Quasi-Classical Model in SU(N) Gauge Field Theory

A.V.KOSHELKIN

Moscow Institute for Physics and Engineering

CONTENTS

1. Introduction.

2. Statement of Problem and Main Goal.

3. Self-Consistent Solution.

4. Fermion and Gauge Field in Developed Model.

5. Application to QCD.

6. Conclusion.

PEOPLE

C.N.Yang, R.L.Mills, Rev. 51 461 (1954).

S.Coleman. Phys. Lett. B 70 59 (1977).

T.Eguchi. Phys. Rev. D 13 1561 (1976).

P.Sikivie, N.Weiss, Phys. Rev. 20 487 (1979).

R.Jackiw, L.Jacobs, C.Rebbi, Phys. Rev. D21 426 (1980 ).

R.Teh, W.K.Koo, C.H.Oh, Phys. Rev. D24 2305 (1981)

V.M.Vyas,T.S.Raju, T.Shreecharan. e-Print: arXiv:0912.3993[het-th].

D.D.Dietrich, Phys Rev. D 80 (2009) 067701.

A.Slavnov, L.Faddeev, Introduction to Quantum Theory of Gauage

Fields, 2nd enl. and rev. ed. Moscow, Nauka, 1988.

E.S.Fradkin, Nucl.Phys. 76 (1966) 588.

and so on, so on …

1. Introduction.

2.Statement of Problem and Main Goal.

)2(0}])([){(

0)(}])([{

mTxAgiix

xmTxAgii

a

a

a

a

�

;],[c

cabbaTfiTT

0)( xAa

)1()()()(

)()()()()(

)()()()(

xTxxJ

xAxAfgxAxAxF

xgJxFxAfgxF

aa

cb

bc

aaaa

ac

bc

aba

abbaTTTr

21},{

Key Approximations

.1)4

;0)()()3

;0)())(()2

));(()()1

YM

FYM

aa

n

x

xx

xAxA

The main goals are

1) to obtained such solutions that both the Yang-Mills and Dirac Equation would be satisfied together;

2) to quantize the fields;

3) to apply the obtained results to QCD .

The Yang-Mills equation

WE ASSUME

)()()()()()())(()()(2 2

ars

bsrc

cabc

bcabc

bcab gJAAAffgAAfkgxAAfkg

)()()( xTxJ aa

)(;;;0)(

)())(sin()())(cos()()(

)1()2()2()1()2()1()1()1(

)2()1(

xkeeeekekeee

xBxexeAA aaaa

The Dirac equation

Provided that

)(2

)()(

0)())(()()(2)( 222

xm

mTigAix

xTAkigTAigTAgm

aa

aa

aa

aa

YMFYM

dx

d ;1

)()()( ,, Fexx ipx

,,,,,0

3

,,,,,0

3

),()(ˆ),()(ˆ2

)(

),()(ˆ),()(ˆ2

)(

pxpbpxpap

pdx

pxpbpxpap

pdx

SOLUTION IS

(Koshelkin,Phys.Lett.,B683 (2010) 205)

)()2(),(),( 3,

*,

3 pppxpxxd

3. Self-Consistent Solution.

a) Gauge field

b) Fermion field

)(;;;0)(

)())(sin()())(cos()()(

)1()2()2()1()2()1()1()1(

)2()1(

xkeeeekekeee

xBxexeAA aaaa

,,,,,0

3

,,,,,0

3

),()(ˆ),()(ˆ2

)(

),()(ˆ),()(ˆ2

)(

pxpbpxpap

pdx

pxpbpxpap

pdx

c) Relation equations

The problem is solvable when the dimension of the gauge group . Thereat, the currents generated by fermions and gauge field exactly compensate each other.

3N

)()()()( xAxTxJ aaa

,,,,,0

322 )(ˆ)(ˆ)(ˆ)(ˆ

2)1(

)cos()cos()cos()sin(2

pbpbpapap

pdNCA

BN

ffff s

bsc

asrbrbsrc

cabba

cab

0)cos()cos( asrbsrc

cab ffC

0)(sin21

0,

2

2

ba

N

ba

CN

(Koshelkin,Phys.Lett.,B696 (2011) 539)

4. Fermion and Gauge Field in Developed Model.

In terms of the multi particle problem,

the solutions correspond to individual states of particles

the solutions correspond to collective states (Fermi liquid-like)

1;2)1(

23

0

2/1

30

2

TC

n

T

mT

TC

n

Ng

NA

1;)1(

2

)1(

2 3/20

2/1

2

2/1

2

T

TnNg

N

Ng

NA

Fermion effective mass.

IN EQUILIBRIUM

1)

2)

1;2)1(

23

0

2/1

30

2/1

2*

TC

n

T

mT

TC

n

Ng

Nm

1;1;3

03/20*

TC

nnm

. 5. Application to QCD

6.Final remarks and conclusion.

1.The self-consistent solutions of the non-homogeneous YM equation and the Dirac equation in the external YM field is derived in the quasi-classical model when the YM field is assumed to be in form of the eikonal wave.

2. The quantum theory of the considered model is developed in the quasi-classical approximation.

3. The considered model is solvable when the dimension of the gauge group and assumes that the fermion and gauge fields have to exist together .That is an alternative to Glasma model by L.D.McLerran and R.Venugopalan.

4. The relation of the developed model to the generally accepted point of view on the matter generated in collisions of heavy ions of high energies is considered.

5. The fermion and gauge fields derived in the explicit form allow to develop diagram technique beyond perturbative consideration.

3N