Inhomogeneous Vacuum States in Nambu-Jona-Lasinio Model

0.13 MB·

by  Sergii Kutnii

lNHOMOGENEOUS VACUUM STATES IN
NAMBU-JONA-LASINIO MODEL
1
1 Sergii Kutnii
0
2
Bogolyubov Institute for Theoretical Physics, Kyiv, Ukraine
n
a
J
January 26, 2011
5
2

]
h Abstract
t
-
p The mean field approach to the Nambu-Jona-Lasinio model is de-
e veloped systematically. Approximate mean field action is obtained,
h
based on the study of divergencies in the mean field action. A spe-
[

cial scalar case of the approximate motion equations is studied and
2
v inhomogenouous solutions are discussed. It is shown that the model
8 can have inhomogeneous vacuum configurations which leads to bound
3
fermionic states.
2
4
. 1
1 INTRODUCTION
0
1
The Nambu-Jona-Lasinio model[1] has been proposed in 1961, long before
1
: QCD arose. It is a common belief that the model can be derived from QCD
v
i in a low-energy limit. An example of such derivation is presented in [2].
X
However, it relies on unproven (though plausible) assumptions about the
r
a structure of gluon propagator in the low-energy limit.
In a previous paper[3] it has been shown that NJL can be derived from
QCD using the mean field approach developed by Kondo [4, 5]. It’s notable
that in this approach NJL is obtained naturally together with a cutoff pa-
rameter Λ which is just the effective gluon mass generated by a dynamic
mechanism suggested by Kondo in the works mentioned. This solves the
regularization problem in a way that is in perfect agreement with the belief
that non-renormalizable models like NJL can only be effective field theories.
It has been demonstrated by Nambu and Jona-Lasinio that a simple
model with only scalar and pseudoscalar quartic terms in its lagrangian has
1
nontrivial vacuum state with fermionic condensate. Thus, initially massless
fermions gain mass dynamically. This result was obtained in the mean field
approach assuming spacial homogeneity, i.e. that the mean field has a con-
stant valueover thespacetime. However ithasbeenpointedout recently that
there can exist inhomogeneous vacuum configurations in this model. basar,
Dunne and Thies have done this for NJL model in 1+1 dimensions [6].
However, if it comes to vacuum inhomogeneities, mean field approach
becomes tricky since it’s necessary to solve the non-simplified gap equation.
Therefore, it becomes interesting to develop some approximations that would
be less tough but allowing to grasp vacuum inhomogeneities. This is done in
this paper.
In the first part of this work NJL derivation from QCD is sketched briefly.
Then the mean field approach to the model obtained is developed systemat-
ically. An approximate mean field action is obtained, based on the study of
divergencies in the mean field action. Afterwards a special scalar case of the
approximate motion equations is studied and inhomogenouous solutions are
discussed.
2
NJL MODEL IN QUANTUM CHROMODYNAMICS
We start with a plain (i.e.) flavorless SU(N) model
1
= ψ¯ iD m ψ TrF Fµν
µν
L − − 2
(cid:16) (cid:17)
Dµ = ∂µ bigAµ
−
F = ∂ A ∂ A ig[A ,A ] (1)
µν µ ν ν µ µ ν −
− −
The cubic gluon self-interaction term can be removed by introducing an
antisymmetric auxiliary field Ba in such a way that leads to the following
µν
lagrangian:
1+σ−2 (1+σ2)g2
= Tr (∂ A ∂ A )2 + [A ,A ]2
µ ν µ ν µ ν
L (cid:20)− 2 − 2 −
1
σ−1Bµν (∂ A ∂ A ) iσgBµν [A ,A ] B2 , (2)
µ ν µ ν µ ν
− − − − 2 (cid:21)
where σ is an arbitrary numeric parameter and fermionic terms have been
omitted for a moment.
