“Low-x” Physics J. Manjavidze & A. Sissakian Introduction


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“Low-x” Physics J.Manjavidze & A.Sissakian

  • Introduction

  • Models for soft processes

  • Hard processes

  • Saturation

  • Equilibrium

  • “tQCD” -- new type perturbation theory

  • Conclusions


Introduction (1)

  • “Low-x” problem in pQCD

  • transverse dimension:

  • expansion parameters:

  • DIS structure function in LLA:

  • Froissart limit:

  • Screening effects are essential!

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



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) 101

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



Introduction (3)

  • Phenomenology of VHM processes

  • Generating function

  • inverse problem:

  • equation “of state”:

  • asymptotic estimation:

  • “chemical potential”:



Introduction (5)

  • VHM: definition

  • “Big partition function”

  • Statement: at ,

  • is the leftist singularity of

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



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



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



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:

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



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



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;...



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



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

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



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





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) 5463

  • NNN, present Conference











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.



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)



tQCD: Dirac measure

  • Unitary definition of measure



tQCD: YM theory on manifold

  • Differential measure :

  • Perturbations generating operator

  • Interactions generating functional



tQCD: generator of events



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



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