strong>GPDs</strong> <strong>in</strong> amplitudes are <strong>in</strong> <strong>in</strong>tegrated<br />
form. Direct <strong>GPDs</strong> extraction is impossible .<br />
Our way: parameterization of <strong>GPDs</strong> <strong>and</strong> comparison with experiment.
Modified PA amplitudes<br />
The hard scatter<strong>in</strong>g amplitudes-transverse quark motion<br />
F a(g)<br />
µ ′ ,µ (x, ξ) = 8πα ∫<br />
s(µ R ) 1<br />
√ 2Nc<br />
0<br />
dτ<br />
∫ d 2 k ⊥<br />
16π 3 φ V µ ′(τ, k2 ⊥ ) fa(g) µ ′ ,µ (k ⊥, x, ξ, τ) ˆD(τ, Q, b) . (12)<br />
φ V (k ⊥ , τ) = 8π 2√ [<br />
2N c f V a 2 V exp −a 2 k 2 ]<br />
⊥<br />
V . (13)<br />
τ ¯τ<br />
We consider Sudakov suppression of large quark-antiquark separations. These <strong>effects</strong> suppress<br />
contributions from the end-po<strong>in</strong>t regions , where factorization breaks down . S<strong>in</strong>ce the Sudakov<br />
factor is exponentiate <strong>in</strong> the impact parameter space- we have to work <strong>in</strong> this space.<br />
The helicity amplitudes for vector-meson electro<strong>production</strong> read<br />
M H µ ′ +,µ+ = √ e ∫<br />
∫<br />
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)] .<br />
2Nc<br />
The renormalization scale µ R is taken to be the largest mass scale appear<strong>in</strong>g <strong>in</strong> the hard scatter<strong>in</strong>g<br />
amplitude, i.e. µ R = max (τQ, ¯τQ, 1/b).<br />
(14)
Cross sections of VM <strong>production</strong><br />
Q 2 dependence of cross sections of ρ <strong>and</strong>φ <strong>production</strong> at W = 75GeV. H1 <strong>and</strong> ZEUS data.<br />
10 3<br />
10 3<br />
σ(ρ) [nb]<br />
10 2<br />
10 1<br />
σ(φ) [nb]<br />
10 2<br />
10 1<br />
10 0<br />
10 -1<br />
10 0<br />
4 6 10 20 40 60 100<br />
5 10 20 30 40<br />
10 -1<br />
Q 2 [GeV 2 ]<br />
Q 2 [GeV 2 ]<br />
Cross sections of ρ <strong>production</strong> with errors<br />
from uncerta<strong>in</strong>ty <strong>in</strong> parton distributions<br />
at W = 75GeV/10 <strong>and</strong> W = 90GeV.<br />
Dashed l<strong>in</strong>e lead<strong>in</strong>g twist results.<br />
Cross sections of φ <strong>production</strong> with errors<br />
from uncerta<strong>in</strong>ty <strong>in</strong> parton distributions<br />
at W = 75GeV. Dashed l<strong>in</strong>e lead<strong>in</strong>g<br />
twist results.<br />
⋆ Power corrections ∼ k 2 ⊥ /Q2 <strong>in</strong> propagators are important at low Q 2 –1/10 suppression at Q 2 ∼<br />
3GeV 2
Q 2 dependence of ρ <strong>and</strong> φ <strong>production</strong> cross sections.<br />
10 2<br />
σ L<br />
(γ * p->ρp) [nb]<br />
10 2<br />
10 1<br />
σ L<br />
(γ * p->φp) [nb]<br />
10 1<br />
10 0<br />
2 4 6 8<br />
Q 2 [GeV 2 ]<br />
2 4 6 8<br />
Q 2 [GeV 2 ]<br />
The longitud<strong>in</strong>al cross section for ρ electro<strong>production</strong><br />
at W = 5 GeV-full l<strong>in</strong>e<br />
<strong>and</strong> W = 10 GeV -dashed l<strong>in</strong>e. Open<br />
triangle- E665<br />
The same for φ electro<strong>production</strong>.<br />
Solid circles-HERMES data.<br />
• All data are described f<strong>in</strong>e. • Predictions for COMPASS should be f<strong>in</strong>e too.
