Oleg V. Selyugin

Oleg V. Selyugin

11

Spin-flip amplitude in the impact parameter represantation

ASI-2008 ASI-2008

« SYMMETRY and SPIN »« SYMMETRY and SPIN »PRAHA -2008

BLTPh,JINR

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

2

Total cross-sectionTotal cross-section Understand the asymptotic

behavior of tot

new (precise) data to constraint the fit: tot vs (ln s)

1% error ~1mb

3

Model el/ 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

PredictionsPredictions 14s TeV

Pomerons

Regge theory

Simple pole – (s/s0)Double pole – Ln (s/s0)

Triple pole – Ln2(s/s0) +C

Landshoff 1984 – soft

1988 HERA data (Landshoff ) – hard 1976 BFKL (LO) -

2005 – Kovner – 7

tot(s)~

66

1 1 1 1;

2 2 2 2

1 2 1 2 1

2 2 1 2 1

3 2 5 1 2 5 1

4 2 5 1 2 5 1

5 2 1 2 1 2 1 2 1

6

( , ) ( , ) ( ) ( ) ( ) ( )

( , ) ( ) ( ) ( ) ( )

( , ) ( ) ( ) ( ) ( ) ( ) ( )

( , ) ( ) ( ) ( ) ( )

( , )[ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )]

M s t T s t u p u p u k u k

T s t u p K u p u k P u k

T s t u p K u p u k P u k

T s t u p u p u k u k

T s t u p K u p u k u k u p u p u k P u k

T

2 1 2 1 2 1 2 1

7 2 5 1 2 5 1

8 2 5 1 2 5 1

( , )[ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )]

( , ) ( ) ( ) ( ) ( ) ( )

( , ) ( ) ( ) ( ) ( ) ( ) ;

s t u p K u p u k u k u p u p u k P u k

T s t u p u p u k P u k

T s t u p K u p u k u k

5

77

3 1 4 2

3 4 1 2 3 1 4 2

| | | |

0

( , ) [ ] ( ) (1 exp[ ])iB i ii

i

ss t g g t i

s

for fixed t

;s The crossing matrix factorized in the limit

the Regge-pole contributions to the helicity amplitudes in the s-chanel

( , ) ( , ) 0B Bs t s t

when exchanged Regge poles have natural parity

( , ) ( , ).B Bs t s t

Regge limit

Impact parameter representation

2

0( , ) ( , )

2iqbip

M s t s b e d b

00

1( , ) ( ) ( , )

2M s t b J b q s b db

Unitarity in impact parameter representation

S S+ = 14 2

21 2 1 2

1 1

(2 )Im , , | | , , ( ( )| |

2

n

r rp p out T p p in d p q T

0

0

( , ) ( ) ( , )T s t is b db J bq b s

( , ) 1;s b

2 2Im ( , ) | Im ( , ) | | Re ( , ) | inels b s b s b g

9

Saturation bound

1010

( , )00

1( , ) ( ) [1 ]

2s bT s t b J b q e db

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

00

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

2 21( , ) [ ]

2s b dzV z b

k

Non-linear equation (K-matrix)

(1 );dN

N Ndy

( , )( , ) ;

1 ( , )

K s bs b

i K s b

[ ] ( ,0);N y s

( , )[ ] ;

1 ( , )

i s bN y

s b

11

0ln( / );y s s

( )[ ] ;

1 ( )

y

y

s f bN y

s f b

Non-linear equation (eikonal)

( , ) 1 exp[ ( , )];N s b i s b

( [1 ]) (1 );dN

Ln N Ndy

12

Eikonal and K-matrixEikonal and K-matrix

dN

dy

13

Eikonal and K-matrixEikonal and K-matrix

14

Interpolating form of unitarization

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

15

Normalization and forms of unitarization

0

1( ) 8 (1 exp[ ( , )])

2tot s b db s b

( , ) 2 ( , );s b s b

0

( ) 4 (1 exp[ ( , )])tot s b db s b

0

( , )( ) 8 ;

