“Low-x” Physics J.Manjavidze & A.Sissakian
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Transcript of “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
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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
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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
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n
n
n /),(),( max1
max
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max)},,(lnexp{ nnsnzn cn
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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
•ISMD-02, Alushta, Crimea
)(),(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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•ISMD-02, Alushta, Crimea
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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
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•ISMD-02, Alushta, Crimea
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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
0
sz
1sz0
1sz
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)(),(
);(),(2
2
snneO
snneO
nn
nn
)( nn eO
•ISMD-02, Alushta, Crimea
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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
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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
•ISMD-02, Alushta, Crimea
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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
);,()(2
),...,,()();,( 22
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•ISMD-02, Alushta, Crimea
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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
J.Manjavidze and A.Sissakian, Phys. Rep., 346 (2001) 1
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•ISMD-02, Alushta, Crimea
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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
•ISMD-02, Alushta, Crimea
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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
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~A
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)(/~),( 22sssA QAQxxG
•ISMD-02, Alushta, Crimea
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.
•ISMD-02, Alushta, Crimea
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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
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1
)(21 )(),...,,(
•ISMD-02, Alushta, Crimea
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22
tQCD: Dirac measure
• Unitary definition of measure
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),(2)(221 i
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)(
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•ISMD-02, Alushta, Crimea
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tQCD: YM theory on manifold• Differential measure :
• Perturbations generating operator
• Interactions generating functional
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24
tQCD: generator of events
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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
25
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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