W dependence of ρ <strong>production</strong> cross section at high energies.<br />
10 3 13.5<br />
σ(ρ) [nb]<br />
10 2<br />
10 1<br />
Q 2 [GeV] 2<br />
3.7<br />
6.0<br />
8.3<br />
10 0<br />
32<br />
50 100 150 200<br />
W [GeV]<br />
Cross sections of ρ <strong>production</strong> at different Q 2 . New ZEUS data.
Ratio of cross sections of φ/ρ <strong>production</strong>.<br />
σ L<br />
(φ)/σ L<br />
(ρ)<br />
0.30<br />
0.25<br />
0.20<br />
0.15<br />
0.10<br />
2/9<br />
Show strong violation of σ φ /σ ρ = 2/9 at<br />
HERA energies <strong>and</strong> low Q 2 is caused by<br />
the flavor symmetry break<strong>in</strong>g between ū<br />
<strong>and</strong> ¯s<br />
ū(x) = ¯d(x) = κ s s(x)<br />
0.05<br />
0.00<br />
2 4 6 8 10 20 40<br />
The flavor symmetry break<strong>in</strong>g factor is<br />
from the fit of CTEQ6M PDFs<br />
Q 2 [GeV 2 ]<br />
Q 2 dependence of σ φ /σ ρ at HERA is determ<strong>in</strong>ed by κ s factor completely.<br />
At HERMES energies we have valence quarks contribution which gives additional suppression of<br />
σ φ /σ ρ ratio.
Analyses of different contributions to the ρ <strong>production</strong>.<br />
σ L<br />
(γ * p->ρp) [nb]<br />
10 2<br />
10 1<br />
Full l<strong>in</strong>e- cross section.<br />
Red-gluon contribution<br />
Green- gluon+ sea contribution.<br />
Blue -Valence- gluon+sea <strong>in</strong>terference<br />
4 6 8 10 20 40 60 80100<br />
W[GeV]<br />
At HERMES energies valence quarks contribution is essential. At energies higher 5GeV 2 it<br />
decreases rapidly with energy grow<strong>in</strong>g . As a result, COMPASS (unpolarized cross secto<strong>in</strong>) is<br />
closed to HERA physics- gluon <strong>and</strong> sea is essential .
Cross section of ρ <strong>and</strong> φ <strong>production</strong> cross<br />
σ L<br />
(γ * p->φp) [nb]<br />
10 2<br />
10 1<br />
σ L<br />
(γ * p->ρp) [nb]<br />
10 2<br />
10 0<br />
10 1<br />
4 6 810 20 40 60 100<br />
4 6 810 20 40 60 100<br />
W[GeV]<br />
W[GeV]<br />
The longitud<strong>in</strong>al cross section for φ at<br />
Q 2 = 3.8 GeV 2 . Data: HERMES (solid<br />
circle), ZEUS (open square), H1 (solid<br />
square), open circle- CLAS data po<strong>in</strong>t<br />
The longitud<strong>in</strong>al cross section for ρ at<br />
Q 2 = 4.0 GeV 2 . Data: HERMES<br />
(solid circle), ZEUS (open square), H1<br />
(solid square), E665 (open triangle), open<br />
circles- CLAS, CORNEL -solid triangle<br />
Such problem appears <strong>in</strong> all the cases when valence quark distributions are essential at low W : ρ 0 ,<br />
ρ + , ω <strong>production</strong>.- Break <strong>in</strong> DD, h<strong>and</strong>bag, other <strong>effects</strong>???
