Nambu-Jona-Lasinio approach to realization of confining medium

15 Pages·0.16 MB·English

by  M. Faber

Nambu–Jona–Lasinio approach to realization of
confining medium
M. Faber∗, A. N. Ivanov† ‡, N.I. Troitskaya‡
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9 Institut fu¨r Kernphysik, Technische Universit¨at Wien,
1
Wiedner Hauptstr. 8-10, A-1040 Vienna, Austria

n
a
J

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Abstract

2
The mechanism of a confining medium is investigated within the Nambu–Jona–
v
0 Lasinio (NJL) approach. It is shown that a confining medium can berealized in the
9 bosonized phase of the NJL model due to vacuum fluctuations of both fermion and
1
Higgs (scalar fermion–antifermion collective excitation) fields. In such an approach
0
1 there is no need to introduce Dirac strings.
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9
PACS: 11.15.Tk, 12.39.Ki, 12.60.Rc, 14.80.Cp.
/
h
Keywords: phenomenological quark model, technifermions, Higgs field, confining
t
-
medium, confinement, composed states, linear potential
p
e
h
:
v
i
X
r
a
∗E–mail: [email protected], Tel.: [phone],Fax: [phone]
†E–mail: [email protected], Tel.: [phone],Fax: [phone]
‡Permanent Address: State Technical University, Department of Theoretical Physics, 195251 St. Pe-
tersburg, Russian Federation
1
1 Introduction
Mechanism of confining medium suggested in Abelian Higgs model by Narnhofer and
Thirring [1] and in a dual QCD by Baker, Ball and Zachariasen [2] can be rather useful
for the realization of quark confinement by a way avoiding the problem of the inclusion
of Dirac strings. The dielectric constant of a confining medium ε(k2) vanishes at large
distances like ε(k2) ∼ k2 which leads to a linearly rising interquark potential realizing
confinement of quarks [3].
The approaches to confining medium [1,2] are based on the similarity between dual su-
perconductivity and the Higgs mechanism inducing non–perturbative vacuum with prop-
erties of superconductor. The Nambu–Jona–Lasinio (NJL)model [4–10],being arelativis-
tic extension of the BCS (Bardeen–Cooper–Schrieffer) theory of superconductivity [11],
gives the alternative mechanism of the realization of a superconducting non–perturbative
vacuum. Recently in Ref.[10]we have investigated the mechanism ofquark confinement in
the Abelian monopole NJL model with dual Dirac strings. In this model quarks and anti-
quarks are classical particles joined to the ends of dual Dirac strings, while monopoles are
quantum massless fermion fields which become massive due to monopole–antimonopole
condensation induced by four–monopole interaction. The monopole–antimonopole con-
densation accompanies itself the creation of the monopole–antimonopole scalar and dual–
vector collective excitations. The ground state of the dual–vector field induced by the
interaction with a dual Dirac string has the shape of the Abrikosov flux line. The latter
leads to linearly rising potential and confinement of quarks and antiquarks joined to the
ends of a dual Dirac string.
This paper is to apply the NJL model analoguous the Abelian monopole NJL model
[10] to the realization of confining medium and consider the way avoiding the inclusion of
dual Dirac strings. Since we do not include dual Dirac strings, the quarks and antiquarks,
confinement of which we are investigating, are described now by an external field ψ(x)
¯
related to an external electric current j (x) = ψ(x)γ ψ(x). Then, due to the absence
µ µ
of dual Dirac strings instead of the monopole fermion fields we use fermion fields like
technifermions introduced in the technicolour approach [12] to the standard electroweak
model for the description of the appearance of the W and Z–boson masses without Higgs
mechanism. The NJL model in our consideration is an Abelian one, and we need to intro-
duce only one sort of technifermions. For the definitness we would call them electroquarks
and define by the field χ(x). The electroquarks are massless and acquire the mass due to
strong local four–electroquark interactions.
