On the Quark-Quark Superconducting Pairing Induced by Instantons in QCD

1 βc
exp {iεm τ }
m
1 (2π )
3
dp exp {ipx} π , βc
(−i) [(εm + iµ) γ4 + pγ ]αβ (εm + iµ)2 + p2
,
(20)
εm = (2m + 1)
m being the integers, and introducing the 4 × 4 matrices ∆0 (ai)(bj ) with the elements ∆0 (ai)(bj )
DC ∆0 AB = UAB A TB dxSC (−x) ∆0 (−x) . A′ B ′ S D
′ ′
(19)
Using the expression of the two - point Green function of the free quarks at β → βc
B SA (x) =
P S ∆0 (αai)(βbj ) = ∆(ai)(bj ) (γ5 C )αβ + ∆(ai)(bj ) (C )αβ , (δdℓ)(γck)
(22)
3
U(αai)(βbj )
(δdℓ)(γck)
=
1 (dℓ)(ck) 1 (dℓ)(ck) δγ δγ (γ5 C )αβ C −1 γ5 S + P(ai)(bj ) (C )αβ C −1 4 (ai)(bj ) 4 1 (dℓ)(ck) 1 (dℓ)(ck) δγ + V(ai)(bj ) (γµ γ5 C )αβ C −1 γ5 γµ + A(ai)(bj ) (γµ C )αβ C −1 γµ 16 16 1 (dℓ)(ck) δγ . + T(ai)(bj ) (σµν γ5 C )αβ C −1 γ5 σµν 24
The superconducting pairing of quarks in QCD was investigated since more than twenty years[1−5]. At the present time a strong interest to this problem is motivated by recent works of Alford, Rajagopal and Wilczeck[6] , Sch¨ afer and Wilczek[7] , Rapp, Sch¨ afer, Shuryak and Velkovsky[8] and other [9−16] authors . In this work the functional integral method[15] is applied to the study of the superconducting pairing of quarks induced by instantons[8,17,18] , and the integral equation determining the critical temperature of phase transition is established. We work in the imaginary time formalism and use the notations
2
From the variational principle δIef f Φ, Φ δΦ we derive the system of field equations ∆A1 B1 (x1 )
D1 C1 = UA 1 B1 T B2 A2 dx2 SC (x2 − x1 ) (x1 − x2 ) ∆A2 B2 (x2 ) SD 1 1 AB
=
dx1
dx2
dx3
D3 C3
dx4 ∆
D1 C1
B2 (x1 ) SC (x1 − x2 ) ∆B2 A2 (x2 ) 1
T A2 SD (x2 − x3 ) ∆ 3
B4 T A4 (x3 ) SC (x3 − x4 ) ∆B4 A4 (x4 ) SD (x4 − x1 ) , 5 1
(14)
........
Z0 being the partition function of the system of the free quarks, Z0 = [Dψ ] Dψ exp −
B dxψ (x) DA ψB (x) . A
(10)
The effective action of the diquark system equals Ief f Φ, Φ W ∆, ∆ W (2) ∆, ∆ = 1 2 − 1 4 = =
CD UBA
= =
1 A B CD ψ (x) ψ (x) UBA ψD (x) ψC (x) , 2 CD DC DC −UAB = −UBA = UAB .
(1) (2)
The partition function of the system equals Z = [Dψ ] Dψ exp − exp with
DC
= =
∆0 AB = const, ∆
0DC
(x)
= const
of this system of equations are the order parameters of the superconducting phase transition with the formation of the Bose condensate of diquarks - the Cooper pairs of quarks. At the critical temperature T → Tc , β → βc , these order parameters tend to vanishing limits and in the r.h.s. of the equations (18) we can neglect the high order terms. Then we obtain the system of linear equations
CD AB
CD ∆BA (x) = UBA ΦDC (x) , ∆
(x) = Φ
CD (x) UBA .
(8)
Applying this transformation in the r.h.s. of the relation (3) and performing the functional integration over fermionic variables, we rewrite the partition function Z in the form of a functional integral AB over the bosonic diquark fields ΦDC (x) and Φ (x) : Z= Z0 Φ Z0 [DΦ] DΦ exp Ief f Φ, Φ , (9)
Institute of Physics, P. O. Box 429, Boho, Hanoi 10000, Vietnam.
Abstract The superconducting pairing of quarks induced by instantons in QCD is studied by means of the functional integral method. The integral equation determining the critical temperature of the superconducting phase transition is established. It is shown that the Bose condensate of diquarks consists of color antitriplet (¯ 3) scalar diquarks.
2
p2
(εm − iµ) +
2
.
p2
(21)
The order parameter matrix with the elements ∆0 (αai)(βbj ) and the coupling constant matrix with the elements U(αai)(βbj ) have the general forms[15]
CD ψD (x) ψC (x) dx ψ (x) ψ (x) UBA A B
=
1 Φ Z0
[DΦ] DΦ exp − 1 2 dx ∆
CD
1 2
Φ
AB
CD (x) UBA ΦDC (x) A B
(7) ,
exp − where
(x) ψD (x) ψC (x) + ψ (x) ψ (x) ∆BA (x)
B DA = γ4 B dxψ (x) DA ψB (x) B A
1 2
CD ψD (x) ψC (x) dx ψ (x) ψ (x) UBA
A
(3)
∂ − µ + γ∇ ∂τ
β b j δa δi , α AB
(4) (x) , (5)
µ being the chemical potential. Introducing the antisymmetric bi-spinor fields ΦDC (x) , Φ ΦCD (x) = −ΦDC (x) , Φ 1
=0
(17)
(x)
−
dx2 ∆
D3 C3
dx3
T B2 A2 dx4 SC (x2 − x3 ) (x1 − x2 ) ∆A2 B2 (x2 ) SD 3 1
(18)
T B4 A4 (x3 ) SC (x4 − x1 ) + ... . (x3 − x4 ) ∆A4 B4 (x4 ) SD 1 3