Ratio of longitud<strong>in</strong>al <strong>and</strong> transverse cross sections<br />
R(V ) = σ L(γ ∗ p → V p)<br />
σ T (γ ∗ p → V p) , (15)<br />
R(ρ)<br />
12<br />
10<br />
8<br />
6<br />
4<br />
2<br />
0<br />
4 6 8 10 20 40<br />
Q 2 [GeV 2 ]<br />
The ratio of longitud<strong>in</strong>al <strong>and</strong> transverse<br />
cross sections for ρ <strong>production</strong> versus Q 2<br />
at W ≃ 75 GeV.<br />
R(φ)<br />
10<br />
8<br />
6<br />
4<br />
2<br />
0<br />
5<br />
10 20 30<br />
Q 2 [GeV 2 ]<br />
The ratio of longitud<strong>in</strong>al <strong>and</strong> transverse<br />
cross sections for φ <strong>production</strong> versus Q 2<br />
at W ≃ 75 GeV.
5<br />
4<br />
R(ρ)<br />
4<br />
3<br />
2<br />
1<br />
R(φ)<br />
3<br />
2<br />
1<br />
0<br />
2 3 4 5 6 7 8 9 10<br />
Q 2 [GeV 2 ]<br />
The ratio of longitud<strong>in</strong>al <strong>and</strong> transverse<br />
cross sections for ρ <strong>production</strong> at low Q 2<br />
Blue l<strong>in</strong>e- HERA , red-COMPASS <strong>and</strong><br />
black- HERMES.<br />
0<br />
2 3 4 5 6 7 8 9 10<br />
Q 2 [GeV 2 ]<br />
The ratio of longitud<strong>in</strong>al <strong>and</strong> transverse<br />
cross sections for φ <strong>production</strong> at low Q 2<br />
Blue l<strong>in</strong>e- HERA , black- HERMES.<br />
Squares- H1 <strong>and</strong> ZEUS data, open diamond- COMPASS, filled circles- HERMES.
Sp<strong>in</strong> density matrix elements<br />
Some essential <strong>sp<strong>in</strong></strong> density matrix elements read<br />
N L = 2|M V (g)<br />
0 +,0 + |2 ,<br />
N T = ∑ [<br />
]<br />
|M V +ν,+ν| (g) 2 + |M V (g)<br />
0 ν,+ν |2 ,<br />
ν<br />
r00 04 = 1 ∑<br />
(|M V (g)<br />
0ν,,+ν<br />
N T + εN |2 + ε |M V (g)<br />
L<br />
r 1 1−1 = −Im r 2 1−1 =<br />
Re r 5 10 = −Im r 6 10 =<br />
ν<br />
1<br />
1 + ɛ<br />
√ ˜R<br />
˜R<br />
1 + ɛ ˜R<br />
1<br />
N T<br />
∣ ∣M H<br />
++,++<br />
∣ ∣<br />
2<br />
,<br />
1<br />
√ 2NL N T<br />
Re<br />
[<br />
0ν ,0ν |2 )<br />
]<br />
M H ++,++ M H∗<br />
0 +,0 +<br />
.........................................................................
Sp<strong>in</strong> density matrix elements-ρ <strong>production</strong> at HERA<br />
0.9<br />
0.8<br />
0.7<br />
<br />
<br />
0.20<br />
0.15<br />
0.10<br />
<br />
<br />
0.00<br />
-0.05<br />
-0.10<br />
<br />
<br />
0.6<br />
0.05<br />
-0.15<br />
0.5<br />
4 6 810 20 40<br />
0.00<br />
4 6 810 20 40<br />
-0.20<br />
4 6 810 20 40<br />
0.20<br />
0.18<br />
0.16<br />
0.14<br />
0.12<br />
0.10<br />
<br />
<br />
4 6 810 20 40<br />
-0.10<br />
-0.12<br />
-0.14<br />
-0.16<br />
-0.18<br />
<br />
<br />
4 6 810 20 40<br />
The SDME sensitive to LL <strong>and</strong> TT transitions via Q 2 .