The Lagrangian of the starting system should read [10]
L(x) = χ¯(x)iγµ∂ χ(x)+G[χ¯(x)χ(x)]2
µ
−G [χ¯(x)γ χ(x)+j (x)][χ¯(x)γµχ(x)+jµ(x)], (1.1)
1 µ µ
where G and G are positive phenomenological coupling constants that we fix below.
1
The Lagrangian Eq.(1.1) is invariant under U(1) group. Due to strong attraction in
the χ¯χ–channels produced by the local four–electroquark interaction Eq.(1.1) the χ–fields
become unstable under χ¯χ condensation, i.e. < χ¯χ >6= 0. Indeed, in the condensed phase
the energy of the ground state of the electroquark fields is negative, i.e.
3
W = − < L (x) >= − G+G [< χ¯χ >]2 < 0, (1.2)
χ 1
4 !
2
whereas in the non–condensed phase, when < χ¯χ >= 0, we have W = 0. This means that
the condensed phase is much more advantageous to the electroquark system, described
by the Lagrangian (1.1). The electroquark fields become condensed without breaking of
U(1) symmetry as well.
In the condensed phase the electroquark fields acquire a mass M satisfying the gap–
equation [4–10]
GM
M = −2G < χ¯(0)χ(0) >= J (M) =
2π2 1
GM d4k 1 GM Λ2
= = Λ2 −M2ℓn 1+ , (1.3)
2π2 π2i M2 −k2 2π2 " M2!#
Z
where Λ is the ultra–violet cut–off.
The condensation of the electroquark fields accompanies the creation of χ¯χ collective
excitations with the quantum numbers of a scalar Higgs meson field ρ and a vector field
A .
µ
The main aim of the approach is to show that in the tree approximation for the A –
µ
field and after the integration over the electroquark and the scalar fields the effective
Lagrangian of the A –field acquires the form [1,2]
µ
1
L [A(x)] = − F (x)ε(2)Fµν(x)−j (x)Aµ(x), (1.4)
eff µν µ
4
where F (x) = ∂ A (x) − ∂ A (x) and ε(2) is the operator of the dielectric constant
µν µ ν ν µ
which inthemomentum representation ε(k2)vanishes atlargedistances, i.e. ε(k2) ∼ k2 at
k2 → 0. Due to the tree A –field approximation integrating over the electroquark and the
µ
scalar fields we can keep only the terms proportional to F (x)Fµν(x) and A (x)Aµ(x).
µν µ
The paper is organized as follows. In Sect. 2 we derive the effective Lagrangian for
collective excitations of electroquarks in one–loop approximation keeping only the main
divergent contibutions and leading order in long–wavelength expansion. That is in accor-
dance with the standard approximation accepted in the NJL model approaches. We show
that such a way does lead to the realization of a non–trivial medium. In Sect. 3 we take
into account convergent contributions of the one–electroquark loop diagrams that means
the step beyond the standard approximation. This has led to the effective Lagrangian
for the vector collective excitations coupled to scalar collective excitations. Integrating
out the scalar collective excitations, since they are much heavier than the vector ones, we
arrive at the effective Lagrangian for the massless vector collective excitation field in a
non–trivial medium with the dielectric constant vanishes at large distances. In Sect. 4 we
show that the external ψ–quarks couple to each other and the medium via the exchange of
themassless vector collective excitations becomeconfined dueto alinearly rising potential
induced by the medium. In Sect. 5 we discuss the obtained results.
2 Effective Lagrangian. Standard approximation
Following [5–10] we define the effective Lagrangian of the ρ and A fields as follows
µ
κ2 g2
L (x) = L˜ (x)− ρ2(x)+ A (x)Aµ(x)−j (x)Aµ(x), (2.1)
eff eff µ µ
4G 4G
1
3
where
ˆ
Det(i∂ − M + Φ)
L˜ (x) = −i x ℓn x . (2.2)
eff ˆ
* (cid:12) Det(i∂ − M) (cid:12) +
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Here we have denoted Φ = −gγµA −(cid:12)κσ, and σ = ρ−M/κ,(cid:12)where g is the electric charge
µ
of the electroquark field that we fix below. In the tree approximation the σ–field has a
vanishing vacuum expectation value (v.e.v.), i.e. < σ >= 0, whereas the v.e.v. of the
ρ–field does not vanish, i.e. < ρ >= M/κ 6= 0.
