SATURATION 2005

( mini-review)

Eugene Levin, Tel Aviv University

DIS'05, April - May 2005 The Goal:

Brief review of ups and downs of highdensity QCD approach in the saturation

domain - my personal selection Saturation E. Levin 1

Outline:

* Saturation at HERA and RHIC?

* Reliable predictions for LHC:

- Saturation scale; - Modified Balitsky-Kovchegov equation; - Antishadowing effect; - Excess for heavy quark and di-jet production at LHC ;

* Theory developments:

- Colour dipoles versus B-JIMWLK approach; - Hunt for Pomeron loops; - Saturation and stochastic processes;

* Problems, ideas, solutions ... t^ bright future ;

Saturation E. Levin 2

What we have learned about saturation at HERA and RHIC

Low Q2: Large Q2:

0.1 0.2 0.3 0.4 0.5

0.2 0.4 0.6 0.8

0.5

1

0.5

1

0.5

1 1.5

0.5

1 1.5

0.5

1 1.5

0.5

1 1.5

0.5

1 1.5

HERA

* E. Gotsman, E. Levin, M. Lublinsky and U. Maor (2002);

* E. Iancu, K. Itakura and S. Munier (2003)

Saturation E. Levin 3

A^2 : 10-4 10-2

x

Q 2 (GeV2 )1 2 4

31

2

A : 1; B : 1 + 3; C : 1 + 2; D: 1 + 2 + 3 + 4 ;

Region 2 ? 2.7 ?

A^2/d.o.f. points A^2/d.o.f. points A 0.55 51 0.75 53 B 0.55 55 0.8 58 C 0.75 84 1.2 91 D 0.7 103 1.1 112

Saturation E. Levin 4

dNch/dh W = 130 GeV

 0 - 6 % 15 - 25 % 35 - 45 %

h 0

100 200 300 400 500 600 700

-5 -4 -3 -2 -1 0 1 2 3 4 5

2( dNch/dh)/Npart W = 130 GeV  h

 < 1 2 - 2.4

3 - 3.4 4 - 4.4

Npart 0.5

1 1.5

2 2.5

3 3.5

4 4.5

50 100 150 200 250 300 350 400 450

dNch/dh W = 200 GeV

 0 - 6 %

15 - 25 % 35 - 45 %

h 0

100 200 300 400 500 600 700 800

-5 -4 -3 -2 -1 0 1 2 3 4 5

2( dNch/dh)/NpartW = 200 GeV  h

 < 1 2 - 2.4

3 - 3.4

4 - 4.4

Npart 0.5

1 1.5

2 2.5

3 3.5

4 4.5

50 100 150 200 250 300 350 400 450 500

* RHIC Multiplicity ( Kharzeev, Levin & Nardi (2001))

pT (GeV) RCP

0 0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2

0 1 2 3 4 5

pT (GeV)

RCP

0 0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2

0 1 2 3 4 5 p

T (GeV)

RCP

0 0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2

0 1 2 3 4 5 pT (GeV)

RCP

0 0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8

2

0 1 2 3 4 5

* RHIC d-A dN/dy (Kharzeev, E.L. and McLerran (2002); Kharzeev, Kovchegov & Tuchin (2004))

Saturation E. Levin 5

Prediction for LHC:

Saturation scale Saturation momentumQ

s (GeV)

Y

1 2 3 4 1

10

6 8 10 12 14

1 - LO BFKL 2 - Our kernel; 3 - NLO BFKL (Durham); 4 - BGW model;

Saturation E. Levin 6

Theory progress: Energy dependence ( Munier & Peschanski (2000))

Saturation momentum

Qs (GeV)

Y

1 2

3 4

1

10 10 2

6 8 10 12 14

1 - High energy behaviour (fixed o""S ); 2 - Low energy corrections (fixed o""S ); 3 - High energy behaviour (running o""S ); 4 - Low energy corrections (running o""S );

Saturation E. Levin 7

Transferred momentum dependence

* Phenomenology and models : (Munier,Stasto & Mueller (2001), Munier

& Wallon (2003), Kowalski & Teaney (2003))

* Semiclassical approach: (Bondarenko, Kozlov & Levin (2003))

* Numeric solution to BK equation: (Golec-Biernat & Stasto (2003),

Gotsman, Kozlov, Levin, Maor & Naftali (2004))

* Analytic approach: (Ikeda and McLerran (2002), Marquet, Peschanski &

Soyez (2005))

Q2s(Y ) = q2 exp s, _o""S A^(r^cr)1-r^cr Y - 32(1-r^cr) lnY t^ = (q2/Q20) Q2s(Y , q = 0)