2
It’s easy to demonstrate that
1+σ−2
Bexp i d4xTr (∂ A ∂ A )2+
µ ν µ ν
Z D (cid:26)− Z (cid:20)− 2 −
(1+σ2)g2 1
+ [A ,A ]2 σ−1Bµν (∂ A ∂ A ) iσgBµν [A ,A ] B2 =
µ ν µ ν µ ν µ ν
2 − − − − 2 (cid:21)(cid:27)
1 1
= exp i d4x TrF2 Bexp i d4x TrF2
(cid:26)− Z (cid:20)−2 (cid:21)(cid:27)Z D (cid:26)− Z (cid:20)−2 −
1
Tr B +σ−1(∂ A ∂ A )+iσg[A ,A ] 2 =
µν µ ν µ ν µ ν
− 2 − (cid:21)(cid:27)
(cid:8) (cid:9)
1
= Cexp i d4x TrF2 (3)
(cid:26)− Z (cid:20)−2 (cid:21)(cid:27)
In order to get rid of of the fourth order gluon term we note that
Tr[A ,A ]2 = TrA [A ,[Aµ,Aν]] = TrA ( A , Aµ,Aν Aµ, A ,Aν ) =
µ ν µ ν µ ν ν
{ { }}−{ { }}
= Tr A ,A 2 Tr A ,Aµ 2 =
µ ν µ
{ } − { }
η 2 3
= Tr A ,A µν A ,Aλ Tr A ,Aµ 2 (4)
µ ν λ µ
{ }− 4 − 4 { }
h i
(cid:8) (cid:9)
Then we introduce auxiliary fields Θ = θ + τa Ta(Θλ = 0) and
µν µν µν λ
Φ = φ+ϕaTa in order to arrive at
1+σ−2 η
= Tr (∂ A ∂ A )2 Θµν A ,A µν A ,Aλ
µ ν µ ν µ ν λ
L (cid:20)− 2 − − { }− 4 −
(cid:16) (cid:17)
(cid:8) (cid:9)
Φ A ,Aµ σ−1Bµν (∂ A ∂ A ) iσgBµν [A ,A ]
µ µ ν µ ν µ ν
− { }− − − −
1 1 2
B2 Θ2 + Φ2 (5)
− 2 − 2g2(1+σ2) 3g2(1+σ2) (cid:21)
It’s obvious now that the Faddeev-Popov lagrangian for gluon fields can
be transformed into
= Aµa abAνb+ a(B)Aaµ+ + (Φ,Θ,B)+ψ¯ iD m ψ (6)
L Kµν Jµ LFP LMF − 0
(cid:16) (cid:17)
b
, where is the ghost part of the lagrangian and (Φ,Θ,B) depends
FP MF
L L
on auxiliary fields only. The operator is
K
3
1+σ2 δab dabc δab
ab = η δab∂2 1+σ2 +ξ−1 ∂ ∂ τc θ
Kµν 2 µν − 2 µ ν − 2 µν − 2 µν −
η (cid:0)σg (cid:1)
µν dabcϕc +δabφ + fabcBc , (7)
− 2 2 µν
(cid:0) (cid:1)
where ξ is a gauge-fixing parameter.
The lagrangian now is quadratic in gluon fields. Thus the gluons can be
integrated out and the effective action becomes
S = iTrlni +
eff
− K
1 1 2
+ d4xTr B2 Θ2 + Φ2
Z (cid:20)−2 − 2g2(1+σ2) 3g2(1+σ2) (cid:21)−
a −1 ab,µν b. (8)
−Jµ K Jν
(cid:0) (cid:1)
The current is
a = Ja(B,C)+gψ¯γ Taψ, (9)
Jµ µ µ
where Ja(B,C) is made of the B field and Faddeev-Popov ghosts .
µ
It’s easy to find the gluon condensate now. Let us put B = 0,Θ = 0,ϕa =
0,φ = φ = 0 which is the case with highest unbroken symmetry. Let us also
0
6
fix the gauge as follows:
1+σ2 +ξ−1 = 0. (10)
Then the equation for φ follows immediately:
0
φ d4p
0 i = 0 (11)
3g2(1+σ2) − Z (1+σ2)p2 +φ
0
Putting
σ = iρ,ρ > 1 (12)
we obtain a nontrivial solution
ρ2 1
φ = − . (13)
0
±πg√6I
Wick rotation was made here so
4
∞
z3dz
I = (14)
Z z2 +1
0
This integral is divergent and needs regularization but we won’t go into it.