Sp<strong>in</strong> density matrix elements-ρ <strong>production</strong> at energies from<br />
HERMES to HERA<br />
0.7<br />
<br />
<br />
0.3<br />
<br />
<br />
-0.1<br />
<br />
<br />
0.6<br />
0.5<br />
0.2<br />
-0.2<br />
0.4<br />
3 4 5 6 7 8<br />
0.1<br />
3 4 5 6 7 8<br />
-0.3<br />
3 4 5 6 7 8<br />
0.22<br />
0.20<br />
0.18<br />
0.16<br />
0.14<br />
0.12<br />
<br />
<br />
3 4 5 6 7 8<br />
-0.12<br />
-0.14<br />
-0.16<br />
-0.18<br />
-0.20<br />
-0.22<br />
<br />
<br />
3 4 5 6 7 8<br />
The SDME sensitive to LL <strong>and</strong> TT transitions via Q 2 .<br />
Blue- HERA , red- COMPASS , black- HERMES.<br />
Open diamond- COMPASS, filled circles- HERMES.
Sp<strong>in</strong> density matrix elements-φ <strong>production</strong> at HERMES<br />
<br />
<br />
0.7<br />
0.6<br />
0.5<br />
0.4<br />
3 4 5 6 7 8<br />
0.3<br />
0.2<br />
<br />
<br />
0.1<br />
3 4 5 6 7 8<br />
<br />
<br />
0.22<br />
0.20<br />
0.18<br />
0.16<br />
0.14<br />
0.12<br />
3 4 5 6 7 8<br />
The SDME for φ meson via Q 2 at W = 5GeV .<br />
Relative phase φ between TT <strong>and</strong> LL amplitudes for <strong>light</strong> VM:<br />
φ model ∼ 3 o φ experim ∼ 22<br />
(Results from HERA, HERMES).
Hierarchy of cross sections for various meson <strong>production</strong>.<br />
σ (γ * p->Vp) [nb]<br />
10 2<br />
10 1<br />
10 0<br />
10 -1<br />
2 3 4 5 6 7 8<br />
Q 2 [GeV 2 ]<br />
σ (γ * p->Vp) [nb]<br />
10 2<br />
10 1<br />
10 0<br />
10 -1<br />
2 3 4 5 6 7 8<br />
Q 2 [GeV 2 ]<br />
Our predictions for HERMES energy<br />
W = 5GeV.<br />
Our predictions for COMPASS energy<br />
W = 10GeV<br />
• Red- dot-dashed l<strong>in</strong>e ρ 0 ; Black-full l<strong>in</strong>e ω. –gluon contribution here - grow<strong>in</strong>g with energy .<br />
• Blue- dotted l<strong>in</strong>e ρ + ; Green-dashed l<strong>in</strong>e K ∗0 . K ∗0 - sea contribution .
Effects of ˜H <strong>in</strong> σ U & A LL asymmetry.<br />
˜H determ<strong>in</strong>es amplitudes with unnatural parity <strong>and</strong> cross section<br />
σ U ∝ |M ˜H| 2<br />
The lead<strong>in</strong>g term <strong>in</strong> A LL is an <strong>in</strong>terference between the H <strong>and</strong> the ˜H terms.<br />
[<br />
]<br />
A LL [ep → epV ] = 2 √ Re M H<br />
1 − ε 2 ++,++ M ˜H∗<br />
++,++<br />
ε|M H 0+,0+ |2 + |M H ++,++| . (16)<br />
2<br />
This ratio is of order 〈 ˜H g 〉/〈H g 〉 .<br />
A LL asymmetry can be measured with longitud<strong>in</strong>ally polarized beam <strong>and</strong> target.<br />
Gluon <strong>and</strong> sea polarized distributions compensate each other essentially.<br />
The dom<strong>in</strong>ant contribution comes from quark polarized distributions We construct ∆u <strong>and</strong> ∆d<br />
distributions. The st<strong>and</strong>ard values δ val = 0.48; α ′ val<br />
= 0.9GeV −2 are used. <strong>GPDs</strong> ˜H is estimated<br />
us<strong>in</strong>g double distribution.