˜
The effective Lagrangian L (x) can be represented by an infinite series
eff
∞ n ∞
i 1
L˜ (x) = tr x Φ x = L˜(n)(x). (2.3)
eff n L* (cid:12) M − i∂ˆ ! (cid:12) + eff
nX=1 (cid:12) (cid:12) nX=1
(cid:12) (cid:12)
(cid:12) (cid:12)
The subscript L means the comput(cid:12)ation of the trac(cid:12)e over Dirac matrices. The effective
Lagrangian L˜(n)(x) is given by [5–10,13]
eff
n−1 d4x d4k 1 1 d4k
L˜(n)(x) = ℓ ℓ e−ik1·x1−...−ikn·x −
eff (2π)4 n 16π2 π2i
Z ℓ=1 (cid:18) (cid:19) Z
Y
1 1
×tr Φ(x ) Φ(x ) ...
L ˆ 1 ˆ ˆ 2
( M − k M − k − k
1
1
×...Φ(xn−1) ˆ ˆ ˆ Φ(x) (2.4)
M − k − k1 −...− kn−1 )
at k +k +...+k = 0. The r.h.s. of (2.4) describes the one–electroquark loop diagram
1 2 n
with n–vertices. The one–electroquark loop diagrams with two vertices (n = 2) determine
the kinetic term of the σ–field and give the contribution to the kinetic term of the A –
µ
field, while the diagrams with (n ≥ 3) describe the vertices of interactions of the σ and
A fields. According to the NJL prescription the effective Lagrangian L˜ (x) should
µ eff
be defined by the set of divergent one–electroquark loop diagrams with n = 1,2,3 and 4
vertices [5–10]. The compution of these diagrams we perform at leading order in the long–
wavelength expansion approximation accepted in the NJL model [5–10] when gradients
of A and σ fields are slowly varying fields. This approximation is fairly good established
µ
for sufficiently heavy electorquark fields χ(x) that we assume following the technicolour
extension of the standard electroweak model [12]. As a result we get
L (x) =
eff
g2 g2 g2
= − J (M)F (x)Fµν(x)+ − [J (M)+M2J (M)] A (x)Aµ(x)
48π2 2 µν (4G1 16π2 1 2 ) µ
1 κ2 κ κ
+ J (M)∂ σ(x)∂µ sigma(x)−M − J (M) σ(x)
28π2 2 µ "2G 4π2 1 #
1 κ2 κ2 κ2 κ2
+ − + J (M)−4M J (M) σ2(x)−2M κ J (M)σ3(x)
2" 2G 4π2 1 8π2 2 # 8π2 2
1 κ2
− κ2 J (M)σ4(x)−j (x)Aµ(x). (2.5)
2 8π2 2 µ
4
In order to get correct factors of the kinetic terms of the σ and A fields we have to set
µ
[5–10]
g2 κ2
J (M) = 1 , J (M) = 1, (2.6)
12π2 2 8π2 2
that arranges the relation κ2 = 2g2/3 [5–10], and J (M) is a logarithmically divergent
2
integral defined by
d4k 1 Λ2 Λ2
J (M) = = ℓn 1+ − . (2.7)
2 π2i (M2 −k2)2 M2! M2 +Λ2
Z
The relations (2.6) can be represented in a more comprehensible way in terms of con-
straints for the renormalization constants of the wave–functions of the A and σ fields.
µ
For this aim we rewrite the Lagrangian (2.5) as follows
L (x) = (2.8)
eff
1 1
= − (1−Zχ)F (x)Fµν(x)+ (1−Zχ)∂ σ(x)∂µσ(x)+... ,
4 A µν 2 σ µ
where
g2 κ2
Zχ = 1− J (M) , Zχ = 1− J (M) (2.9)
A 12π2 2 σ 8π2 2
are the renormalization constants of the wave–functions of the A and σ fields procreated
µ
by vacuum fluctuations of the electroquark fields. In Eq.(2.8) we have kept only kinetic
terms of the A and σ fields, other terms are irrelevant at the moment. Since A and σ
µ µ
are bound χ¯χ–states, the renormalization constants Zχ and Zχ should vanish due to the
A σ
so–called compositeness condition Zχ = Zχ = 0 [14], i.e.