Having this solution, we can build the mean field approximation which is
equivalent to leaving the zero order term only in the saddle-point expansion
of the path integral that corresponds to lagrangian (6). The mean field
lagrangian is
= TrAµ η ρ2 1 ∂2 + 1 ρ2 +ξ−1 ∂ ∂ +η φ Aν +
µν µ ν µν 0
L − − −
(cid:2) (cid:0) (cid:1) (cid:0) (cid:1) (cid:3)
+ ψ¯ iD m ψ + (15)
0 FP
− L
(cid:16) (cid:17)
b
It’s clear now that our solution describes dynamic gluon mass generation:
φ
m2 = 0 (16)
A ρ2 1
−
Note that the effective mass doesn’t depend on the free parameter ρ.
Let us fix the gauge (10) in the action (15) and integrate out the gluons
omitting the Faddev-Popov ghosts. The resulting action will be
S = d4xψ¯ i∂ m ψ +
0
Z (cid:16) − (cid:17)
+ 1 db4xd4yψ¯(x)γ Taψ(x) x ∂2 +m2 −1 y ψ¯(y)γµTaψ(y)
2(ρ2 1) Z µ h | A | i
− (cid:0) (cid:1)
(17)
The Green’s function in the second term of the action can be expanded
as follows:
∞
x ∂2 +m2 −1 y = δ(x−y) (−1)n∂2n (18)
h | A | i m2 m2n
(cid:0) (cid:1) A Xn=0 A
Therefore, we can conclude that the higher order terms will be small and
the expansion can converge if
p < m Λ (19)
µ A
| | ≡
5
This condition defines the low energy limit of the theory.
The zero order term of the expansion above gives a four-fermion term:
1
(0) = ψ¯γ Taψ ψ¯ γµTaψ (20)
Lqq 2(ρ2 1)Λ2 i µ ij j k kl l
−
Using the technique developed in [3], this term can be simplified further
which leads to the following fermionic lagrangian:
= ψ¯ i∂ˆ m ψ
q i 0 i
L − −
(cid:16) (cid:17)
1
+ ψ¯ψ ψ¯ ψ ψ¯γ5ψ ψ¯ γ5ψ
i i k k i i k k
4(ρ2 1)Λ2 −
− (cid:2)
N +2 1
ψ¯γ ψ ψ¯ γµψ ψ¯γ5γ ψ ψ¯ γ5γµψ (21)
i µ i k k i µ i k k
− 2N − 2 (cid:21)
Diagonality with respect to color indices is a notable feature of this la-
grangian. It allows us to define colorless mean fields which will be done
below.
3
DYNAMICS OF THE FERMIONIC MEAN FIELDS
Let us now study the NJL model itself. We’ll work within the mean field
approachtoitdeveloped first byNambuandJona-Lasiniotofindthevacuum
condensate of the model. Let us start with the following path integral that
corresponds to the lagrangian (21):
Z = ψ¯ ψexp i d4x ψ¯ i∂ˆ m ψ +α ψ¯ψψ¯ψ ψ¯γ5ψψ¯γ5ψ
0
Z D D (cid:26)− Z h (cid:16) − (cid:17) −
(cid:2)
1
βψ¯γ ψψ¯γµψ + ψ¯γ5γ ψψ¯γ5γµψ , (22)
µ µ
− 2 (cid:21)(cid:21)(cid:27)
Where we’ve put
1
α = (23)
4(ρ2 1)Λ2
−
N +2
β = (24)
2N
for the sake of brevity.