0.20<br />
0.2<br />
0.15<br />
0.1<br />
σ u<br />
/σ (ρ)<br />
0.10<br />
A LL<br />
(ρ)<br />
0.0<br />
0.05<br />
-0.1<br />
0.00<br />
2 3 4 5 6 7 8<br />
Q 2 [GeV 2 ]<br />
-0.2<br />
2 3 4 5 6 7 8<br />
Q 2 [GeV 2 ]<br />
The ratio of σ U <strong>and</strong> σ for ρ <strong>production</strong> versus Q 2 at W = 5 GeV. Data taken from HERMES .<br />
The solid (dashed, dash-dotted) l<strong>in</strong>e represents our estimate (obta<strong>in</strong>ed with e u ˜Hu val − e d ˜Hd val , with<br />
e u ˜Hu val<br />
+ e d ˜Hd val<br />
). .<br />
Larger result is expected for this ratio for ω cross section ratio.<br />
The A LL asymmetry for ρ <strong>production</strong> at W = 5 (10) GeV dashed (dash-dotted) l<strong>in</strong>e.<br />
Data taken from COMPASS <strong>and</strong> HERMES
⋆ Sp<strong>in</strong>-flip contribution. Effects of E <strong>GPDs</strong>.<br />
The proton <strong>sp<strong>in</strong></strong>-flip amplitude is associated with E <strong>GPDs</strong>.<br />
√ ∫ −t 1<br />
M µ ′ −,µ+ ∝ dx E a (x, ξ, t) Fµ a 2m<br />
′ ,µ(x, ξ)<br />
−1<br />
Double distribution model is used to construct E <strong>GPDs</strong>. E parameters- from Pauli form factor.<br />
St<strong>and</strong>ard connection with ord<strong>in</strong>ary distribution :<br />
E a (x, 0, 0) = e a (x)<br />
M. Diehl, ...,P.Kroll ’04<br />
And ∫ 1<br />
dxe a val(x) = κ a , κ u ∼ 1.67, κ d ∼ −2.03<br />
κ a -anomalous magnetic moments of valence quarks.<br />
0<br />
This means that E a ∝ κ a <strong>and</strong> E u <strong>and</strong> E d have different signs.<br />
⋆ ρ <strong>production</strong>- M ρ ∝ e u Eval u − e dEval d = 2 3 Eu val + 1 3 Ed val<br />
different signs- compensation .<br />
⋆ ω <strong>production</strong>- M ω ∝ e u Eval u + e dEval d = 2 3 Eu val + 1 3 Ed val<br />
same signs- enhancement.
A UT asymmetry for ρ <strong>production</strong>.<br />
A UT = −2 Im[M ∗ +−,++ M ++,++ + εM ∗ 0−,0+M 0+,0+ ]<br />
∑ ′<br />
ν [|M +ν ′ ,++| 2 + ε|M 0ν ′ ,0+| 2 ]<br />
∝ Im < E >∗ < H ><br />
| < H > | 2<br />
E distributions needed to calculate M +−,++ – from Pauli FF of nucleon. +DD form for GPD.<br />
A UT<br />
(ρ 0 )<br />
0.1<br />
0.0<br />
-0.1<br />
-0.2<br />
A UT<br />
(ρ 0 )<br />
0.20<br />
0.15<br />
0.10<br />
0.05<br />
0.00<br />
-0.05<br />
-0.10<br />
W=8 GeV<br />
Q 2 =2 GeV 2<br />
-0.3<br />
0.0 0.2 0.4 0.6<br />
-t' [GeV 2 ]<br />
-0.15<br />
-0.20<br />
0.0 0.2 0.4 0.6<br />
-t'[GeV 2 ]<br />
Predictions for HERMES energy W =<br />
5GeV, Q 2 = 3GeV 2 . Prelim<strong>in</strong>ary HER-<br />
MES data are shown.<br />
Predictions for COMPASS energy W =<br />
10GeV. Prelim<strong>in</strong>ary COMPASS data at<br />
W = 8GeV S<strong>and</strong>acz, Photon 09, Bradamante, TPSH 09
A UT asymmetry for ω <strong>and</strong> ρ + <strong>production</strong>.<br />
A UT<br />
(ω)<br />
0.05<br />
0.00<br />
-0.05<br />
-0.10<br />
-0.15<br />
2 4 6 8<br />
Q 2 [GeV 2 ]<br />
A UT<br />
(ρ + )<br />
0.6<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0.0<br />
2 4 6 8<br />
Q 2 [GeV 2 ]<br />
Our predictions for ω at HERMES energy<br />
W = 5GeV.<br />
Not small negative asymmetry: no compensation<br />
<strong>in</strong> e u E u + e d E d .<br />
Our predictions for ρ + at HERMES<br />
energy W = 5GeV .<br />
Large positive asymmetry- no compensation<br />
<strong>in</strong> E u − E d .