A σ
g2 κ2
Zχ = 1− J (M) = 0 , Zχ = 1− J (M) = 0. (2.10)
A 12π2 2 σ 8π2 2
The compositeness conditioncanbeappliedtoboundstatesinnon–perturbative quantum
field theory [14]. As a result we arrive at Eq.(2.6). Picking up Eq.(1.3) and Eq.(2.6) we
bring up the effective Lagrangian (2.5) to the form
1 1
L (x) = − F (x)Fµν(x)+ M2A (x)Aµ(x)−jµ(x)A (x)
eff 4 µν 2 A µ µ
2
1 1 σ(x)
+ ∂ σ(x)∂µσ(x)− M2σ2(x) 1+κ −j (x)Aµ(x), (2.11)
2 µ 2 σ " Mσ # µ
where M = 2M is the mass of the σ–field and
σ
g2 g2
M2 = − [J (M)+M2J (M)] (2.12)
A 2G 8π2 1 2
1
is the squared mass of the A –field. Without loss of generality one can assume that
µ
M ≪ M = 2M. This should allow one to integrate over the heavy scalar collective
A σ
excitations and derive an effective Lagrangian only for the vector ones.
The Lagrangian Eq.(2.11) describes the effective Lagrangian of the collective exci-
tations (χ¯χ–bound states) derived within the standard procedure of the NJL approach
5
[4–10], i.e. at leading order in long–wavelength expansion of the one–electroquark loop
diagrams.
The effective Lagrangian describing the propagation of the A –field in a dielectric
µ
medium should take the form Eq.(1.4). If it is a confining medium, the Fourier transform
of a dielectric constant ε(k2) should vanish at k2 → 0, i.e. ε(k2) → 0 at k2 → 0 [1,2].
It is seen that the Lagrangian Eq.(2.10) describes a dielectric medium with ε = 1. This
shouldimplythatthe realization of the confining medium in the NJL approach goes beyond
the standard procedure of the derivation of the effective Lagrangian describing collective
excitations. The simplest extension of the NJL prescription is to take into account the
contributions of convergent electroquark loop diagrams and the Higgs field loops. As has
been shown in Ref.[9] contributions of convergent electroquark loop diagrams play an
important role for processes of low–energy interactions of low–lying hadrons.
3 Effective Lagrangian for confining medium
Thus, in order to obtain non–trivial contributions to the dielectric constant we should
leave a standard approximation for the derivation of the effective Lagrangians in the NJL
models.
Since we are interested in the kinetc terms of the A –field, the simplest step leading
µ
beyond the standard approximation is to take into account convergent contributions of
the one–electroquark loop diagrams to the kinetic term of the vector field A . The exact
µ
calculation of the electroquark loop diagram with two–vector vertices alters the effective
Lagrangian (2.11) as follows
1 1
L (x) = − F (x)ε(2)Fµν(x)+ M2A (x)Aµ(x)−jµ(x)A (x)
eff 4 µν 2 A µ µ
2
1 1 σ(x)
+ ∂ σ(x)∂µσ(x)− M2σ2(x) 1+κ , (3.1)
2 µ 2 σ " Mσ #
where ε(2) is given by
g2 1 2
ε(2) = 1− dηη(1−η)ℓn 1+ η(1−η) , (3.2)
2π2 Z0 " M2 #
the Fourier transform of which yields
g2 1 k2
ε(k2) = 1− dηη(1−η)ℓn 1− η(1−η) . (3.3)
2π2 Z0 " M2 #
We have dropped the convergent contributions of the electroquark loop diagrams to the
other terms of the effective Lagrangian Eq.(3.1), since they are less important for the
problem under consideration.
