“low-x” physics j.manjavidze & a.sissakian

1

• Introduction • Models for soft processes

• Hard processes

• Saturation

• Equilibrium • “tQCD” -- new type perturbation theory • Conclusions

““Low-x” PhysicsLow-x” PhysicsJ.Manjavidze & A.Sissakian

ISMD-02, Alushta, Crimea

2

Introduction (1)

“Low-x” problem in pQCD

• transverse dimension:

• expansion parameters:

• DIS structure function in LLA:

•Froissart limit:

Screening effects are essential! Screening effects are essential!

L.V.Gribov,E.M.Levin and M.G.Ryskin, Phys. Rep., 100 (1983) 1;...

•ISMD-02, Alushta, Crimea

sxsq ||

1~)/ln(~)/1ln( 20

2 qqx ss

)ln(ln)/1ln(~),(ln 22 qxxqF

))/(ln(),( 2222 xqqOxqF

3

Introduction (2)

VHM and the “low-x” problem

• Very High Multiplicity: - mean multiplicity

• multiplicity is not too high:

• inelasticity coefficient for VHM processes is large:

J.Manjavidze, El. Part. At. Nucl., 16 (1985) 101J.Manjavidze and A.Sissakian, JINR Rap. Comm., 5/31 (1988) 5


1/1 max E

)(snn

mEnn /max

4

Introduction (3)

Phenomenology of VHM processes

• Generating function

inverse problem:

equation “of state”:

asymptotic estimation:

“chemical potential”:

ISMD-02, Alushta, Crimea

msnszszt n

n

n

n /),(),( max1

max

),(2

1|),(

!

110 szt

z

dz

iszt

zn nzn

),(),(ln snzsztz

zn c

max)},,(lnexp{ nnsnzn cn

))(/)(ln()/(),(ln),( EEnEnzEn totnc

5

Introduction (5)

VHM: definition

“Big partition function”

StatementStatement: at ,

is the leftist singularity of

J.Manjavidze and A.Sissakian, JINR Rap. Comm., 5/31 (1988) 5


)(),(1

szszT nn

n

n)(),( szsnz sc

),( szT

max)},,(lnexp{ nnsnzn cn

)(szs

6

Introduction (6)

Classification of asymptotics

• -- multiperipheral models:

• -- (semi)hard processes:

• -- unstable vacuum (first order ph. tr.)

Asymptotic classes:

• -- multiperipheral models

• -- (semi)hard processes

• -- unstable vacuum

J.Manjavidze and A.Sissakian, Phys. Rep., 346 (2001) 1

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1sz

)( nn

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)( nn

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)( nn

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),( ENzc

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01),( ENzc

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7

Models

Multiperipheral kinematics:

• longitudinal momenta:

• transverse momenta:

•Multiperipheral (Regge) anzats:

• the «superpropagator»

The LLA of pQCD gives:

•E.Kuraev, L.Lipatov and V.Fadin, Sov. Phys. JETP, 44 (1976) 443•V.Fadin, talk at present Conference•L.Lipatov, talk at present Conference

nppp ...21

.constq

}ln))1()0(')(exp{(),( 2 szqozqP

2.0,55.02ln12

1)0( ss


8

VHM Solutions: Multiperipheral Model

Critical Pomeron,

• “Pomeron weak coupling model” leads to

• “Pomeron strong coupling model” leads to

Above-critical Pomeron,

• Model may predict singularity at

Dual resonance model.

• Mass spectrum of resonances

• Resonance decay onto hadrons is assumed Poissonian

The model gives :

Conclusion: MP kinematics predicts:


0

sz

1sz0

1sz

2/5,,)( 00 constemm m

)(),(

);(),(2

2

snneO

snneO

nn

nn

)( nn eO


9

Range of validity of MPM

Range of validity of the multiperipheral anzats

• to produce the multiplicity one

must use Pomerons

• mean impact parameter

• to exclude short distance interactions:

J.Manjavidze, El. Part. At. Nucl., 16 (1985) 101

nsnsnb /)('/)('~22

)(2

snn

,...3,2,1,)(~ snn


10

Hard processes

Definitions

• DIS kinematics:

• structure function with gluons is

• LLA describes Brownian motion over coordinate

the time is

The mobility must be high:

Yu.L.Dokshitcer, D.L.Dyakonov and S.I.Troyan, Phys. Rep., 58 (1980) 271;...