Saturation E. Levin 8

B-K non-linear equation: x1x

0

x 1 x0

x2

NN ( x ) ( x12 ) N ( x02 )01 N ( x12 ) . N( x 02 )

@ N (y, ~x01, ~b)

@ y =

CF o""S

2ij2 Z d

2x2 x

201

x202 x212 t,2N (y, ~x12,

~b - 1

2 ~x02)

- N (y, ~x01, ~b) - N (y, ~x12, ~b - 12 ~x02) N (y, ~x02, ~b - 12 ~x12)u""

Saturation E. Levin 9

Deficiencies of B-K equation:

* Correct only in LLA approximation of pQCD with BFKL kernel

in LO;

* The mean field approximation to the JIMWLK equation;

* It is not correct in the saturation region;

* The region where we can neglect the non-linear corrections

should be specified by conditions beyond the BK equation;

Saturation E. Levin 10

B-K equation versus NLO BFKL: Nnon-linear term / o""4S s2\Delta BF KL ; Nlinear term / o""2S s\Delta BF KL;

with \Delta BF K L = o""S A^LO BF K L + o""2S A^N LO BF K L

Correct strategy (theory point of view ):

For 1/o""S > y = ln s > 1 N LOBF K Llinear term

For y = ln s > (2/o""S ) ln(1/o""S ) N LOBF K Llinear term + Nn-l term For y = ln s > (1/o""2S ) N N LOBF K Llinear term + Nn-l term

Saturation E. Levin 11

Our suggestion:( Ellis, Kunszt and Levin 1994) _o""S A^N LO (r^ ) = - ! _o""S A^LO (r^ ) ; !(r^ ) = _o""S (1 - !) _o""S A^LO (r^ ) ; r^ DGLAP = _o""S a, 1! - 1c' ;

Saturation E. Levin 12

Modified B-K equation:

* @N (r, Y ; b)@ Y = CF o""Sij2 Z d

2r0 r2

(~r - ~r 0)2 r02

"

2N "r

0, Y ; ~b - 1

2 (~r

- ~r

0)<< - N "r0, Y ; ~b - 1

2 (~r

- ~r

0)<< N "~r - ~r 0, Y ; b - 1

2~r

0<< << B

-Kterm -

- @@ Y " 2N "r

0, Y ; ~b - 1

2 (~r

- ~r

0)<< - N "r0, Y ; ~b - 1

2 (~r

- ~r

0)<< N "~r - ~r 0, Y ; b - 1

2~r

0<<<< new

_o""S ! A^LO (r^ ) has the following form in Y , r representation

_o""S ! A^LO (r^ ) ! _o""S Z KLO (r, r

0) d2 r0 @ N (Y , r

0)

@ Y

Saturation E. Levin 13

Modified B-K equation versus B-K equation:

 0  0.2  0.4  0.6  0.8

 1

 6  8  10  12  14 N

y

b=0

r/R = 0.5

 0  0.2  0.4  0.6

 6  8  10  12  14

y

b=10 BKBK-modified

 0.1

 1  10  100  1000

 0  2  4  6  8  10  12 Qsat

b

y=14

 0.1

 1  10  100

 0  1  2  3  4  5  6  7

b

y=11

BKBK-modified

Saturation E. Levin 14

 0.1

 1  10  100  1000

 10  11  12  13  14 Qsat

y

b=0

 0.1

 1  10  100  1000

 10  11  12  13  14

y

b=5

BKBK-modified

 10  20  30  40  50  60  70  80

 4  6  8  10  12  14  16 < b 2 >

y

r/R=0.5

 10  20  30  40  50  60  70  80  90

 4  6  8  10  12  14  16

y

r/R=1

BKBK-modified

Saturation E. Levin 15

d? dyd2pt /

o""S

p2t Z d

2kt A'(k2

t ) A'((~p - ~k)

2 t )

10-3 10-4 10-5 10-6

x

1

5 10

50 100

500

kt2

D z^ \Phi NL Hx, kt LR^\Phi L Hx, kt L

85

% 90

% 95

%

100

% 110

% 120

% 110 %

120 %

Qs2HxL

Saturation E. Levin 16

Antishadowing effects: Z

d x A'N L(k2, Y ) = C onst(k2) = Z d x A'L(k2, Y )