The other non–trivial contributions to the kinetic term of the A –field come from the
µ
convergent one–electroquark loop diagrams inducing the interactions between σ and A
µ
fields. They read
κg2 σ(x)
δL (x) = M σ(x) 1+κ A (x)Aµ(x)+
eff 8π2 " 2M # µ
6
g2 σ(x)
+ ℓn 1+κ F (x)Fµν(x). (3.4)
µν
24π2 " M #
This is the most general interaction of two vector mesons with scalar fields derived in
leading order of the expansion in powers of gradients of the σ–field, ∂ σ(x), which are
µ
slowly varying fields.
By appending the interaction Eq.(3.4) to the Lagrangian (3.1) we arrive at the follow-
ing effective Lagrangian
2
1 1 σ(x)
L (x) = L [A(x),σ(x)]+ ∂ σ(x)∂µσ(x)− M2σ2(x) 1+κ , (3.5)
eff eff 2 µ 2 σ " Mσ #
where we have denoted
1 1
L [A(x),σ(x)] = − F (x)ε(2,σ)Fµν(x)+ M2(σ)A (x)Aµ(x)−jµ(x)A (x). (3.6)
eff 4 µν 2 A µ µ
The dielectric constant ε(2,σ) reads
g2 σ(x)
ε(2,σ) = 1− ℓn 1+κ −
6π2 " M #
g2 1 2
− dηη(1−η)ℓn 1+ η(1−η) , (3.7)
2π2 Z0 " M2 #
and M2(σ) is defined
A
g2 g2 κg2 σ(x)
M2(σ) = − [J (M)+M2J (M)]+ Mσ(x) 1+κ . (3.8)
A 2G1 8π2 1 2 4π2 " 2M #
It is seen that a dielectric constant has become a functional of the σ–field. Thereby, one
canexpectasubstantialinfluenceofthevacuumfluctuationsoftheσ–fieldonthedielectric
constant. Since the σ–field is much heavier than the A –field, i.e. M = 2M ≫ M , at
µ σ A
low energies only vacuum fluctuations of the σ–field are important. Therefore, in order
to pick up the contribution of vacuum fluctuations of the σ–field we have to integrate it
out. The effective Lagrangian L [A (x)] can be defined [13]
eff µ
ei d4xLeff[Aµ(x)] = Dσei d4x{Leff[Aµ(x),σ(x)]+Leff[σ(x)]}, (3.9)
R Z R
where L [σ(x)] reads
eff
2
1 1 σ(x)
L [σ(x)] = ∂ σ(x)∂µσ(x)− M2σ2(x) 1+κ −V [σ(x)], (3.10)
eff 2 µ 2 σ " Mσ # eff
where V [σ(x)] describes the contribution of the convergent electroquark loops yielding
eff
self–interactions of the σ–field at leading order of the expansion in powers of the gradients
of the σ–field. One can expect that V [σ(x)] ∼ σ6(x)+.... The integrals over the σ–field
eff
can be normalized by the condition
Dσei d4xLeff[σ(x)] = 1. (3.11)
Z R
7
Of course, the exact integration over the σ–field cannot be done and we should develop
an approximate scheme.