Hierarchy of A UT asymmetry for various meson <strong>production</strong>.<br />
A UT<br />
(V)<br />
0.6<br />
0.4<br />
0.2<br />
0.0<br />
-0.2<br />
-0.4<br />
-0.6<br />
0.0 0.2 0.4 0.6<br />
-t' [GeV 2 ]<br />
A UT<br />
(V)<br />
0.5<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0.0<br />
-0.1<br />
-0.2<br />
2 3 4 5 6 7 8<br />
Q 2 [GeV 2 ]<br />
Predictions for HERMES energy W =<br />
5GeV, Q 2 = 3GeV 3 . ρ 0 , ω. ρ + , K ∗0 Predictions for COMPASS energy W =<br />
10GeV.<br />
At small −t A UT ∝ √ −t. For ρ + flatness caused by large <strong>sp<strong>in</strong></strong>-flip contribution E to cross section<br />
∝ t. (No compensation between u <strong>and</strong> d quarks here) .
Parton angular momenta.<br />
Evaluation from Ji’s sum rule<br />
< J a > = 1 2 [ea v<br />
20 + ha v<br />
20 ] + eā20 + hā20<br />
< J g > = 1 2 [eg 20 + hg 20 ] (17)<br />
e(h) a 20 - second moments of correspond<strong>in</strong>g parton distributions.<br />
In our model we f<strong>in</strong>d the angular momenta for valence quarks:<br />
< J u v >= 0.222, < Jd v >= −0.015, < Jg >= 0.214. (18)<br />
These results are closed to other estimations (Lattice) ...<br />
Orbital angular momenta:<br />
Polarized distributions from BB fit.<br />
< L i >=< J i > − 1 2˜h i 10 (19)<br />
< L u v >= −0.241, < L d v >= 0.155, < L g >=< J g > . (20)
π + <strong>production</strong> at HERMES.<br />
Pion pole <strong>and</strong> h<strong>and</strong>bag contributions to π + <strong>production</strong>.<br />
Calculation of M 0−,++ is a special case.<br />
M µ ′ ν ′ ,µν ∝ √ −t ′|µ−ν−µ′ +ν ′ |<br />
from angular momentum conservation.<br />
M 0−,++ ∝ √ −t ′ 0 ∝ const but h<strong>and</strong>bag amplitude ∝ t ′<br />
M 0−,++ -is determ<strong>in</strong>ed by twist 3 contribution → const .
Cross section of π + <strong>production</strong> at HERMES.<br />
300<br />
250<br />
W=3.83 GeV<br />
300<br />
250<br />
W=3.99 GeV<br />
Q 2 =2.45 GeV 2<br />
dσ/dt (π + ) [nb/GeV) 2 ]<br />
200<br />
150<br />
100<br />
50<br />
dσ/dt (π + ) [nb/GeV) 2 ]<br />
200<br />
150<br />
100<br />
50<br />
0<br />
LT<br />
T<br />
TT<br />
L<br />
0<br />
0.2 0.4 0.6<br />
Q 2 =3.44 GeV 2 0.0 0.2 0.4 0.6<br />
-50<br />
-t'[GeV 2 ]<br />
-t'[GeV 2 ]<br />
Cross section of π + <strong>production</strong> at<br />
HERMES with HERMES data.<br />
The partial cross section<br />
dσ L /dt, dσ T /dt, dσ LT /dt, dσ TT /dt .<br />
Information about different amplitudes<br />
contribution.
