13th International Conference on Elastic & Diffractive Scattering (13th "Blois Workshop") CERN, 29th June - 3rd July 2009

GPDs and hadron elastic scattering

O.V. SelyuginDubna, JINR

Contents

1. Introduction

2. GPDs and hadrons form-factors

3. t-dependence of the GPDs

6. Conclusion

4 Unitarization of the elastic scattering amplitude

5. The differential cross sections

Elastic scattering amplitude

pp pp pp pp

( )0,577... the Euler constant 1 2and are small correction terms

2 2 2 2 21 2 3 4 5

dσ=2π[|Φ| +|Φ| +|Φ| +|Φ| +4|Φ| ]

dt( , ) ( , ) ( )h e i

i i is t s t t e

1 2( , ) [ ln ( B (s, t) | t | / 2 ) ]s t

Unitarity

0

0

( , ) ( ) (1 exp[ ( , )])T s t is bdb J bq i s b

Factorization

( , ) ( ) ( );s b h s f b ( )h s s

4

Impact parameters dependence

Soft and hard Pomeron

Donnachie-Landshoff model;

Schuler-Sjostrand model

1 01 ln( / )1 1

0

( , ) [ ( ) t s ssT s t h e

s

2 02 ln( / ) 22

0

( ) ] ( )t s ssh e F t

s

Second part of scattering amplitude

1 22 1 2

0 0

( , ) [ ( ) ( ) ) ( )grs sT s t h h A b

s s

ASSUMPTION

1 2( ) ( ); ( ) ( );em grf b F b f b A b

General Parton Distributions -GPDs

Electromagnetic form factors(charge distribution)

Energy momentum tensor form factors (matter distribution)

Fallowing to A. Radyushkin

Phys.ReV. D58, (1998) 114008GPDs

0limit

1

1 1( ) , , ;q qF t dx H x t

1

2 1( ) , , ;q qF t dx E x t

q(x;t) = Hq(x,0,t) + H

q(-x,0,t) q(x;t) = Eq(x,0,t) + E

q(-x,0,t)

1

1 0( ) , , ;q qF t dx x t H

1

2 0( ) , , ;q qF t dx x t ε

Energy-momentum tensor

1 2 2

1[ , , , , ] ( ) ( ) ;q q

q qdx x H x t E x t A B

1

0( ) , ;q qA t dx x x t H

1

0( ) , ;q qB t dx x x t ε

Our ansatz

2

0.4

(1 ), ( ) ex ;p[ ]q x

x t q x a tx

H

q(x) is based on the MRST2002 and A. Radyushkin (2005)

0.69 3.50 0.5( ) 0.262 (1 ) (1 3.83 37.65 );u x x x x x 0.65 4.03 0.5( ) 0.061 (1 ) (1 49.05 8.65 );d x x x x x

1. Simplest;

2. Not far from Gaussian representation

4. Valid for large t

3. Satisfy the (1-x)n (n>= 2)

F1p *t2

Gep/Gd

GMp/(pGd)

GEn

GMn

Gravitational and Dirac form-factors

1 2 2

0( ) , ( 1.8, )q q

DA t dx x x t G GeV H

2 3

2 2 2 21 2

( ) ( ) ( )

*(1 ) / ( );

grf b A b K b

r b r b

PARAMETERS

Fixed:

1 1 2 2( , ) ( , ) ( , );s b k s b k s b

1 2 1 20.08; 0.45; 0.25; 0.1;

1 2/ 0.008 / 4.47.h h Free:

1 2 1 21.09; 1.57; 2.4 30.9.6;k k r r

2 2731/ 947 3.

( 52.8 GeV )pp p sm l tp s al

( 52.8 GeV ) argpp p l e tp s

( 62.1 GeV )p p sp p

( 52.8 GeV )pp p sm l tp s al

( 53 GeV )spp pp

( 62.7 GeV )p p sp p

( 541 GeV )spp pp

( 541 546 GeV )s andpp pp

( 1800 GeV )p p sp p

PREDICTIONS ( 10, 14 TeV )pp pp s

1.8 TeV; (0) 0.208; 80.3 ;tots mb

10 TeV; (0) 0.2 1 ;38; 32tots mb

14 TeV; (0) 0.2 1 ;35; 46tots mb

Lower energy

There is a small place for the secondary reggeons (with the intercept > 0,5 )

Non-linear equation (K-matrix)

(1 );dN

N Ndy

[ ] ( ,0);N y s

( , )[ ] ;

1 ( , )

i s bN y

s b

31

0ln( / );y s s

( )[ ] ;

1 ( )

y

y

s f bN y

s f b

2 5400 / 947 6;

Interpolating form of unitarization

( , ) / [1 (1 ( , ) / ) ];G s b i s b

1; 1 ( , )( , ) 1 ;

(1 ( , )) 1 ( , )

s bG s b

s b s b

.

( , ) 1 ( ( , );G s b exp s b

1/(1 [1 ] ) (1 );dN

N Ndy

32

Fit:

;

Experiment data choose the eikonal form

Summary

1. Proposed a new simple model of proton-proton and

proton-antiproton elastic scattering .

3 . Both distributions are obtained from our model of GPDs of the

hadrons. The corresponding electromagnetic and gravitational form factors of

the proton are calculated with our proposed t-dependence of the GPDs.

2. The model is based on the assumption that the scattering amplitude is a sum of terms proportional to the charge distribution of the hadron (dominant at small t) and terms proportional

to the matter distribution of the hadron (dominant at large t).

Summary

4. The model includes the contributions of the soft and hard pomerons with intercepts = 1+0.08 and 1+0.45 and slopes = 0.25 and 0.1 GeV-2.

5, The model describes all high-energy experimental data beginning from sqrt(s)=52.8 GeV in the Coulomb hadron interference region and at large |t| =10 GeV2.

(with 4 free parameters , 2 = 3 per point)6. We note the essential contribution of the hard pomeron at the LHC energy at small and large t. This leads to a large value of rho at small t.

7. We do not see the odderon contribution at small and large t.

END

END

THANKSFOR YOUR ATTANTION

Saturation bound

37

00( , ) 2 ( ) ( , )Bs b q J bq M s q dq

2 21( , ) [ ]

2s b dzV z b

k

G(s,b)

1 1 2 2( , ) ( , ) ( , );s b k s b k s b

1 1 2 2( , ) ( , ) ( ); ( , ) ( ) ( );s b h s b f b s b h s f b

1 21 2

0 0

( ) [ ( ) ( ) ]s s

h s h hs s