1 ( , )tot

U s bs b db

U s b

1616

( , ) ( , ) ( , ) / 2;U s b s b s b

0

( , )( ) 4 ;

1 ( , ) / 2tot

s bs b db

s b

Low energiesLow energies ( , ) ( , ) ( , );eik U m Borns b s b F s t

Inelastic cross sections

0

( ) 2 (1 exp[ 2 ( , )])eikinel s b db s b

20

( , )( ) 2 ;

(1 ( , ) / 2)U m

inel

s bs b db

s b

17

( )1;

( )el

tot

s

s

eikonaleikonal

U-matrixU-matrix

(1 / );uu n u

dNN N c

dy 2;uc

Asymptotic features of unitarization schemes

Low energiesLow energies( , ) ( , );eikG s b s b

( , ) ( , );U mG s b s b

High energiesHigh energies ( , ) 1;eik satG s b b

( , ) 2;U m satG s b b

18

Effect of antishadowing

0

( ) 8 (1 exp[ ( , )])tot s b db s b

( , ) ( , )

0

( ) 8 e (1 e )s b s binel s b db

« Black Disk Limit is a direct consequence of the exponential unitarization with an extra assumption on the pure imaginary nature of

the phase shift »

S. Troshin (hep-ph/0701241 v4 June 4S. Troshin (hep-ph/0701241 v4 June 4

Renormalized eikonalRenormalized eikonal

19

( )1;

( )el

tot

s

s

2 TeV

20

14 TeV

21

Corresponding phases

1( , ) 2 { ( , )};

2U m eiks b tanh s b

1 ( , ) / 2( , ) { };

1 ( , ) / 2U m

eikU m

s bs b log

s b

1( ) (1 1 4 ( ) ),

2 2l l l

if s s f

1( ) (1 1 4 ( ) ), 1.

2 2l l l

if s s f

22

Renormalization of the U-matrix - K-matrix

0

( , )( ) 4 ;

1 ( , )tot

s bs b db

s b

2

2

( , ) ( , ) / 2( , )

(1 ( , ))inel

s b s bG s b

s b

( ) 1;

( ) 2el

tot

s

s

1 ( , ) / 2;

1 ( , ) / 2

s bS

s b

1

1 ( ,;

)S

s b

23

( , )( , ) 1 ;

1 ( , ) / 2

s bG s b S

s b

( , )( , ) 1 ;

1 ( , )

s bG s b S

s b

2424

RHIC beams +internal targets fixed target modes ~ 14 GeV

Analysing power

*2

4Im[ ]N nfl fl

dA F F

dt s

2| ( ) ( , ) |iC N

de F t F s t

dt

*1 22

4| || | sin( )N nfl fl

dA F F

dt s

NA

2626

2

0( , ) ( , )

2iqbip

M s t s b e d b

ˆ ( , )00

1( , ) ( ) [1 ]

2s bT s t b J b q e db

1 2( , ) ( , ) ( , )( ) ( )c LSs b s b i s b b l

2727

( , )00

( , ) ( ) [1 ]c s bnfF s t ip b J b q e db

( , )210

( , ) ( ) c s bsf LSF s t p b J b q e db

Eikonal case

2828

3 4 1 2 3 4 1 2 3 4 3 4( , ) ( , ) ( , ) ( , )

8B B

kF p q U p q i d U p k F k q

2 2 21 1 2 3 4 1 2 5

1 21 2 1 2 3 4 5

1

1

( )(1 ) 2(1 2 2 )( , )

(1 )(1 )(1 ) 4

;1

u u u u u u u us b

u u u u u u u

u

u

5 55 2 2

1 2 3 4 5 1

( , ) ;(1 )(1 ) 4 (1 )

u us b

u u u u u u

2 51 20

1

( , ) ( )(1 )sf

uF s t p b J b q db

u

2929

2 2

2

(1 )1( , ) 1 1

(1 ) (1 ) (1 ) ( )