For this aim it is convenient to rewrite Eq.(3.9) as follows
ei d4xLeff[Aµ(x)] =
1 1
=Rei d4x − 4 Fµν(x)ε(2)Fµν(x)+ 2MA2Aµ(x)Aµ(x)−jµ(x)Aµ(x)
h i
R 1 1
Dσei d4x − 4Fµν(x)[ε(2,σ)−ε(2)]Fµν(x)+ 2[MA2(σ)−MA2]Aµ(x)Aµ(x)
n o
R
Z
ei d4xLeff[σ(x)], (3.12)
R
Recall that the effective Lagrangian Eq.(3.6) has been calculated keeping the quadratic
terms in the A –field. This means that in the integrand we can expand the exponential
µ
keeping only quadratic terms in the A –field expansion
µ
ei d4xLeff[Aµ(x)] =
1 1
=Rei d4x − 4 Fµν(x)ε(2)Fµν(x)+ 2MA2Aµ(x)Aµ(x)−jµ(x)Aµ(x)
h i
R 1
Dσ 1− i d4xF (x)[ε(2,σ)−ε(2)]Fµν(x)
µν
4
Z n Z+1i d4x[M2(σ)−M2]A (x)Aµ(x) ei d4xLeff[σ(x)]
2 A A µ
1 Z 1 o R
= ei d4x − 4 Fµν(x)ε(2)Fµν(x)+ 2MA2Aµ(x)Aµ(x)−jµ(x)Aµ(x)
h i
R1
1− i d4xF (x) < ε(2,σ)−ε(2) > Fµν(x)
µν
4
n Z 1
+ i d4x < M2(σ)−M2 > A (x)Aµ(x)
2 A A µ
1 Z o
i d4x − Fµν(x) < ε(2,σ) > Fµν(x)+
≃ e 4
h
R 1
+ < M2(σ) > A (x)Aµ(x)−j (x)Aµ(x) . (3.13)
2 A µ µ
i
Thus, up to the quadratic terms in the A –field expansion the effective Lagrangian
µ
L [A (x)] is given by
eff µ
1
L [A(x)] = − F (x) < ε(2,σ) > Fµν(x)
eff µν
4
1
+ < M2(σ) > A (x)Aµ(x)−jµ(x)A (x). (3.14)
2 A µ µ
The derivation of the Lagrangian Eq.(3.14) has been performed in the A –field tree ap-
µ
proximation. In this case the operators F (x)Fµν(x) and A (x)Aµ(x) are not affected by
µν µ
the contributions of higher powers in the A –field expansion. Therefore, in the A –field
µ µ
tree approximation the justification of the validity of the derivation of the Lagrangian
Eq.(3.14) does not need the smallness of higher power terms in the A –field expansion.
µ
8
The expectation values < ε(2,σ) > and < M2(σ) > read
A
< ε(2,σ) >= Dσε(2,σ) expi d4zL [σ(z)] =
eff
Z Z
g2 σ(x) g2 1 2
= 1− ℓn 1+κ − dηη(1−η)ℓn 1+ η(1−η)
*6π2 " M #+ 2π2 Z0 " M2 #
g2 1 2
= Zσ − dηη(1−η)ℓn 1+ η(1−η) , (3.15)
A 2π2 Z0 " M2 #
and
< M2(σ) >= DσM2(σ) expi d4zL [σ(z)] =
A A eff
Z Z
g2 g2 κg2 σ(x)
= − [J (M)+M2J (M)]+ Mσ(x) 1+κ = (3.16)
2G1 8π2 1 2 *4π2 " 2M #+
g2 g2
= − [J (M)+M2J (M)]+M2Zσ ,
2G 8π2 1 2 M
1
where we have denoted
g2 σ(x)
Zσ = 1− ℓn 1+κ =
A *6π2 " M #+
g2 σ(x)
= 1− Dσ ℓn 1+κ expi d4zL [σ(z)], (3.17)
6π2 " M # eff
Z Z
and
κg2 σ(x) σ(x)
Zσ = 1+κ =
M *2π2 2M " 2M #+
κg2 σ(x) σ(x)
= Dσ 1+κ expi d4zL [σ(z)]. (3.18)
2π2 2M " 2M # eff
Z Z
It is seen that the quantities Zσ and Zσ are just constants, and Zσ has the meaning
A M A
of the renormalization constant of the wave–function of the A –field caused by vacuum
µ
fluctuations of the σ–field. The vacuum expectation values entering the constants Zσ and
A
Zσ can be represented in the form of time–ordered products [15]
M
g2 σ(x)
Zσ = 1− ℓn 1+κ =
A *6π2 " M #+
g2 σ(x)
= 1− Dσ ℓn 1+κ expi d4zL [σ(z)] =
6π2 " M # eff
Z Z
g2 σ(x)
= 1− 0 T ℓn 1+κ expi d4zL [σ(z)] 0 (3.19)
