1. in the impact parameter represantation
ASI-2008
« SYMMETRY and SPIN »
PRAHA -2008
Spin-flip amplitude
in the impact parameter represantation
Oleg V. Selyugin
BLTPh,JINR

2. Structure of hadron elastic scattering amplitude
Introduction
Structure of hadron elastic scattering amplitude
Non-linear equations and saturation
Basic properties of the unitarization schemes
b-dependence of the phases
Analysing power in different unitarization schemes
Summary
P. , B. Nicolescu, O.V. Selyugin

3. Total cross-section Understand the asymptotic behavior of stot
new (precise) data to constraint the fit: stot vs (ln s)g
1% error  ~1mb

4. Predictions Model stot sel/stot r(t=0) B (t=0) COMPET 111 0.11 Marsel
103
0.28
0.12 19
Dubna
128
0.33
0.19
21.
Pomer.(s+h)
150
0.29
0.24
21.4
Serpukhov
230
0.67

5. Pomerons stot(s)~ Landshoff 1984 – soft P a = 0.08
Regge theory
stot(s)~
Landshoff 1984 – soft P
a = 0.08
Simple pole – (s/s0)a
Double pole – Ln (s/s0)
Triple pole – Ln2(s/s0) +C
1976 BFKL (LO) - P -a2 = 0.4
a2 = 0.45
1988 HERA data (Landshoff ) – hard P
2005 – Kovner – 7 P -ai =

7. Regge limit for fixed t The crossing matrix factorized in the limit
the Regge-pole contributions to the helicity amplitudes in the s-chanel
when exchanged Regge poles have natural parity

8. Impact parameter representation

9. Unitarity in impact parameter representation
S S+ = 1

10. Saturation bound

11. Non-linear equation (K-matrix)

12. Non-linear equation (eikonal)

13. Eikonal and K-matrix

14. Eikonal and K-matrix

15. Interpolating form of unitarization

16. Normalization and forms of unitarization
Low energies

17. Inelastic cross sections
eikonal
U-matrix

18. Asymptotic features of unitarization schemes
Low energies
High energies

19. Effect of antishadowing
S. Troshin (hep-ph/ v4 June 4
« Black Disk Limit is a direct consequence of the exponential unitarization with an extra assumption on the pure imaginary nature of the phase shift »
Renormalized eikonal

20. 2 TeV

21. 14 TeV

22. Corresponding phases

23. Renormalization of the U-matrix - K-matrix

24. RHIC beams +internal targets º
fixed target mode
Ö s ~ 14 GeV

25. Analysing power

27. Eikonal case

29. K-matrix
U-matrix

30. A N - eikonal

31. A N - eikonal

32. A N – Born term

33. A N – K-matrix

34. A N – K-matrix

35. A N – UT-matrix

36. A N – UT-matrix

38. A N – UT-matrix (New fit of from

39. Summary
Unitarization effects is very important for the description spin correlation parameters at RHIC
Non-linear equations correspond to the different forms of the unitarization schemes. They can have the same asymptotic regime.
An interpolating form of unitarization can reproduce both the eikonal
and the K-matrix (extended U-matrix} unitarization

40. Summary
The true form of the unitarization, which is still to be determined,
can help to determine the form of the non-linear equation,
which describes the non-perturbative processes at high energies.

41. END

42. The experiments on proton elastic scattering occupy
an important place in the research program at the LHC.
It is very likely that BDL regime will be reached at LHC energies It will be reflected in the behavior of B(t) and r(t).

43. Double Spin Correlation parameter

44. The Elastic Process: Kinematics
scattered
proton
(polarized)
proton beam
RHIC beams +
internal targets º
fixed target mode
Ö s ~ 14 GeV
polarized
proton target
or Carbon target
recoil proton
or Carbon
essentially 1 free parameter:
momentum transfer t = (p3 – p1)2 = (p4 – p2)2 <0
+ center of mass energy s = (p1 + p2)2 = (p3 – p4)2
+ azimuthal angle j if polarized !
Þ elastic pp kinematics fully constrained by recoil proton only !
С П И Н 0 5
Alessandro Bravar

45. Complex eikonal and unitarity bound

46. Some AN measurements in the CNI region
pC Analyzing Power
p = 21.7 GeV/c
PRL89(02)052302
pp Analyzing Power
p = 200 GeV/c
PRD48(93)3026
no hadronic
spin-flip
with hadonic
spin-flip
AN(%)
no hadronic
spin-flip
r5pC µ Fshad / Im F0had
Re r5 = ± 0.058
Im r5 = ± 0.226
highly anti-correlated -t
С П И Н 0 5
Alessandro Bravar

47. Spin correlation parameter - AN
ANbeam (t ) = ANtarget (t )
for elastic scattering only!
Pbeam = Ptarget . eB / eT

48. Soft and hard Pomeron Donnachie-Landshoff model;
Schuler-Sjostrand model

49. Such form of U-matrix does not have the BDL regime
Predictions at LHC (14 TEV)

50. Corresponding phases

51. Radius of the saturation
Low s

53. momentum transfer t = (p3 – p1)2 = (p4 – p2)2 <0
+ center of mass energy s = (p1 + p2)2 = (p3 – p4)2
+ azimuthal angle j if polarized !
Þ elastic pp kinematics fully constrained by recoil proton only

54. Soft and hard Pomeron Donnachie-Landshoff model;
Schuler-Sjostrand model
Cudell-Lengyel-Martynov-Selyugin

56. Corresponding phases 2

57. Extended forms of the unitarization
Ter-Martirosyan-Kaidalov
Giffon-Martynov-Predazzi
eikonal
U-matrix

58. Interpolating form of unitarization

59. Elastic scattering amplitude

60. Impact parameter representation and unitarization schemes

61. General form of Unitarization
Ter-Martirosyan, Kaidalov
Desgrolard, Martynov
Miettinen, Thomas
eikonal
K-matrix

64. Eikonal

65. Impact parameters dependence
The Froissart bound
Factorization

66. Bound on the region of validity of the U-matrix phase
Corresponding phases
Bound on the region of validity of the U-matrix phase

67. U-matrix

68. TANGH

69. Total cross sections at the LHC

70. The r parameter linked to stot via dispersion relations
sensitive to stot beyond the energy
at which is measured
predictions of stot beyond LHC energies
Or, are dispersion relations still valid at LHC energies?

71. If all particles are the same the amplitude do not changes with
2 last combination don’t satisfied the cgarge invariants and change signe of time
If all particles are the same the amplitude do not changes with
Change з1бз2 on л1бл2 and P->K K->P
So the 6 cobination also disappear.

72. Saturation of the Gluon dencity

73. Interpolating form of unitarization

74. Interpolating form of unitarization

75. A N – UT-matrix

76. Eikonal and renormalized U-matrix

77. The nuclear slope parameter b
t-region of 10-2  10-1 GeV2
The b parameter is sensitive to the exchange process
Its measurement will allow to understand the QCD based models of hadronic interactions
“Old” language : shrinkage of the forward peak
b(s)  2 ’ log s ; where ’ is the slope of the Pomeron trajectory  0.25 GeV2
Not simple exponential - t-dependence of local slope
Structure of small oscillations?

78. An-UfT
AN UmTei

79. Non-linear equations
U-matrix
eikonal