;1 (1 )

c sf c sf

c sf c sf c sf

sfc

c c

s b

K-matrix

U-matrix

2( , ) ;

(1 / 2 / 2) 1 / 2 (1 / 2)c sf sfc

c sf c c

s b

3030

A N - eikonal

( , ) (1 exp[ ( , )])Eiknfs b s b

( , ) ( , ) exp[ ( , )])Eiksf nfs b s b s b

50 ;GeVs

3131

A N - eikonal

( , ) (1 exp[ ( , )])Eiknfs b s b

( , ) exp[ ( , )])Eiksf nfs b s b

50 ;GeVs

500 ;GeVs

3232

A N – Born term

( , ) ( , )Bornnfs b s b

( , ) ( , )Bornsfs b s b

500 ;GeVs

3333

A N – K-matrix

( , )( , )

1 ( , )nfK m

nf

s bs b

s b

2

( , )( , )

(1 ( , ))sfK m

nf

s bs b

s b

50 ;GeVs

3434

A N – K-matrix

( , )( , )

1 ( , )nfK m

nf

s bs b

s b

2

( , )( , )

(1 ( , ))sfK m

nf

s bs b

s b

500 ;GeVs

50 ;GeVs

3535

A N – UT-matrix

( , )( , )

1 ( , ) / 2nfU m

nf

s bs b

s b

2

( , )( , )

(1 ( , ) / 2)sfU m

nf

s bs b

s b

50 ;GeVs

3636

A N – UT-matrix

( , )( , )

1 ( , ) / 2nfU m

nf

s bs b

s b

2

( , )( , )

(1 ( , ) / 2)sfU m

nf

s bs b

s b

50 ;GeVs 500 ;GeVs

3737

1( , ) 2 { ( , )};

2U m eiks b tanh s b

50 ;GeVs

500 ;GeVs

( , )/2 ( , )/2

( , )/2 ( , )/2( , ) 2 ;eik eik

eik eik

s b s b

U m s b s b

e es b

e e

3838

A N – UT-matrix (New fit of from

( , )( , )

1 ( , ) / 2

fitnfU m

fitnf

s bs b

s b

2

( , )( , )

(1 ( , ) / 2)sfU m

fitnf

s bs b

s b

d

dt

50 ;GeVs

500 ;GeVs

;( , ) ( , )fit Bornnf nfs b s b

( , )fitnf s b

Summary

Non-linear equations correspond to the different forms of the unitarization schemes. They can have the same asymptotic regime.

39

Unitarization effects is very important for the description spin correlation parameters at RHIC

An interpolating form of unitarization can reproduce both the eikonal

and the K-matrix (extended U-matrix} unitarization

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.

40

ENDEND

41

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 (t).

Double Spin Correlation parameter

4343

NNA

4*32

*1

252 Re2

4),(

sdtd

tsANN

С П И Н 0 5 Alessandro Bravar

RHIC beams +internal targets fixed target modes ~ 14 GeV

The Elastic Process: Kinematics

recoil protonor Carbon

(polarized)proton beam

scatteredproton

02 inout pptpolarized

proton targetor Carbon target

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 if polarized !

elastic pp kinematics fully constrained by recoil proton only !

Complex eikonal and unitarity bound

0.50.75

1

1.25

1.5

x

0

0.5

1

1.5

2

y

1

1.25

1.5

1.75

2

Im Geik

1

1.25

1.5

1.75

2

Im Geik

1

2

3

x

0

0.5

1

1.5

2

y

0

0.5

1

1.5

2

Im Geik

0

0.5

1

1.5

2

Im Geik

( , ) (1 exp[ ( , ) ( , )])G s b Y s b i X s b

45

С П И Н 0 5С П И Н 0 5 Alessandro BravarAlessandro Bravar

Some AN measurements in the CNI region

pp Analyzing Power

no hadronicspin-flip

-t

AN

(%)

E704@FNALp = 200 GeV/c

PRD48(93)3026

E950@BNLp = 21.7 GeV/c

PRL89(02)052302

with hadonicspin-flip

no hadronicspin-flip

pC Analyzing Power

r5pC Fs

had / Im F0had

Re r5 = 0.088 0.058

Im r5 = 0.161 0.226

highly anti-correlated

Spin correlation parameter - AN

4747

ANbeam (t ) AN

target (t )

for elastic scattering only!