6
Now we introduce one scalar, one pseudoscalar, one vector and one pseu-
dovector auxiliary fields ξ,η,vµ,wµ to get rid of the fourth-order terms. We
arrive at the following path integral:
Z = ψ¯ ψ ξ η v wexp i d4x ψ¯ i∂ˆ m ψ +ξψ¯ψ +ηψ¯γ5ψ
0
Z D D D D D D (cid:26)− Z h (cid:16) − (cid:17)
1
+ v ψ¯γµψ +w ψ¯γ5γµψ ξ2 η2 β−1v2 +2w2 =
µ µ
− 4α − − (cid:21)(cid:27)
(cid:0) (cid:1)
= ψ¯ ψ ξ η v wexp i d4x ψ¯ i∂ˆ m +ξ +ηγ5 +v +γ5w ψ
0
Z D D D D D D (cid:26)− Z h (cid:16) − (cid:17) −
1 b b
ξ2 η2 β−1v2 +2w2 (25)
− 4α − − (cid:21)(cid:27)
(cid:0) (cid:1)
Let us now define the matrix mean field to be
Ω = ξ +ηγ5 +v +γ5w. (26)
It’s obvious that the tebrms in (??) thbat arebquadratic in auxiliary fields
canbeunderstood assome bilinear functionof Ω, therefore we’ll denote them
simply as Φ (Ω).
2
The fermions can be integrated out now and we get the following effective
b
action for Ω:
b 1
S [Ω] = iNTrlni i∂ˆ m +Ω d4xΦ (Ω) (27)
eff − (cid:16) − 0 (cid:17)− 4α Z 2
b b
The equation of motion that can be derived from it is
∂Φ Ω
1 2
(cid:16) (cid:17)(x) = G (x,x) (28)
4α ∂Ωb Ω
b
whereG (x,y)istheGreen’sfunbctionforoperatori∂ m +Ω. Thisequation
Ω 0
−
can be called gap equation or Gor’kov equation, based on the analogy to
b b
superconductivity. By putting Ω = const it can be reduced to a system
of algebraic equations on the components of Ω which was in fact done by
b
Nambu and Jona-Lasinio themselves in the very first article on the subject.
b
Studying the inhomogeneities, however, is much more complicated since
one has to deal with a nonlinear integral equation (28). Another difficulty
is that deriving the Green’s function in the right-hand side of it encounters
7
the problem of ultraviolet divergencies. Fortunately enough, we’ve got the
cutoff, but it should be applied explicitly which is what we’re going to do
now.
Let us expand the logarithm in the effective action (27). We’ve got
1 1
S [Ω] = iNTrlni i∂ˆ m iNTrln 1+ Ω d4xΦ (Ω) =
eff − (cid:16) − 0(cid:17)− (cid:18) i∂ˆ m0 ◦ (cid:19)− 4α Z 2
−
1 b b
= d4xΦ (Ω) iNSpS(0) d4xΩ(x)+
−4α Z 2 − Z
iN b b b
+ Sp d4xd4yS(x y)Ω(y)S(y x)Ω(x)+... (29)
2 Z − −
b b b b
, where S(x y) is the plain Dirac propagator.