);,( 2 nqxDabn),/1ln( x

2lnln q

1}lnln/)/1{ln( 2 qx

niconstxqqq in ,...,2,1,,... 222

21


11

LLA in VHM kinematics• One may introduce

The evolution (DGLAP) equation gives:

Considering jets creation,

probability to produce gluons in the gluon jet of the mass:

In result:

• The mobility depends on multiplicity:

LLA has finite range of validity in the VHM region LLA has finite range of validity in the VHM region

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),...,,()();,( 22

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1)},(/)/1{ln( zx


}2/exp{|| k

12

pQCD jets

• Evolution equation:

If

then the solution:

Introduction of infrared cut-off does not alter the estimation!

Jet «generate» moving singularity at

Conclusion: the tendency to weighting of pQCD jets in the VHM

domain is predicted


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0

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13

Renormalons• «Renormalons» reflect an uncertainty to pQCD

The infrared renormalon uncertainty

indicates necessity to include the non-perturbative effects

G.t’Hooft, «The why’s of subnucl. phys.» Erice, 1977; B.Lautrup, Phys.Lett., B69 (1978) 109

A.H.Mueller, Nucl.Phys., B250 ( 1985) 327, V.I.Zacharov, Nucl. Phys., B385 (1992)

R.Akhouri and V.I.Zakharov, hep-ph/9610492, V.I.Zakhrov, hep-ph/9811294

222 )/(~ q


14


15

Saturation: definitions

• Saturation: the occupation number of «low-x» gluons can not be arbitrary large

• Saturation scale:

Number of gluons:

becomes large in the weak coupling limit:

The dynamics becomes essentially classical!D.Kharzeev, E.Levin and M.Martin, hep-ph/0111315; A.H.Mueller, hep-ph/0111244; L.McLeran and R.Venugopalan, Phys.Rev., D49 (1994) 2233; D50 (1994) 2225; Yu.Kovchegov, Phys.Rev., D54 (1996) 5463NNN, present Conference

2

22 ),(

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16


17


18


19


20

Equilibrium

• Statement: multiple production hadron final state in the deep VHM region is equilibrium.

J.Manjavidze and A.Sissakian, Phys. Rep., 346 (2001) 1; See also: Proceedings of the Int. Workshops on «Very High Multiplicity

Physics»Dubna, 2000, 2001, 2002.


21

tQCD: references+idea

• Vacuum expectation value

N.N.Bogolyubov & S.Tyablikov, Zh. Eksp. Teor. Phys.,19 (1949) 256

R.Jackiw, C.Nohl and C.Rebbi, 1978; R.Jackiws, 1977

G.W.Mackey, 1969

N.P.Landsman & N.Linden, 1991; N.P.Landsman, 1991

C.J.Isham, 1984

Faddeev, in “Solitons”

L.D.Faddeev & V.E.Korepin, Phys. Rep., 42 (1978) 1

(Gauge invariance) (Unitarity condition) (Time reversibility)

k

ii

iSk peDpppG

1

)(21 )(),...,,(


22121 |),...,,(|),...,,( kk pppGpppG

22

tQCD: Dirac measure

• Unitary definition of measure

)(|),...,,(|1

),(2)(221 i

k

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)()(Re)(2

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)(

)()(Re2)()(),(

))()())(()(()( peppepp

x

xjx

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)(

)(()(


23

tQCD: YM theory on manifold• Differential measure :

• Perturbations generating operator

• Interactions generating functional

)2()4(/)2,4( OOO

),()(|))()(

)(()(

,

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•ISM D-02, Alushta, Crimea

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24

tQCD: generator of events

)]([|),...,,(| ),(2)(221 i

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)()(}){(1 3

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J.Manjavidze, J.Math.Phys. 41 (2000) 5710; J.Manjavidze and A.Sissakian, J.Math.Phys. 42 (2001) 641, 42 (2001) 4158; Theor. Math. Phys.; 123 (2000) 776; 130 (2002) 153; Phys. Rep., 346 (2001) 1; hep-ph/0201182

•ISM D-02, Alushta, Crimea

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Conclusions• It is impossible to consider the “low-x” domain without VHM

effects.

• tQCD

--- presents expansion over inverse interaction constant:

(i) no divergences

(ii) no phenomenological dimensional constant of -type

(iii) no “asymptotic freedom” (?)

--- each order is gauge invariant

(i) no Faddeev-Popov gauge fixing conditions

• tQCD “works” at arbitrary distances

• tQCD includes the pQCD as a “small distance” approximation

• tQCD presents expansion over Plank constant


“low-x” physics j.manjavidze & a.sissakian

Documents

vhm region ismd

multiperipheral models

phenomenology of vhm

vhm kinematicsone

high multiplicity

vhm domain

mean multiplicity multiplicity

sissakian introduction