Pbeam = Ptarget . B / T

Soft and hard Pomeron

Donnachie-Landshoff model;

Schuler-Sjostrand model

1 01 ln( / )1

0

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

s

2 02 ln( / ) 22

0

( ) ] ( )t s ssh e F t

s

4949

Such form of U-matrix does not have the BDL regimeSuch form of U-matrix does not have the BDL regime

Predictions at LHC Predictions at LHC (14 TEV)(14 TEV)

( ) 230 ;tot s mb ( )

0.67;( )

el

tot

s

s


( , ) {1 ( , )};eik U ms b log s b

( , )( , )( , ) {1 };eik s bs bU m s b e e

50

Radius of the saturation

2( ) ( );R s Ln s

2 2( ) ( ).R s Ln ss Low s

/ 0.1*0.3 0.03soft IPomeron

/ 0.4*0.1 0.04hard IPomeron

'2 / 2

0 0( ) 4[ ( ) [ ( )] ( ) ( )]2 2( ( ))

h hR s R Ln R Ln Ln s Ln s

B d Ln s

1ing 2 22 Im ( , ) | Im ( , ) | | Re ( , ) | 0ins b s b s b g

( , ) [1 (1 ) ( )]ins b i g exp i

( , ) [1 ( )]s b i exp i

( , ) 1 exp[ ( , )]

1 exp[ Re ( , )][ ( ) ( )]

s b i s b

s b cos i sin

52

5353

momentum transfer t = (p3 – p1)2 = (p4 – p2)2 <0+ center of mass energy

s = (p1 + p2)2 = (p3 – p4)2 + azimuthal angle if

polarized !

elastic pp kinematics fully constrained by recoil proton

only

Soft and hard Pomeron

Donnachie-Landshoff model;

1 01 ln( / )1

0

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

s

2 02 ln( / ) 22

0

( ) ] ( )t s ssh e F t

s

Schuler-Sjostrand model

Cudell-Lengyel-Martynov-Selyugin

54

55

[ ];dN

Expdy

Corresponding phases 2

56

Extended forms of the unitarization

0

1( ) 8 (1 exp[ 2 ( , )])

2tot s b db C s bC

0

( , )( ) 8 ;

1 2 ( , )tot

s bs b db

C s b

Ter-Martirosyan-KaidalovTer-Martirosyan-Kaidalov

Giffon-Martynov-PredazziGiffon-Martynov-Predazzi

1;

2C

1.2;C

0

1( ) 8 (1 exp[ 2 ( , )])

2tot s b db C s bC

1

1 ( )( , ) 2 ( , )

2 !n

n

G ns b i s b

i n

1( ) nG n C

1( ) ! nG n n C

eikonaleikonal

U-matrixU-matrix

57


24

68

10

g

00.25

0.5

0.75

1b

0.4

0.5

0.6

0.7

Re Geik

0.4

0.5

0.6

0.7

Re Geik

58

Elastic scattering amplitude

2 2 2 2 21 2 3 4 52 [| | | | | | | | 4 | | ]

d

dt

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

( , ) ( , ) ( )h e ii i is t s t t e

pp pp pp pp

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

59

0

0

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

0

0

( , )( , ) ( ) ;

1 ( , )