−
The first term in the expansion can be rewritten as follows:
b
d4xd4p pˆ+m
SpS(0) d4xΩ(x) = Sp 0 Ω(x) (30)
Z Z (2π)4 p2 m2
− 0
b b b
Let us now demonstrate that each term in the series can be reduced to
∞
d4pd4x Ξ (x,p)
i
Z
Xi=0
b
. For this, it’s enough to demonstrate that
d4yS(x y)Ω(y) d4pe−ip(y−z)Ξ(p,z) = d4pe−ip(x−z)Ξ(p,z)
Z − Z Z
b
b b b e
8
So
d4yS(x y)Ω(y) d4pe−ip(y−z)Ξ(p,z) =
Z − Z
d4byd4qd4pb qˆ+m b
= e−iq(x−y) 0 Ω(y)e−ip(y−z)Ξ(p,z) =
Z (2π)4 q2 m2
− 0
d4yd4qd4p qˆ+m b b
= e−iq(x−y) 0
Z (2π)4 q2 m2 ×
− 0
∞
(y z)α1..(y z)αn ∂ ∂
− − .. Ω(z) e−ip(y−z)Ξ(p,z) =
× n! (cid:20)∂zα1 ∂zαn (cid:21)
Xn=0
b b
d4yd4qd4p qˆ+m
= e−iq(x−y) 0
Z (2π)4 q2 m2 ×
− 0
∞
in ∂ ∂ ∂ ∂
.. Ω(z) .. e−ip(y−z) Ξ(p,z) =
×Xn=0 n! (cid:20)∂zα1 ∂zαn (cid:21)(cid:20)∂pα1 ∂pαn (cid:21)
b b
d4yd4qd4p qˆ+m
= e−iq(x−y) 0 e−ip(y−z)
Z (2π)4 q2 m2 ×
− 0
∞
( i)n ∂ ∂ ∂ ∂
− .. Ω(z) .. Ξ(p,z) =
×Xn=0 n! (cid:20)∂zα1 ∂zαn (cid:21) ∂pα1 ∂pαn
b b
qˆ+m d4y
= d4qd4pe−i(qx−pz) 0 e−i(p−q)y
Z q2 m2 Z (2π)4 ×
− 0
∞
( i)n ∂ ∂ ∂ ∂
− .. Ω(z) .. Ξ(p,z) =
×Xn=0 n! (cid:20)∂zα1 ∂zαn (cid:21) ∂pα1 ∂pαn
b ∞ b
qˆ+m ( i)n ∂ ∂ ∂ ∂
= d4qd4pe−i(qx−pz) 0 δ(p q) − .. Ω(z) .. Ξ(p,z) =
Z q2 −m20 − Xn=0 n! (cid:20)∂zα1 ∂zαn (cid:21) ∂pα1 ∂pαn
∞ b b
pˆ+m ( i)n ∂ ∂ ∂ ∂
= d4pe−ip(x−z) 0 − .. Ω(z) .. Ξ(p,z) Q.E.D(31)
Z p2 −m20 Xn=0 n! (cid:20)∂zα1 ∂zαn (cid:21) ∂pα1 ∂pαn
b b
Therefore, it’s now obvious that if we define the operator
∞
( i)n ∂ ∂ ∂ ∂
V(x,p) = − .. Ω(z) .. (32)
Xn=0 n! (cid:20)∂zα1 ∂zαn (cid:21) ∂pα1 ∂pαn
b b
9
it’s possible to express the expansion (29) as
1 d4xd4p ∞ ( 1)k p+m k
S [Ω] = d4xΦ (Ω)+iN Sp − 0 V(x,p) 1
eff −4α Z 2 Z (2π)4 k (cid:20)p2 m2 (cid:21) ◦
Xk=1 b − 0
b b
(33)
We can easily read from this that all the integrations in (29) reduce to
expressions of type
p ..p
I(m,n) = d4p α1 αm (34)
α1..αm Z (p2 m2)n
− 0
First of all, it’s easy to see that integrals like this are nonzero for even
values of m only and are finite if
d(n,m) = 2n m 4 > 0 (35)
− −
. It’s obvious that differentiation ∂ of the integrand in (34) increments
∂pµ
its d(n,m) by 1 while leaving the difference n m unchanged. Thus we
−
can conclude from (33) that the series contains only the integrals (34) with
m n. But it’s also true that a k-th order in Ω term can contain only the
≤
integrals with n k.
≥ b
Thereforeweimmediatelyconcludethattheexpansion(33)hasonlyfinite
number of divergent terms which are
1. first order term;
2. second order up to second derivatives of Ω;
3. third order up to first derivatives; b
4. fourth order with no derivatives of Ω.
Regularization by cutoff will replacebthe divergencies with some finite
factors of order Λn in the cutoff parameter. Therefore, the finite part can be
treated as a small correction to the divergent one. So by omitting the finite
terms we can build an approximation of the effective action (27) that has
remarkable features. First, it’s similar in structure to the Ginzburg-Landau
functional in superconductivity, since it contains derivatives up to second
and nonlinearities up to fourth order; second, it requires no extra conditions
being imposed on the model to be valid.
10