int
* (cid:12) 6π2 " M # !(cid:12) +
(cid:12) Z (cid:12)
(cid:12) (cid:12)
and (cid:12) (cid:12)
(cid:12) (cid:12)
κg2 σ(x) σ(x)
Zσ = 1+κ =
M *2π2 2M " 2M #+
κg2 σ(x) σ(x)
= Dσ 1+κ expi d4zL [σ(z)] =
eff
2π2 2M " 2M #
Z Z
κg2 σ(x) σ(x)
= 0 T 1+κ expi d4zL [σ(z)] 0 , (3.20)
* (cid:12) 2π2 2M " 2M # int !(cid:12) +
(cid:12) Z (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
9
where L [σ(z)] reads
int
2
1 1 σ(x)
L [σ(x)] = ∂ σ(x)∂µσ(x)− M2σ2(x) 1+κ −V [σ(x)]−
int 2 µ 2 σ " Mσ # eff
1 1 1
− ∂ σ(x)∂µσ(x)+ M2σ2(x) = −κM σ3(x)− σ4(x)−V [σ(x)]. (3.21)
2 µ 2 σ σ 2 eff
In order to understand the structure of the constants Zσ and Zσ we suggest to calculate
A M
them in the one–loop approximation of the σ–field exchange. One can expect that the
computation of the constants Z and Z accounting for multi–loop contributions should
A M
not change the one–loopresult substantially but only provide a redefinition of parameters.
In the one–loop approximation Z and Z are given by
A M
g2κ2 1 κ2g2 ∆ (M )
Zσ = 1+ < 0|σ2(x)|0 >= 1+ 1 σ ,
A 3π2 M2 48π4 M2
σ σ
g2κ2 1 κ2g2 ∆ (M )
Zσ = < 0|σ2(x)|0 >= 1 σ , (3.22)
M 2π2 M2 32π4 M2
σ σ
where ∆ (M ) is a quadratically divergent momentum integral
1 σ
d4k 1
∆ (M ) = . (3.23)
1 σ π2i M2 −k2
Z σ
The magnitude of ∆ (M ) depends on the regularization procedure. For example, within
1 σ
dimensional regularization ∆ (M ) is negative. The computation of ∆ (M ) by means of
1 σ 1 σ
a cut–off regularization is not unambiguous. The result of the computation depends on a
shift of the virtual σ–field momentum [16]. Indeed, one can define ∆ (M ;Q) instead of
1 σ
∆ (M ) [16]
1 σ
d4k 1 1
∆ (M ;Q) = = ∆ (M )+ Q2, (3.24)
1 σ π2i M2 −(k +Q)2 1 σ 4
Z σ
where Q is an arbitrary 4-vector that can be a space–like one, i.e. Q2 < 0. Thus, ∆ (M )
1 σ
is an arbitrary quantity on both magnitude and sign. Below we fix ∆ (M ) using the
1 σ
compositeness condition for the A –field.
µ
The momentum integral ∆ (M ) resembles the quadratically divergent integral J (M)
1 σ 1
which we encounter for the computation of the one–electroquark loop diagram defining
the vacuum expectation value < χ¯(0)χ(0) >. However, as ∆ (M ) is related to the
1 σ
contribution of one–loop σ–field exchange diagrams, the cut–off parameters applied to
the regularization of ∆ (M ) and J (M) can arbitrarilly differ on magnitude. This makes
1 σ 1
∆ (M ) and J (M) independent each other. The integral J (M) suffers the same problem
1 σ 1 1
as ∆ (M ) and can be also considered as an arbitrary parameter of the approach fixed
1 σ
by the gap–equation Eq.(1.3), i.e. J (M) = 2π2M/G.
1
The Lagrangian Eq.(3.14) can be rewritten as follows
1
L [A(x)] = − F (x)[Zσ +ε (2)]Fµν(x)
eff 4 µν A eff
1
+ M2 A (x)Aµ(x)−jµ(x)A (x), (3.25)
2 eff µ µ
10