U s bT s t is b db J bq

iU s b

Impact parameter representation and unitarization schemes

0

0

( , ) ( ) ( , )T s t is b db J bq b s

60

General form of UnitarizationGeneral form of Unitarization

1( , ) [1 exp( ( , ))];s b n s b

n

Ter-Martirosyan, KaidalovTer-Martirosyan, Kaidalov

Desgrolard, MartynovDesgrolard, Martynov

1 1( , ) [1 ]

(1 ( , ))n ns b

n s b

( , )( , ) 1 [1 ]

s bs b

Miettinen, ThomasMiettinen, Thomas

eikonaleikonal

1 K-matrixK-matrix

0.4 2

0.4 2

2

( , )[ ] ;

1 ( , )

0.1 [( / 3) ]

1 0.1 [( / 3) ]

1 [ 0.1 [( / 3) ]]

i s bN y

s b

s Exp bN

s Exp b

N Exp s Exp b

1.3 1.3 1.3 1.3

0

(1 ) [ (1 ) ]z

x x Exp x x dx 6262

3 1 4 2

3 4 1 2 3 4 1 2 3 1 4 2

| | | |

, 0

( , ) ( , ) ( ) [ ] ( ) (1 exp[ ])iB i ii

ss b U s b i s g g t i

s

1 3( , ) ( , ) ( )

( , ) ( ) iN C

F s t s t t

F s t F t e

2( ) 2 ( ) / | |CF t G t t

( , ) (ln ( , )),d d

B s t s tdt dt

Re ( , )( , ) ;

Im ( , )N

N

F s ts t

F s t ( ) 4 Im ( , )tot s F s t

63

Eikonal

( , ) / 2{1 ( [ / 2])}s b i Exp b Exp i

{1 [ [ / 2]]*[cos( sin( / 2)) sin( sin( / 2)]2

iExp b cos b i b

64

Impact parameters dependence

0

0

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

The Froissart bound

2( ) log ( )tot s a s

Factorization

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

65


Bound on the region of validity of Bound on the region of validity of the U-matrix phasethe U-matrix phase

66

U-matrix

( , ) exp[ / 2)] /(1 exp[ / 2)]2

is b b i b i

2

1{1 (1 [ / 2] sin( / 2)]}

2 1 2 [ / 2]

ib cos i b

b bcos

67

TANGH

2 2(1 );dN

N Ndy

( , ) [ ( , )]s b Tanh s b

68

Total cross sections at the LHC

69

linked to tot via dispersion relations

sensitive to tot beyond the energy

at which is measured

predictions of tot beyond LHC energies

Or, are dispersion relations still valid at LHC energies?

The The parameter parameter

70

7171

2 last combination don’t satisfied the cgarge invariants and change

signe of timeIf 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.

Saturation of the Gluon dencity

[ ];dN

Expdx

( ) (1 )xG x x x

72


73


74

7575

A N – UT-matrix

( , )( , )

1 ( , ) / 2

fitnfU m

fitnf

s bs b

s b

2

( , )( , )

(1 ( , ) / 2)sfU m

fitnf

s bs b

s b

Eikonal and renormalized U-matrix0

2.5

5

7.5

10

g

0

2500

5000

7500

10000s, GeV ^H- 2L

0.2

0.4

0.6

GHs,bL

0.2

0.4

0.6

GHs,bL

76

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?

The nuclear slope parameter bThe nuclear slope parameter b

77

7878

AN UmTeiAn-UfT

( , )( , )

1 ( , )eik fitU

fit

s bs b

s b

( , )( , )

1 ( , ) / 2eik fitU

fit

s bs b

s b

2

( , )( , )

(1 ( , ))eik

sU

fit

s bs b

s b

2

( , )( , )

(1 ( , ) / 2)eik

sU

fit

s bs b

s b

eikonaleikonal U-matrixU-matrix

0.1

1 0.1

sN

s

(1 );dN

N Ndy

( log[1 ]) (1 );dN

N Ndy

1 [ 0.1 ]N exp s

Non-linear equationsNon-linear equations ( ) ( , )fixN y s b

79

Documents

Oleg V. Selyugin