Plot of R - 1 = (A'N L - A'L)/A'L

-1 -0.8 -0.6 -0.4 -0.2

 0  0.2

 6  8  10  12  14  16 (f NL-f

L )/f NL

y

kt = 1 -1 -0.8 -0.6 -0.4 -0.2

 0  0.2  0.4

 6  8  10  12  14  16 (f NL-f

L )/f NL

y

kt = 10

-1 -0.8 -0.6 -0.4 -0.2

 0  0.2  0.4

 6  8  10  12  14  16y

kt = 5 -1 -0.8 -0.6 -0.4 -0.2

 0  0.2  0.4  0.6

 6  8  10  12  14  16y

kt = 20

Saturation E. Levin 17

Excess of heavy quarks: (Eskola, Honkanen, Kolhinen, J-W Qiu & Salgado (2002- ))

* Eskola, Kolhinen & Vogt (2004)

In GLR-MQ approach:

@2xG @ ln Q2 d@y =

@ 2xGDGLAP

@ ln Q2 @ y - _o""

2 S C

xG2

Q2

Saturation E. Levin 18

Theory development: Dipoles versus B-JIMWLK Dipoles (Mueller (94))

* Z (Y ; [u]) = Pi R Pn(...xi, yi...) Qni=1 u(xi, yi) dVi

* @ Pn@ Y = -Pi \Gamma (1 ! 2)N (Pn(...xi, yi...) -Pn-1(...xi, yi...))

* @ Z (Y ,[u])@ Y = \Gamma (1 ! 2)Nu(1 - u) s's'u Z (Y , [u])

( E.L. & Lublinsky (2002)) Initial conditions:

* Z (Y , [u = 1]) = 1 ;

* Z (Y = Y0, [u]) = u ;

Saturation E. Levin 19

CGC and JIMWLK equation:

\Gamma  \Gamma  \Gamma  \Gamma \Gamma  \Gamma  \Gamma  \Gamma  \Gamma  \Gamma  \Gamma  \Gamma \Gamma  \Gamma  \Gamma  \Gamma 

\Delta  \Delta  \Delta  \Delta \Delta  \Delta  \Delta  \Delta  \Delta  \Delta  \Delta  \Delta \Delta  \Delta  \Delta  \Delta 

time(z)

t  E/m2 (t1 , z 1 )(t , z

2 (t

i , z )i(t , z )

( t' , z'1 1 ) )2 (t'2 , z'2 )

i i( t' , z' )( t' , z' )

interaction time

i+1 i+1 i+1 i+1

t=t0

: L(r) + ju .Au +L(A) At t=t 0 : L(r ) + ju Au +L(A).

Classical Fields Quantum  Fields

t1 - t'1 >> t 2 2- t' >> ... >> t i - t'i >> t i+1 - t'i+1

' At t=t0

Quantum  Fieldst=t'FieldsClassical 0

Saturation E. Levin 20

B- JIMWLK A~! BFKL Pomeron Calculus: ( Iancu & Mueller, Mueller & Shoshi, Iancu & Triantafyllopoulos, McLerran & . . . , . . . , . . . , Kovner & Lublinsky)

Good news:

* B- JIMWLK = BFKL Pomeron Calculus;

* B- JIMWLK ! all BFKL Pomeron vertices;

* B- JIMWLK ! the first operator proof of Pomeron Calculus;

* B- JIMWLK does not lead to merging of Pomerons ;

* B- JIMWLK t^ dipole approach; Saturation E. Levin 21

Bad news:

* No progress has been achieved in Pomeron calculus;

* My nightmare: Lipatov is correct with his effective

Lagrangian;

Probabilistic interpretation: a hope?! " Reggeon field theory is equivalent to a chemical process where a radical can undergo diffusion, absorption, recombination, and autocatalytic production. Physically, these "radicals" are wee partons (colour dipoles)."

( P. Grassberger & K. Sundermeyer: "Reggeon Field

Theory and Markov processes" (1978))

( Grassberger & Sundermeyer (1978), E.L. (1992), Boreskov (2004))

Saturation E. Levin 22

Probabilistic interpretation: Colour dipoles: 1. the wee partons of the BFKL Pomeron; 2. the correct degrees of freedom at high energy;

The typical death-birth process ( Markov chain ):

* @ Pn@ Y = -Pi \Gamma (1 ! 2)N (Pn(...xi, yi...) -Pn-1(...xi, yi...))

Saturation E. Levin 23

DOF: 1/Nc2

1/Nc2

leading Nc order 1/N2c corrections 1/Nc2 correctionsx

y

1

1 x2

2y

x1 y1

x y

2

x1 y1

2

x 2 y2

a) b) c) topology 1 BFKL Pomeron 2 BFKL Pomeron

topology

Multi Reggeon

MRP\Delta >>  \Delta  2 

Pomeron

P -> 2 P 2P -> MRP2 P -> 3 P

Saturation E. Levin 24

Hunt for Pomeron loops: y

y

1

2 0

Y

\Gamma 

\Gamma 

( 1 -> 2 ) ( 2 -> 1 ) \Gamma  (1 -> 2) = \Gamma  (2-> 1) In Reggeon Calculus

In probabilistic interpretation  we need to find a correct  normalization for  (2-> 1)\Gamma 

E.L &  Lublinsky, Kovner & Lublinsky, Rembiesta & Stasto, .....................................)

