Oleg V. Selyugin
description
Transcript of 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
Corresponding phases
( , ) {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
Interpolating form of unitarization
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
Corresponding phases
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
Interpolating form of unitarization
73
Interpolating form of unitarization
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