Asymmetry <strong>in</strong> π + <strong>production</strong> at HERMES.<br />
A UT<br />
s<strong>in</strong> (φ- φ S<br />
)(π + )<br />
1.0<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
0.0<br />
-0.2<br />
-0.4<br />
-0.6<br />
-0.8<br />
-1.0<br />
0.0 0.2 0.4 0.6<br />
-t'[GeV 2 ]<br />
A UL<br />
(π + )<br />
0<br />
-1<br />
0.0 0.2 0.4 0.6<br />
-t'[GeV 2 ]<br />
A UT asymmetry of π + <strong>production</strong> at<br />
HERMES with prelim<strong>in</strong>ary HERMES<br />
data.<br />
A UL asymmetry of π + <strong>production</strong> at<br />
HERMES with prelim<strong>in</strong>ary HERMES<br />
data. Dashed l<strong>in</strong>e- twist 3 is omitted.
π 0 <strong>production</strong> at HERMES.<br />
10<br />
8<br />
W=3.99 GeV<br />
Q 2 =2.45 GeV 2<br />
0.6<br />
0.5<br />
s<strong>in</strong> φ S<br />
W=3.99 GeV<br />
Q 2 =2.45 GeV 2<br />
dσ/dt (π 0 ) [nb/GeV) 2 ]<br />
6<br />
4<br />
2<br />
0<br />
T<br />
LT<br />
0.2 0.4 0.6<br />
A UT<br />
(π 0 )<br />
0.4<br />
0.3<br />
0.2<br />
0.1<br />
0.0<br />
s<strong>in</strong> (φ- φ S<br />
)<br />
0.2 0.4 0.6<br />
-t'[GeV 2 ]<br />
-t'[GeV 2 ]<br />
Cross section of π 0 <strong>production</strong>. Unseparated<br />
cross section- black, red-transverse,<br />
blue-longitud<strong>in</strong>al-transverse <strong>in</strong>terference<br />
cross section.<br />
s<strong>in</strong>φ S <strong>and</strong> s<strong>in</strong>φ − φ S moments of A UT<br />
asymmetry for π 0 <strong>production</strong> at<br />
HERMES<br />
No pion pole contribution here → small cross section . At COMPASS π 0 cross section smaller<br />
then 1nb/GeV 2
Conclusion<br />
• We analyse meson electro<strong>production</strong> with<strong>in</strong> h<strong>and</strong>bag approach.<br />
• Modified PA is used to calculate hard subprocess amplitude.<br />
• PDF:<br />
–gluon, sea, valence PDFs -from CTEQ6 parameterization.<br />
–polarized ˜H from BB parameterization.<br />
– E distributions -from Pauli nucleon FF analyses.<br />
• <strong>GPDs</strong> are calculated us<strong>in</strong>g these PDF on the bases of DD representation.<br />
• Direct <strong>GPDs</strong> extraction is impossible .<br />
Our way: parameterization of <strong>GPDs</strong> , amplitudes calculation <strong>in</strong> h<strong>and</strong>bag approach ;<br />
comparison with experiment.<br />
• We describe f<strong>in</strong>e :<br />
–cross section of <strong>light</strong> meson <strong>production</strong>: corrections ∼ k 2 ⊥ /Q2 <strong>in</strong> propagators are important.<br />
–<strong>sp<strong>in</strong></strong> observables: R -ratio , SDME <strong>and</strong> different asymmetries for various meson <strong>production</strong>.<br />
• Vector meson electro<strong>production</strong>-can be an excellent object to study <strong>GPDs</strong>.<br />
In different energy ranges, <strong>in</strong>formation about quark <strong>and</strong> gluon <strong>GPDs</strong> can be extracted from the<br />
cross section <strong>and</strong> <strong>sp<strong>in</strong></strong> observables of the vector meson electro<strong>production</strong>.