( Salam, Mueller & Salam,

Iancu & Mueller,

Mueller, Munier & Shoshi, Iancy & Triantafyllopoulos,

Mueller, Shoshi & Wong,

Saturation E. Levin 25

Markov process with Pomeron loops:

* @Pn@Y = - Pi \Gamma (1 ! 2) N (Pn(...xi, yi...) - Pn-1(...xi, yi...))

P P

P P P

\Gamma (1->2)

- Pi,k \Gamma (2 ! 1) N (Pn(...xi, yi, ..., xk, yk, ...) - Pn+1(...xi, yi, ..., xk, yk, ...))

P P P \Gamma (2->1)

P P

P

- Pi,k \Gamma (2 ! 3) N (Pn(...xi, yi, ..., xk, yk, ...) - Pn-1(...xi, yi, ..., xk, yk, ...))

\Gamma (2->3) P PP

PP PPP

Saturation E. Levin 26

Toy model: @ Z

@ Y = - \Gamma (1 ! 2) u(1 - u)

@ Z

@ u +

\Gamma (2 ! 1) u(1 - u) @

2 Z

(@ u)2 + \Gamma (2 ! 3)u (1 - u)

2 @

2 Z

(@ u)2

\Gamma (1 ! 2) / _o""S ; \Gamma (2 ! 1) / _o""3S /N 2c ; \Gamma (2 ! 3) / _o""S /N 2c ;

\Gamma (1!2) \Gamma (2!1) /

N 2c

_o""2S A` 1 ;

\Gamma (1!2) \Gamma (2!3) / N

2 c A` 1

Saturation E. Levin 27

Asymptotic solution for \Gamma (2 ! 3) = 0: Equation:

* 0 = - \Gamma (1 ! 2) u(1 - u) @ Z (Y =

1, u)

@ u + \Gamma (2

! 1) u(1 - u) @

2 Z (Y = 1, u)

(@ u)2

Solution:

Z (u; Y ! 1) = 1 - B + B eu*(u

-1)

B = 1 from unitarity constraints Saturation E. Levin 28

N(Y) = \Sigma n 11 nn

Y

0 Y' N*(Y')

N (Y-Y') Unitarity constraint:

N (1) = P

1

n=1 (

-1

u* )

n n! !p

n(1) !

t n(1)

!n t^ @ nZ/@ un|u=1

Answer ( Boreskov (2004),E.L. (2005), Rembiesa & Stasto (2005)) :

N (Y = 1) = 1 - e

-u* < but 6= 1

Saturation E. Levin 29

Lessons:

* Asymptotic solution leads to a grey disc (not black!!!);

* Using the large parameters of our theory, the

semiclassical approach can be developed for searching for both the asymptotic solution and the correction to this solution;

* The corrections to the asymptotic solution decrease at

large values of Y and can be found from the Liouville-type linear equation;

* The important region of u are u ! 1 which should be

specified by using the unitarity constraint;

Saturation E. Levin 30

Saturation and sto chastic pro cesses: (Weigert, Blaizot,Iancu & Weigert, Iancu & Triantafyllopoulos, Mueller, Shoshi & Wong, E.L., . . . )

Poisson representation:

* Pn(Y ) = Z do"" F (o"", Y ) t, o""

n

n! e

-o""u"" t^ < Pn(o"") >

* Z (Y ; u) = Z do"" F (o"", Y ) e(u

-1)o""

* @ F (o"", Y )@ Y = - @@ o"" (A(o""F (o"", Y )) + 12 @

2

@ o""2 (B (o""F (o"", Y ))

Saturation E. Levin 31

do"" = A(o"") + pB (o"") dW (Y ) where dW (Y ) is a stochastic differential for the Wiener process

Could lead to a computation of the high energy amplitude using direct methods !!!

Saturation E. Levin 32

My conclusions:

* We have a good chance to make reliable estimates of the

non-linear effects at LHC energies;

* These effects lead not only to a suppression but also to an

increase of the inclusive cross sections for k>=Qs;

* Theory becomes dangerously complicated but very interesting;

* We still have a hope to develop a reliable computer procedure

to calculate the main properties of the new phase of QCD Colour Glass Condensate;

Saturation E. Levin 33